€¦ · acknowledgements my supervisor ezequiel rodrigues barbosa and my co-supervisor marcos...
TRANSCRIPT
Universidade Federal de Minas Gerais
Institute of Exact Sciences - ICEx
Department of Mathematics
PhD Thesis
About a Class of Optimal Sobolev Vector Inequalitiesof Second Order
Aldo Peres Campos e Lopes
Advisor : Prof. Ezequiel Rodrigues Barbosa
Co-Advisor: Prof. Marcos Montenegro
Belo Horizonte - April 9, 2014
To my parents, Alfreu and Eunice.
Acknowledgements
My supervisor Ezequiel Rodrigues Barbosa and my co-supervisor Marcos Montenegro.
I appreciate the encouragement and support, and by the beautiful theme proposed. The
discussions always so enlightening and instructive. I am grateful to my supervisor for his
patience and kindness to address several questions that have arisen, by having their time
even on Saturdays and other non-school days.
Members of the bank examiner who kindly agreed to participate in the completion of
this work, professors Ezequiel Rodrigues Barbosa, Marcos Montenegro, Emerson Jurandir
and Joao Marcos Bezerra do O.
To my parents, Alfreu and Eunice, by ample incentive to the “domain of knowledge”.
Even before entering the University, the great desire of my father was watching me with
the title of Doctor, “Dr. Aldo”,as always told me.
To my many colleagues in the UFMG who helped me in various ways.
To Professor Francisco Dutenhefner by the orientation in the MSc and for under-
standing and flexibilitye. To Susana C. Fornari for her patience and dedication during
my undergraduate research and encouraging academic research.
To the professor Sylvie Marie who without knowing it helped me get into the math
career. I do not forget a phrase she said.: “Many people choose to do engineering because
they like math. Why not choose mathematics since they like math?”
To many professors of UFMG math department who helped answering questions and
teaching courses I attended as a student. In this regard, I am grateful to professors like:
Gastao, Mario Jorge, Rogerio Mol, Marcelo T. Cunha, among several other.
To UFMG math department that helped me, as much as possible, in several events
that I participated . It is certainly well deserved the concept 6 by CAPES..
UNIFEI colleagues who helped in the reduction of my working hours for a few semesters.
To other colleagues, friends and professors who should be included here. However, do
not do a list with names so I could forget to mention someone.
Again, to all, my sincere thanks.And above all, God, Jehovah, the Almighty, Great
Scientist and Designer Creator of the universe and the wonderful human mind.
Science cannot solve the ultimate mystery of nature. And that is because, in the last
analysis, we ourselves are part of nature and therefore part of the mystery that we are
trying to solve.
Max Planck
Abstract
We approach potential elliptic systems involving Paneitz-Branson operators and crit-
ical nonlinearities. First, we present conditions for the existence of regular solutions of
potential systems in Riemannian Geometry, a decomposition in diagonal bubbles to ap-
plications of Palais-Smale and theoretical applications of this decomposition. Then, we
Euclidean space, we present another decomposition in bubbles and apply the decomposi-
tion in bubbles o a result of compactness. Finally, we apply all those results in extremal
applications for optimal Sobolev inequalities on compact manifolds.
6
Contents
General Introduction 7
0.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.2 Proposal and Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.3 Organization and Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 Preliminary Mathematical Material 17
1.1 Curvatures in a Riemannian manifold . . . . . . . . . . . . . . . . . . . . . 17
1.2 The Musical Isomorphism and Divergence of Tensors . . . . . . . . . . . . 21
1.3 Homogeneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Sobolev Spaces of Vector Valued Maps of Second Order . . . . . . . . . . . 26
1.5 Coercivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6 The Scalar AB Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.6.1 Partial Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.7 AB Vector Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.7.1 Partial Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Elliptic Systems of Fourth Order 45
2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Regularidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Bubbles Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.4 Pointwise Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Concentracao L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.6 Compacidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3 Sharp Sobolev Vetorial Inequality of Second Order 99
3.1 Extremal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Final Considerations 103
4.1 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7
General Introduction
0.1 Historical Overview
In 1983, Paneitz [27] introduced the fourth order operator P 4g : C4(M) → C0(M),
defined by
P 4g u := ∆2
gu− divg
((2
3Rgg − 2 Ricg
)(∇u)#
),
for all u ∈ C4(M), where (M, g) is a Riemannian manifold of dimension n = 4, Ricg is the
Ricci tensor Ricci, Rg is the scalar curvature, divg is the divergente and ∆g is the Laplace-
Beltrami operator with respect to the metric g. This operator P 4g has some properties of
conformal invariance. Accurately, if g = e2ϕg is conformal to the metric g, ϕ ∈ C∞(M),
then
P 4g = e−4ϕP 4
g .
Associated with this operator, we have the notion of Q-curvatura, a curvature which also
has conformal properties. For this case n = 4, a Q-curvature is given by
Q4g =
1
6
(∆gRg − 3|Ricg |2g +R2
g
).
Beyond this conformal invariance, the operator P 4g appears in the following relation be-
tween the curvatures Q4g and Q4
g:
P 4g ϕ+Q4
g = Q4ge
4ϕ .
It is noteworthy that the Q-curvature in the dimension n = 4, and for locally conformally
flat manifold, is inside the integral in the Gauss-Bonnet formula for Euler characteris-
tic, and thus has a very important role in the study of topology and geometry of the
Riemannian manifold of dimension 4. We have the following integral identity
4π2χ(M) =
∫M
(Qg +
1
8|Weylg |2
)dvg , (1)
where χ(M) is the Euler characteristic of the manifold M and Weylg denotes the Weyl
tensor with respect to the metric g. As the |Weylg |2dvg is an punctual conformal invariant,
we obtain that the integral of Q-curvatura∫Qg dvg is a conformal invariant. For more
details on this, see the articles by Chang [8] and Chang-Yang [7].
The generalization for the case n ≥ 5 was made by Branson [6] in 1987. Let (M, g) be a
Riemannian manifold of dimension n ≥ 5. We define the operator P ng : C4(M)→ C0(M)
by
8
P ng u := ∆2
gu− divg
((anRgg + bnRicg) (∇u)#
)+n− 4
2Qngu ,
where
an =(n− 2)2 + 4
2(n− 1)(n− 2), bn = − 4
n− 2,
and
Qng =
1
2(n− 1)∆gRg +
n3 − 4n2 + 16n− 16
8(n− 1)2(n− 2)2Rg −
2
(n− 2)2|Ricg |2g
is the Q-curvatura for dimension n ≥ 5. We denote P ng also by Pg. This operator also has
conformal invariance properties. That is, considering u ∈ C∞(M), u > 0, and the metric
g = u4
n−4 g wich is conformal to the metric g, we have
P ng ϕ = u−
n+4n−4P n
g (uϕ), (2)
for all ϕ ∈ C∞(M). In particular, by taking ϕ ≡ 1, we get the following elliptic semilinear
diferencial equation satisfied by the conformal factor u:
P ng u =
n− 4
2Qngu
n+4n−4 , u > 0 .
It is interesting to compare this relationship between the Paneitz-Branson operator
and the Q-curvatura of the metric g with the scalar curvature Rg. Let’s see what we
mean. Initially, we have following
Lgu = Rgun+2n−2 ,
where Lg = 4n−1n−2
∆g +Rg; g = u4
n−2 g and n ≥ 3. We have also a equivalent identity in the
case n = 2. The operator Lg is a conformal operator. An important result in conformal
differential geometry is the resolution of the Yamabe problem , where the operator Lg
plays a key role. That is, we can always find a metric g, in the conformal class of g, such
that the scalar curvature Rg is constant, considering that the manifold is smooth, closed
and dimension n ≥ 2.
Because of similarities of the properties, in the conformal differential geometry, be-
tween the operators Lg e Pg, it is natural to ask whether the Q-curvatura has the same
property, that is, how it would be the Yamabe problem for Q-curvature. Partial results
have been established in response to this question for manifolds of dimensions ≥ 5 (see
[15], [22], [23], [28]).
But these results are limited by the lack of the maximum principle for differential
operators of higher orders. Because of this problem, we need positivity in the conformal
factor u which is determined by Paneitz-Branson equation and is not clear way to guar-
antee that this condition is satisfied more generally. This problem can be solved using
9
the conformal covariance property of the Paneitz-Branson operator, that is, by (2). This
property tells us that uv is a solution for the Paneitz-Branson equation in the geometry of
the metric g if u is a solution in the geometry of g. This condition plays an important role
in the type of functions which can be a local minimum of a functional which is naturally
associated with the Yamabe problem for the Q-curvature.
There is another problem, an analytical problem, when we consider the Yamabe prob-
lem for the Q-curvature. As in the case of the classical Yamabe problem, we have problems
when we try to use variational methods to find solutions of Paneitz-Branson equation be-
cause of the exponent of the nonlinearity, n+4n−4
, where n is the dimension of the manifold,
which is the critical Sobolev exponent W 2,2(M) less one, because the immersion W 2,2(M)
in L2nn−4 is not compact.
One idea contained in [10], by Hebey-Djadli-Ledoux, is that the Sobolev inequalities
of second order can be used to deal with the problem of concentration in approximating
sequences for the solution, since the infimum of the functional associated with the problem,
the Paneitz-Branson functional, is smaller than a critical value. In this method, it is not
clear when the infimum is positive and the Yamabe constant is less than or equal to the
Yamabe constant of the sphere.
A partial solution to this problem was made by F. Robert and P. Esposito in 2002,
see [15]. It has been shown that if n ≥ 8 and the manifold is locally conformally flat, then
there exists a minimizer for the Paneitz-Branson functional.
The effect of this result is that the part of existence of the Yamabe problem for the
Q-curvature is brought to a point analogous to that Aubin led the Yamabe problem for
the scalar curvature. But, is not yet clear when the Green function of the Paneitz Branson
operator is positive, in case the operator be coercive. This excludes try to use the methods
of Schoen to complete the problem.
It was shown by D. Raske (see [29]) that there exists a metric in the conformal class
of the arbitrary metric in a smooth closed Riemannian manifold, of dimension n ≥ 5,
such that the Q-curvature of the metric is constant. Existence of solutions is obtained
through the combination of variational methods, Sobolev inequalities of second order and
the blow-up theory of W 2,2(M). Below is the result.
We define the Paneitz Branson constant as
λg(M) := infw∈C∞+ (M)
∫MwPgw dvg
‖w‖22nn−4
,
where Pg is the Paneitz-Bransonoperator. Let λ(Sn) be the Paneitz-Branson constant of
the unit sphere with the canonical metric. We have the following result (see [29]):
Theorem 1 (David Raske, 2011). Let (M, g) be a Riemannian manifold of dimension
n ≥ 5. Suppose that at least one of the following conditions is valid:
i. The Paneitz-Branson constant is less than λ(Sn);
10
ii. The Yamabe constant of g is greater than or equal to the inverse of the Yamabe
constant of the n-sphere;
iii. n ≥ 8 and g is not locally conformally flat.
then exists a smooth minimizer, positive of Paneitz-Branson functional and there exists a
metric h in the conformal class of g such that Qh = λ, where λ is the Paneitz-Branson
constant of g.
thus, the study of operators like the Paneitz-Branson is very important in the analysis of
geometrical problems as the problem of the prescribed Q-curvature, where the Yamabe
problem for the Q-curvature is a particular case.
Consider the system of equations
−∆2gui + divg
(Ai(∇ui)#
)+
k∑j=1
Aij(x)uj = u2#−1i , (3)
where 2# = 2nn−4
, U = (u1, . . . , uk), A = (Aij) is a continuous map of M to M sk(R) such
that A(x) is positive definit for all x ∈ M , M sk(R) is the space of real symmetric matrices
k × k, and the Ai are symmetric tensor fields of type (2,0). In this context, we consider
ui > 0 for all i. This system can be seen as a natural generalization of the equations
involving operators of type Paneitz Branson. Thus, considering the scalar case, that is
k = 1, and the tensor Ai = f · g, where f is a smooth function, the system (3) can be
rewritten as
−∆2gu+ bα∆gu+ cαu = u2#−1 . (4)
Assume that the constants bα and cα are converging sequences of real positive numbers,
satisfying cα ≤ b2α4
. From 2000 to 2004, many authors studied the case (4) above, for
example F. Robert, E. Hebey, Z. Djadli and M. Ledoux in [10, 15, 22] and [24]. Hebey-
Robert-Wen in [24] discussed the compactness of solutions of (4), precisely when bα and
cα converge respectively to b0 and c0 and the solutions uα converge weakly in H2,2(M).
They found conditions such that the limit uα is nontrivial.
The following theorems have been proved by Hebey-Robert-Wen in 2004 (see [24]).
Let
Ag =(n− 2)2 + 4
2(n− 1)(n− 2)Rg g −
4
n− 2Ricg (5)
be field of (2, 0)-tensor. We denote by λi(Ag)x, 1, ..., k, the g-eigenvalues of Ag(x) and
define λ1 as infimum of the i and x of the λi(Ag)x and λ2 as the supremum of the i and
x of the λi(Ag)x. We denote por Sc the critical set defined by
Sc = λ ∈ R : λ1 ≤ λ ≤ λ2 . (6)
11
We have the following results:
Theorem 2 (Hebey-Robert-Wen, 2004). Let (M, g) be a compact manifold locally confor-
mally flat of dimension n and (bα)α, (cα)α converging sequences of positive real numbers
with positive limits b∞ e c∞ such that cα ≤ b2α4
for all α. We consider equations of type
∆2gu+ bα∆gu+ cαu = u2#−1 , (7)
and we assume that b∞ /∈ Sc, where b∞ is the limit of bα and Sc is the critical set given
by (6). Then the family (7) is pseudo-compact when n ≥ 6 and compact when n ≥ 9.
We say that a family of equations are solutions of (7) is pseudo-compact if, for any
sequences (uα) in H2,2(M) of positive solutions that converges weakly in H2,2(M), the
weak limit u0 of uα is nonzero.
The following theorem is a complement of the above theorem when the dimension is
n = 6, 7 or 8 and b∞ is is below the lower limit λ1 de Sc.
Theorem 3 (Hebey-Robert-Wen, 2004). Let (M, g) be a compact manifold locally con-
formally flat of dimension n = 6, 7 or 8 and (bα)α, (cα)α converging sequences of real
positive numbers with positive limit b∞ and c∞ such that cα ≤ b2α4
for all α. We consider
equations like
∆2gu+ bα∆gu+ cαu = u2#−1 ,
and we assume that b∞ < λ1 = min Sc, where b∞ is the limit of bα and Sc is the critical
set given by (6). Then the family (7) is compact.
Pseudo-compactness has a traditional interest because it seeks nontrivial solutions of
limit equation we obtain from (7) making α→ +∞.
On the other hand, we say that the family of equations (7) is compact if any sequence
(uα)α in H2,2(M) of positive solutions of (7) is limited in C4,θ(M), 0 < θ < 1 and then
converges, if necessary take a subsequence, on C4(M) for some function u0.
Compactness is a concept clearly stronger than pseudo-compactness.
Pseudo-compactness for elliptic equations of second order the type of Yamabe has
been widely studied. Compactness for Yamabe equations of second order was studied by
Schoen from 1988 to 1991 (see [32, 33, 34, 35]).
As application of the results of compactness, we can study also the existence of ex-
tremal in sharp Sobolev scalar inequalities of second order.
From the continuous immersion H2,2(M) → L2#(M), then there are constants A,B >
0 such that:
‖u‖22# ≤ A‖∆gu‖2
2 +B‖u‖2H1,2(M) , ∀ u ∈ H2,2(M). (8)
We are interested in the sharp constants A,B from the above inequality. More pre-
cisely, the best constant A is defined by
12
A0 = inf
A: exists B such that the above inequality (8) is valid ∀ u ∈ H2,2(M). (9)
A natural question that arises is: the infimum A0 is achieved? The answer is yes, the
infimum A0 is achieved in (8). In 2000, Djadli-Hebey-Ledoux [10] proved the result with
the restriction that the metric g is conformally flat. Later, in 2003, Emmanuel Hebey
proved the result for any Riemannian manifold in [25] (see also [26]).
Follow from the Sobolev theorem that the constant A0 is well defined. This constant
was calculated by Lieb [25], Lions [26], Edmunds-Fortunato-Jannelli [14] and Swason [37].
Precisely, we have:
1
A0
=n(n2 − 4)(n− 4)w
4nn
16,
where wn is the volume of the sphere Sn in Rn+1.
The infimum in (9) is achieved and there exists a constant B > 0 such that:
(∫M
|u|2#dvg) 2
2#
≤ A0
∫M
(∆gu)2 dvg +B
∫M
(|∇u|2g + u2
)dvg , (10)
for all u ∈ H2,2(M).
The second best Sobolev constant associated with (8) is defined by:
B0 = inf B ∈ R; (10) is valid . (11)
The second Sobolev Riemannian inequality states that, for any u ∈ H2,2(M), we have
(∫M
|u|2# dvg) 2
2#
≤ A0
∫M
(∆gu)2 dvg +B0
∫M
(|∇gu|2 + |u|2
)dvg . (12)
A nonzero function u0 ∈ H2,2(M) is said extremal for the inequality (12) if
(∫M
|u0|2#
dvg
) 2
2#
= A0
∫M
(∆gu0)2 dvg +B0
∫M
(|∇gu0|2 + |u0|2
)dvg .
You can find some comments about the second best constant in [10] (Djadli, Hebey,
Ledoux, 2000). But for the extremal of the sharp inequalities there are studies only for
the case of first order. To this reasoning, see [11] also [5, 4].
All has been said above is being considered for the case k = 1. The main objective
of this thesis is to extend some of these results for the case k ≥ 2 and apply them to the
study of existence of extremal in sharp Sobolev vector inequalities of second order. We’ll
talk more about this in the next section.
13
0.2 Proposal and Relevance
One of the main goals of this thesis is extend the results on compactness of solutions of
equations involving Paneitz-Branson type operators for systems and apply these results
to obtain results of existence of extremal for a class of sharp Sobolev inequalities of
second order. This is an important question, both the mathematical point of view, for
involve a larger structure and his understanding, as from the point of view of analytical
applications, by allowing the study of several elliptic PDE systems of second order on
Riemannian manifold. Let me be a bit more clear..
Let (M, g) be a compact Riemannian manifold, of dimension n ≥ 5, whose volume
element is dvg. We consider here maps U ∈ H2,2k (M) that are solutions the following
model equation:
−∆2gU + divg
(A(∇U)#
)+∇UG(x, U) = ∇F (U) . (13)
That is,
−∆2gui + divg
(Ai(∇ui)#
)+ ∂iG(x, U) = ∂iF (U) , (14)
for each i = 1, ..., k, where 2# = 2nn−4
is the Sobolev critical exponent for the immersions of
H2,2(M) in Lp(M) spaces. Note that the operator Pg := −∆2g+divg
(Ai(∇·)#
)+∂iG(x, ·)
is Paneitz-Branson type. Here arise some questions, such as:
• There is a nonzero solution U for (13)?
• If there is a nonzero solution U , which regularity we can get??
• The set of the solutions of (13) is compact in some topology?
We will answer these questions in the following chapters for a class of homogeneous
functions F and G.
Answering the above questions, we continue with the study of existence of extremal
in sharp Sobolev inequalities of second order in the vector case. This study, for the
case of second order, is also motivated by the study of the best constants for first order
inequalities in the vector case which was done by E. Barbosa e M. Montenegro (see [5]).
E. Barbosa e M. Montenegro studied inequalities like:
(∫M
F (U) dvg
) pp∗
≤ A0(p, F,G, g)
∫M
|∇gU |p dvg + B0(p, F,G, g)
∫M
G(x, U) dvg , (15)
where A0 and B0 are the best constants, 1 ≤ p < n, U = (u1, · · · , uk), F : Rk → R is a
positive continuous function, and p∗-homogeneous and G : M × Rk → R is a continuous
function, positive and p-homogeneous in the second variable. The vector theory of best
14
constants developed (in [5]) considers a number of questions involving the constants A0
and B0. Some follow directly from scalar theory, others are more complex, for example,
the behavior of B0 for all the parameters involved and problem of existence and C0
compactness of the extremal. A map U is extremal when it achieves equality in (15).
When F and G are C1 maps a extremal is weak solution of the system:
−A0(p, F,G, g)∆p,gui +1
pB0(p, F,G, g)
∂G(x, U)
∂ti=
1
p∗∂F (U)
∂ti, i = 1, · · · , k ,
where ∆p,gu = divg(|∇gu|p−2∇gu) denotes the p-Laplacian operator associated with the
metric g.
Now consider Uα ∈ H1,p(M,Rk) weak solutions of the system:
−∆p,gU +1
p∇UGα(x,U) =
1
p∗∇Fα(U) em M ,
where Fα : Rk → R are positive C1 functions and p∗-homogeneousand the Gα : M×Rk →R are C1 functions and p-homogeneous in the second variable. Consider sequences of
limited solutions (Uα)α whose limite is U ≡ 0, taking a subsequence if it is necessary, we
have the following bubbles decomposition:
Uα =l∑
j=1
Bj,α +Rα ,
for all α > 0, where (Bj,α)α, j = 1, · · · , l, are k-bubbles and (Rα)α ⊂ H1,p(M,Rk) is
such that Rα → 0 in H1,p(M,Rk) when α → +∞. With this result we can add more
properties to the sequences of limited solutions such that Uα 0 in H1,p(M,Rk). We have
also pointwise estimates or C0 for (Uα)α (see [9]). The blow-up points of the sequences
(Uα)α have much of the information of the sequences, this is the Lp concentration property.
Concentration properties were studied by Druet, Hebey and Robert (see [17] and [12])
and extensions of these works were made by G. Souza [9].
In this work, we follow similar ideas of E. Barbosa and M. Montenegro in [5]. We
consider the sharp Sobolev inequalities of second order:
(∫M
F (U)dvg
) 2
2#
≤ A0
∫M
(∆gU)2 dvg +
+ B0
∫M
((A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg , (16)
where F is a 2#-homogeneous map and G is 2-homogeneous on the second variable, and
also the systems of second order are associated to this inequality:
15
−∆2gU + divg
(A(∇U)#
)+
1
2∇UG(x, U) =
1
2#∇F (U) . (17)
We consider sequences of limited solutions (Uα)α whose limite is U ≡ 0. If necessary take
a subsequence, we obtain the following bubbles decomposition :
Uα =l∑
j=1
Bj,α +Rα ,
for all α > 0, where (Bj,α)α, j = 1, · · · , l, k-bubbles and (Rα)α ⊂ H2,2(M,Rk) is such
that Rα → 0 in H2,2(M,Rk) when α → +∞. Added to this result, we have pointwise
estimates or C0 for (Uα)α. The blow-up points or concentration of the sequences (Uα)α
have a lot of information about the sequences and generating L2 concentration properties.
We apply all results for the study of existence of extremal for the inequalities (16).
0.3 Organization and Ideas
This work has three chapters. In Chapter 1, we show some definitions and some basic
results that will be used in the others chapters. In special, we show the Sobolev spaces
of second order and we make fazemos um panorama detalhado da scalar theory de best
constants on sharp Sobolev inequalities of second ordem e destacamos alguns problemas
em aberto. Morover, we describe some important problems of the vector theory of best
constants and we state our main contributions. We include in this chapter, some basic
results about Euclidean and Riemannian Sobolev vector inequalities. Precisely, we show
that the best constant associated to the inequality
(∫RnF (U) dx
) 2
2#
≤ A∫Rn
(∆U)2 dx
is given by
A0(F, n) = M2
2#
F A0(n)
where MF = maxSk−12
and Sk−12 =
t ∈ Rk;
∑ki=1 |ti|2 = 1
. Morover, characterized the
associated extremal as the type of U0 = u0t0, where u0 is a extremal function associated
to the Euclidean Lp-Sobolev scalar inequality and t0 is a maximum point of F in Sk−12 .
Chapter 2 has the largest number of contributions made. Turn our attentionto the
following system:
−∆2gU + divg
(A(∇U)#
)+∇UGα(x, U) = ∇Fα(U) (18)
In this chapter, we study the blow-up decomposition of the solutions (Uα)α of (18).
We put conditions for existence of minimal energy solutions. Then we study the behavior
16
of sequences (Uα)α, given by the solutions of systems (18), as function of the behavior of
Fα and Gα.
With the hypothesis that the (Uα)α is limited, we obtain that these sequences is a
Palais-Smale sequences of the functional J associated to (18). That is, the sequences Uα
is such that:
(J(Uα))α is limited and DJ(Uα)→ 0 in(H2,2k (M)
)′.
From the limitation of Uα we get the existence of a weak limit U0 in H2,2k (M). Working
with the case where U0 ≡ 0, the hypotheses about Fα and Gα let us decompose the
functional J in terms of others functional Lik such that the components of Uα form Palais-
Smale sequences for the functional Lik. This decomposition in functional Lik allow us to
use the results obtained by F. Robert (see [30]) to obtain bubbles decomposition for the
components of Uα, which concludes this part.
Using bubbles decomposition we have the pointwise estimates for a sequences of solu-
tions of (18) that have 0 as a weak limit.
Then, we have the L2 concentration, where we extended for the vector case some
results. In some of these, we assume that G(x, U) =∑k
i,j=1Aij(x)ui(x)uj(x). We use some
estimates and the Bochner-Linerowicz-Weitzenbook formula for the main L2 concentration
result.
In the main part of chapter 2, we proof the compactness. For this, we use several
previous results, as bubbles decomposition and L2 concentration.We also use an important
and useful tool that is the Pohozaev type identity. Our contributions at this point extend
some results ound in studies of E. Hebey, F. Robert, Z. Djadli, M. Ledoux and V. Felli,
made in the scalar case (fourth order equations), for the vector case (systems of fourth
order).
In the chapter 3, we study the existence of etremal maps. The main ideia here is the
following. Suppose that the second order vector Sobolev sharp inequality has no extremal
map. We consider then a numerical sequence (α) such that 0 < α < B0 and α → B0.
Associated to this sequence, we construct functions Fα and Gα and also solutions Uα of
systems of type studied on the previous chapters. The sequence (Uα) converges for U
such that will be a extremal map.
17
Chapter
1Preliminary Mathematical Material
In this chapter, introduce some basic notations and definitions we will use throughout
the remainder of this paper. We remember some basic facts of Riemannian geometry.
Then, there is the Sobolev space theory in Riemannian manifolds. In this section and
in the rest of this paper, we assume that the manifold is compact. We show the norms
that we use, and some properties of these Sobolev spaces.
Some definitions are more usual. But present below the definitions which will be useful
in the following work.
Over the years, about forty years, much attention has been given to the sharp Sobolev
Riemannian inequalities. There is a vast literature with a wide theory of the best constants
that is connected with areas such as analysis, geometry and topology. These inequalities
play an important role in geometric analysis, especially in the study of the existence
and multiplicity of solutions to the Yamabe problem (see [2], [20], [34]), Riemannian
isoperimetric inequalities (see [13]), among other applications. Most of these results shows
the influence of geometry and topology.
Some efforts were made in the study of shapr Sobolev Riemannian inequalities in some
decades. Part of the obtained results is kown as the AB program. Sharp inequalities and
first order Sobolev best constants in the scalar case were studied by Aubin, Druet, Hebey,
Vaugon, among others (see [3], [13], [19], [21]). But sharp inequalities and best Sobolev
constants in the vector case were studied by Hebey, E. R. Barbosa and M. S. Montenegro
(see [17], [18] and [5], [4]). For sharp Sobolev inequalities of second order in the scalar case,
the paper made by Djadli-Hebey-Ledoux show a introduction to study the first constant.
In this chapter, we show the AB program for scalar sharp Sobolev inequalities, and
after in the vector case, of second order. We show here some contributions, that is, we
answer some of the questions of AB program of the inequalities of second order.
1.1 Curvatures in a Riemannian manifold
Let (M, g) be a compact Riemannian manifold of dimension n. Let x ∈ M and X, Y, Z ∈TxM . We define:
18
R(X, Y )(x)Z = ∇X(x)(∇Y (x)Z)−∇Y (x)(∇X(x)Z)−∇[X,Y ](x)Z ,
where X, Y , Z are vector fields on M such that X(x) = X, Y (x) = Y, Z(x) = Z. This
definition is independant of the choice of the extensions X, Y , Z. Given x ∈ M and
X, Y, Z ∈ TxM and η ∈ (TxM)∗, we define the curvature tensor as follows:
R(x)(X, Y, Z, η) = η(R(Y, Z)(x) .X) = 〈R(X, Y )Z,W 〉 .
The function R is a (3,1)-tensor. The coordinates of R in a chart are given by:
R(x)lijk =
(∂Γlki∂xj
)x
−
(∂Γlji∂xk
)x
+ Γljα(x)Γαki(x)− Γlkα(x)Γαji(x) , (1.1)
where Γkij are the Christoffel symbols.
The Riemman tensor is a (4,0)-tensor and the coordinates in a chart are Rijkl :=
giαRαjkl.
The curvature operator R in x ∈ M , R : Λ2x → Λ2
x is defined by the relation
〈R(X ∧ Y ),W ∧ Z〉 = 〈R(X, Y )Z,W 〉 ,
where Λ2x is the space generated by the 2-forms X ∧ Y and
(X ∧ Y ) (Z,W ) = 〈Y, Z〉〈X,W 〉 − 〈X,Z〉〈Y,W 〉 .
Note that, if Xini=1 is a orthonormal basis of TpM , then Xi ∧ Yii<j is a orthonormal
basis of Λ2x. Thus, the symmetric bilinear function R is well defined.
Now let σ ⊂ TpM be a bidimensional space of TpM and X, Y a basis for σ. The
sectional curvature of σ in x is given by:
K(σ) = K (X, Y ) =R(X, Y, Y,X)
‖X ∧ Y ‖2,
where ‖X ∧ Y ‖2 = ‖X‖2‖Y ‖2 − 〈X, Y 〉2.
The Ricci tensor which we denote by Ricg or Ric is a symmetric (2,0)-tensor defined
as the following bilinear function:
Ric : TM × TM → R ,
that associates to each pair of fields (X, Y ) the trace of the function Z 7→ R(X,Z)Y .
Then we have:
Ric (X, Y ) =n∑i=1
〈R(Xi, X)Y,Xi〉 ,
where Xini=1 is a orthonormal basis of TpM .
19
The Ricci curvature in the direction of X ∈ TM , with |X| = 1, is defined as
Ric (X) = Ric (X,X) .
If X ∈ TM is unitary and X(p) = v, p ∈ M and v ∈ TpM , then we write Ricp(v)
instead of Ric(X,X). Let e1, · · · , en be a orthonormal basis with v = ei for some i.
Then:
Ricp(v) =∑j 6=i
〈R(v, ei)v, ei〉
=∑j 6=i
K(v, ei)
The scalar curvature we denote by Rg or Scalg is the trace of the Ricci tensor Rg :=
gijRij, where Rij are the components of Ricg in a chart.
The Riemann curvature Rmg is defined by
Rmg(x)(X, Y, Z, T ) = g(x) (X,R(Z, T )Y ) .
The components of Rmg in a chart are given by Rijkl = giαRαjkl where Rα
jkl are the
components of the curvature R as described above in (1.1).
Now let (M, g) be a Riemannian manifold of dimension n ≥ 3. The Weyl curvature
of g, denoted by Weylg, is a field of tensor of class C∞ four times covariant on M . Such
curvature is defined by:
Weylg = Rmg−1
n− 2Ricgg +
Rg
2(n− 1)(n− 2)g g ,
where is the Kulkarmi-Nomizu product, define below. For all x ∈ M we have:
Weylg(x) = Rmg(x)− 1
n− 2Ricg(x) g(x) +
Rg(x)
2(n− 1)(n− 2)g(x) g(x) .
The components Wijkl of Weylg are given by the relation:
Wijkl = Rijkl −1
n− 2(Rikgjl +Rjlgik −Rilgjk −Rjkgil)
+Rg
(n− 1)(n− 2)(gikgjl − gilgjk)
Let E be a vector space and h, k two tensores twice covariant and symmetric on E,
that is, for all X and Y in E:
20
h(X, Y ) = h(Y,X) and k(X, Y ) = k(Y,X) .
We define the Kulkarni-Nomizu product of h and k, we denote by h k, as the four
times covariant tensor in E, defined for all X, Y, Z, T ∈ E by
h k(X, Y, Z, T ) = h(X,Z)k(Y, T ) + h(Y, T )k(X,Z)− h(X,T )k(Y, Z)− h(Y, Z)k(X,T ) .
We see that the Kulkarmi-Nomizu product is symmetric in the sense that, for all h
and k, we have h k = k h. We also see that the product is distributive with respect
to addition, in that, for all h, k1 and k2 we have h (k1 + k2) = h k1 + h k2.
A Riemannian manifold is locally conformally flat if, in each point of M there exists
a neighborhood conformally equivalent to the Rm. That is, if x ∈ M , there exists a
neighborhood U of x and a function u class C∞, that is, u : U → R, such that a metric
(local) g = e2ug is flat in U .
If a manifold (M, g) has dimension n ≥ 4, then (M, g) is conformally flat if and only
if, Weylg ≡ 0.
21
1.2 The Musical Isomorphism and Divergence of Tensors
Consider x ∈M . # is the musical isomorphism between TxM and (TxM)∗, defined as:
# : TxM −→ (TxM)∗
X 7−→
(TxM) −→ R
Y 7−→ 〈X, Y 〉g(x) ,
This isomorphism is the identification of a Euclidean space with its dual. We denote:
X# the image of X via # and η# the image of η ∈ (TxM)∗ via the inverse of #. This
definition extends naturally vector field (that is, (0,1)-tensors) and to (1,0)-tensors. If X
is vector field and η is a (1, 0)−tensor, the coordinates of their images in a chart are:
Xi := (X#)i = gijXj
ηi := (η#)i = gijηj ,
that is an expression that is independent of the chart. Clearly, we have (X#)# = X e
(η#)# = η.
In what follows, in this work, A is defined as:
A((X)#, (X)#
)=
k∑i=1
Ai((Xi)
#, (Xi)#), (1.2)
where Ai, i = 1, ..., k, are positive and symmetric tensores, that is, A is a sum of continuous
symmetric (2, 0)−tensor. We have X = (X1, . . . , Xk) is such that each Xi, i = 1, . . . , k
is a (1,0)-tensor. In what follows, for simplicity, we say that A is a smooth symmetric
(2,0)-tensor when we refer the sum (1.2) above. Here Al(Xl)# is a (1, 0)-tensor whose
local coordinates are
(A(Xl)
#)i
= Aij((Xl)
#)j
= Aijgik(Xl)k
As the manifold M is compact and A is continuous, there exists a constant C > 0
such that: ∣∣∣∣∫M
A((X)#, (X)#
)dvg
∣∣∣∣ ≤ C
∫M
|X|2g dvg ,
for all X, where
22
∫M
A((X)#, (X)#
)dvg =
k∑i=1
∫M
Ai((Xi)
#, (Xi)#)dvg .
We have that the manifoldM is compact andA is a symmetric and positive (2, 0)−tensor,
then there are positive constants c and C, such that:
c|X|2g ≤ Ai((X)#, (X)#
)≤ C|X|2g , (1.3)
for X a (1,0)-tensor.
Let X be a smooth vector field in M . The divergence of X is a smooth function in M
given by:
divg X : M → R
p 7→ (divg X)(p) = tr v 7→ (∇vX)(p) ,
where v ∈ TpM and tr is the trace of the linear operator. To define on charts, consider
X a (1,0)-tensor in M . The divergence is defined as
divg(X) = gij(∇X)ij = gij(∂iXj − ΓkijXk
).
This expression is independent of the chart.
Throughout this work we use sometimes the following useful theorem.
Theorem 4 (Divergence theorem). Let (M, g) be a compact Riemannian manifold without
boundary. Let η be a smooth (1,0)-tensor. Then we have:∫M
divg(η) dvg = 0 .
in particular, given u, v ∈ C∞(M), we have∫M
u∆gv dvg =
∫M
〈∇u,∇v〉g dvg∫M
(∆gu) v dvg .
where 〈·, ·〉 is the scalar product associated with g for 1-form.
23
1.3 Homogeneous Functions
Consider G : M × Rk → R a continuous function and 2-homogeneous in the second
variable class C1 in the second variable. For example:
G(x, t) =k∑
i,j=1
Aij(x)titj ,
where t = (t1, . . . , tk) ∈ Rk and A = (Aij)M → M sk(R) is continuous, such that (Aij(x))
is positive defined, for all x ∈ M , and M sk(R) is the space of real symmetric matrices
k × k.
Let F : Rk → R be a positive function class C1 and 2#-homogeneous1, where 2# = 2nn−4
.
For example:
F (t) =k∑i=1
|ti|2#
,
where t = (t1, . . . , tk). In this case,
∂iF (t) = |ti|2#−2ti .
Now let F : Rk → R be a continuous function and q-homogeneous. Consider the
unitary sphere in the norm p, that is:
∂Bp [0, 1] =t ∈ Rk; |t|p = 1
,
where |t|p = (|t1|p + · · ·+ |tk|p)1p being t = (t1, . . . , tk). As the set ∂Bp [0, 1] is compact,
then there are constants mF,p > 0 e MF,p > 0 such that:
mF,p ≤ F (t) ≤MF,p ∀ t ∈ ∂Bp [0, 1] .
Now consider any t ∈ Rk \ 0. Then, by the q-homogeneity of F , we have
F
(t
|t|p
)=
1
|t|qpF (t) ,
for all t ∈ Rk \ 0. Thus,
mF,p|t|qp ≤ F (t) ≤MF,p|t|qp ∀ t ∈ Rk . (1.4)
As the space Rk has finite dimension, then all norms are equivalent. Therefore, there
exists a constant c > 0 such that
1F (λt) = λ2#
F (t), ∀ t ∈ Rk and λ > 0
24
|t|p ≤ c|t|q ∀ t ∈ Rk .
then
F (t) ≤M ′F,p|t|qq ∀ t ∈ Rk ,
where M ′F,p = cqMF,p is a positive constant.
In this work, in most situations, we use that F is a 2#-homogeneous function. There-
fore, this case will simplify the notation where F is a 2#-homogeneous function and the
norm is Euclidean: mF,2 = mF e MF,2 = MF . Then
mF |t|2#
2 ≤ F (t) ≤MF |t|2#
2 ∀ t ∈ Rk .
Similarly, if G : M × Rk → R is a q-homogeneous function in the second variable, we
have:
mG,p|t|qp ≤ G(x, t) ≤MG,p|t|qp ∀ t ∈ Rk .
In this work, we use, unless the contrary that G is a 2-homogeneous function on
the second variable. Therefore, this case will simplify the notation where G is a 2-
homogeneous function on the second variable and the norm is Euclidean: mG,2 = mG and
MG,2 = MG. Thereby, we have:
mG|t|22 ≤ G(x, t) ≤MG|t|22 ∀ t ∈ Rk . (1.5)
Let F : Rk → R be a C1 function q-homogeneous. In this work, we use the following
Euler identity:
k∑i=1
∂iF (t)ti = q F (t) .
Part of the vector theory of best constants does not follow directly from scalar theory.
One of the differences is in relation to the nature of vector functions satisfying the condi-
tions of homogeneity. In the case (k ≥ 2), there are examples of homogeneous functions
they are just continuous. Note the following example. Consider
F (t) = |t|2#µ e G(x, t) = β(x)|t|2µ ,
where | · |µ is a µ-norm defined by |t|µ =(∑k
i=1 |ti|µ) 1µ
for 1 ≤ µ < ∞ e |t|∞ =
max |t|i; i = 1, . . . , k.The absence of regularity of F and G creates an obstacle for the study of several
questions, since the arguments are based on Euler equations satisfied by critical points of
functional. That is, the approach in what follows would be restricted without assuming
25
regularity of F and G.
26
1.4 Sobolev Spaces of Vector Valued Maps of Second Order
We consider (M, g) a compact Riemannian manifold and dvg = dv(g) the Riemannian
measure associated with the metric g. Given a function u : M → R class C∞(M) and k
a integer, we denote ∇ku the k-nth covariant derivative of u and |∇ku| the norm of ∇ku,
defined by
|∇ku| = gi1j1 · · · gikjk(∇ku
)i1···ik
(∇ku
)j1···jk
,
where(∇ku
)i1···ik
denotes the components of ∇u in a chart. However this definition does
not depend on the choice of the chart.
If (M, g) is a compact Riemannian manifold, the Sobolev space H2,2(M) is, by defini-
tion, the completion of C∞(M) em L2(M) by the norm
‖u‖′H2,2(M) =2∑j=0
(∫M
|∇ju|2g dvg) 1
2
,
where dvg = dv(g) is Riemannian measure associated with the metric g.
The space H2,2(M) is a Hilbert space, with norm:
‖u‖2H2,2(M) =
∫M
|u|2 dvg +
∫M
|∇u|2 dvg +
∫M
(∆gu)2 dvg
Note that the norms ‖·‖′H2,2(M) and ‖·‖H2,2(M) are equivalent. Indeed, by the Bochner-
Lichnerowitz-Weitzenbock formula, we have that∫M
(∆gu)2 dvg =
∫M
|∇2u|2g dvg +
∫M
Ricg
((∇u)#, (∇u)#
)dvg
for all u ∈ H2,2(M). then, we have
‖∇2u‖22 + ‖∇u‖2
2 + ‖u‖22
= ‖∆gu‖22 −
∫M
Ricg
((∇u)#, (∇u)#
)dvg + ‖∇u‖2
2 + ‖u‖22
≤ |∆gu|22 + C‖∇u‖22 + ‖u‖2
2
for all u ∈ H2,2(M). But, hence we have that there exists a positive constant C such that
‖ · ‖′H2,2(M) ≤ C‖ · ‖H2,2(M). On the other hand, note that, for all function u ∈ C∞(M):
(∆gu)2 ≤ |∇2u|2
thus, as the two norms of H2,2(M) are equivalent.
27
The associate scalar product is defined by:
〈u, v〉 =
∫M
〈u, v〉g dvg +
∫M
〈∇u,∇v〉g dvg +
∫M
〈∇2u,∇2v〉g dvg
=2∑j=0
∫M
〈∇ju,∇jv〉g dvg .
As M is compact, the space H2,2(M) does not depend on a metric Riemannian. If M
is a compact manifold with two metrics g and g, then exist a real number C > 1 such
that, in any point of M :
1
Cg ≤ g ≤ Cg .
These two inequalities we reagrd as inequalities between bilinear forms.
A linear operator T : E → F between two Banach spaces (generally, we use E =
H2,2(M) and F = Rk, with k ∈ N) is compact if, for any sequences (un)n∈N ∈ E
uniformly bounded in the norm of E, then exists u ∈ F and a subsequence (unk) such
that:
limn→∞
T (unk) = u ,
strongly in F .
The space H2,2(M) is reflexive. Thus, all bounded sequence in H2,2(M) have weakly
convergent subsequences. And, if T : H2,2(M) → Rk is compact, then T takes limited
sequences in H2,2(M) in sequences that have convergent subsequences in Rk, for all k ∈ N.
The space H2,2(M) it is also a separable space. Therefore, every bouded sequence in
(H2,2(M))∗
has convergent subsequences (in the weak∗ topology).
Consider (Tn)n∈N ∈ (H2,2(M))∗
and T ∈ H2,2(M). We say that (Tn) converges weakly
for T if
limn→∞
Tn(u) = T (u) for all u ∈(H2,2(M)
)∗,
or
Tn T em(H2,2(M)
)∗,
where n→∞. (H2,2(M))∗
is the space of continuous linear forms of H2,2(M).
If M is a Riemannian manifold of dimension n ≥ 5, then the immersion
H2,2(M) → Lq(M) ,
is compact for q ∈(
1, 2nn−4
]. The immersion
28
H2,2(M) L2nn−4 (M) ,
is continuous, but not compact. That is, there exists a constant C > 0 such that
‖u‖L
2nn−4 (M)
≤ C‖u‖H2,2(M) .
We denote by H2,2k (M) = H2,2
(M,Rk
)the vector Sobolev space H2,2(M) × · · · ×
H2,2(M), that is
H2,2k (M) =
U = (u1, · · · , uk);ui ∈ H2,2(M) for all i = 1, · · · , k
.
This space has the norm:
||U||H2,2k (M) =
(∫M
(∆gU)2 dvg +
∫M
(|∇gU|2 + |U|2
)dvg
) 12
,
where U = (u1, . . . , uk) and,
∫M
|U|22dvg =k∑i=1
∫M
|ui|2 dvg,∫M
|∇gU|22dvg =k∑i=1
∫M
|∇gui|2dvg
∫M
(∆gU)2 dvg =k∑i=1
∫M
(∆gui)2 dvg .
Also:
|U|p = (|u1|p + · · ·+ |uk|p)1p .
To simplify, we use that:
|U| = |U|2 .
The vector-valued Sobolev space H2,2k (M) has different properties due to the properties
of the space H2,2(M). The space H2,2k (M) is a Hilbert space and has the following scalar
product:
〈U ,V〉H2,2k (M) =
k∑i=1
〈ui, vi〉 ,
where 〈·, ·〉 is the usual scalar product in H2,2(M) and U = (u1, · · · , uk) and V =
(v1, · · · , vk).The space H2,2
k (M) is separable and reflexive. Thus, the unit ball of H2,2k (M) is weakly
compact. In other words, for any sequences (Un)n∈N ∈ H2,2k (M) such that
29
‖Un‖H2,2k (M) ≤ C for all n ∈ N ,
there exists a subsequence (Uni)n∈N ∈ H2,2k (M) and exists U ∈ H2,2
k (M) such that
Uni U , (1.6)
weakly in H2,2k (M) when n→∞.
Let us return to the same bounded sequences (Un)n∈N in H2,2k (M). Let T : H2,2
k (M)→F be a compact operator and F a Banach space. Then the sequences (T (Un))n∈N converge
to T (U) (note the weak convergence of (1.6)). Furthermore, if Tn → T in(H2,2k (M)
)∗,
then
Tn(Un)→ T (U) .
We define the spaces Lqk(M) = Lq(M,Rk), for each q with 1 ≤ q < ∞, as the space
Lq(M) × · · · × Lq(M). That is,
Lqk(M) = U = (u1, · · · , uk); ui ∈ Lq(M), i = 1, · · · , k ,
with norm
‖U‖Lqk(M) =
(k∑i=1
‖ui‖q
) 1q
,
where U = (u1, . . . , uk) and ‖ · ‖p is the norm of Lp(M), defined by
‖u‖p =
(∫M
|u|p dvg) 1
p
.
To simplify, when there is no ambiguity,
‖U‖p
to indicate the norm of U = (u1, . . . , uk) in Lpk(M)
The Sobolev immersion H2,2k (M) → Lqk(M) is compact for 1 ≤ q < 2n
n−4and continuous
if q = 2nn−4
. So we say that 2∗ = 2nn−4
is the critical exponent with respect to the immersion
H2,2k (M) → Lqk(M).
30
1.5 Coercivity
Consider U = (u1, ..., uk) ∈ H2,2k (M). We define the functional Φ : H2,2
k (M) → R and
Ψ = ΨG : H2,2k (M)→ R, respectively, by:
Φ(U) =
∫M
F (U) dvg (1.7)
and
Ψ(U) = ΨG(U) =
∫M
(∆gU)2 dvg +
∫M
A((∇U)#, (∇U)#
)dvg +
∫M
G(x, U) dvg(∫M
F (U) dvg
) 2
2#
,
where A is as defined at the beginning (a sum of smooth (2,0)-tensors). We also define
Ψ : H2,2k (M)→ R by
Ψ(U) =
∫M
(∆gU)2 dvg +
∫M
A((∇U)#, (∇U)#
)dvg +
∫M
G(x, U) dvg .
Thus,
Ψ(U) = ΨG(U) =Ψ(U)(∫
M
F (U) dvg
) 2
2#
. (1.8)
We denote by L2k(M) the Sobolev space of L2(M)×L2(M)× · · · ×L2(M) with norm:
‖U‖2L2k(M) =
∫M
|U |2 dvg =k∑i=1
∫M
|ui|2 dvg ,
where U = (u1, · · · , uk). Similarly, we define the space L2#
k (M).
Using the Sobolev immersions H2,2k (M) → L2
k(M) and H2,2k (M) → L2#
k (M) the
functional ΨG and Φ are well defined.
Definition 1 (coercivity). We say that Ψ is coercive if exists α > 0 such that:∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ α
∫M
|U |2 dvg ,
for all U ∈ H2,2k (M), where A and G : M × Rk → R are as defined before.
We have the following proposition that show equivalent definitions of coercivity.
Proposition 5. The definitions below are equivalent:
(i) Ψ is coercive.
31
(ii) Exists α > 0 such that:∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ α‖U‖2
2#
≥ αM− 2
2#
F
(∫M
F (U) dvg
) 2
2#
,
for all U ∈ H2,2k (M).
(iii) Exists α > 0 such that:∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ α‖U‖2
H2,2k (M)
,
for all U ∈ H2,2k (M).
Proof. (iii)⇒ (ii)
Follows from the immersion:
H2,2k (M) → L2#
k (M) .
(ii)⇒ (i)
We obtain using the immersion:
L2#
k (M) → L2k(M) .
(i)⇒ (iii)
With (i) we have:∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ α‖U‖2
L2k(M) ,
for some α > 0. Consider 0 < ε < 1, such that ε ≤ αα+k
, where k > 0 is a constant
presented below. We have:
L(U) :=
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg , (1.9)
is equal to
L(U) = εL(U) + (1− ε)L(U) .
Hence,
L(U) ≥ ε
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg + (1− ε)α
∫M
|U |2 dvg .
32
Using the homogeneity of G and that A is bounded, we have:
L(U) ≥ ε
∫M
((∆gU)2 + c|∇U |2 +mG|U |2
)dvg + (1− ε)α
∫M
|U |2 dvg .
Consider ε0 = min ε, εc, εmG, that is, ε0 = rε, where r = 1, c, or mG. Then
L(U) ≥ ε0
∫M
((∆gU)2 + |∇U |2 + |U |2
)dvg + (1− ε)α
∫M
|U |2 dvg ,
take (1− ε)α ≥ ε0. Hence:∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ ε0‖U‖2
H2,2k (M)
.
33
1.6 The Scalar AB Program
In Rn, there exists a constant A > 0 such that
‖u‖2
L2# (Rn)≤ A
∫Rn
(∆ξu)2 dx , (1.10)
for all u ∈ C∞c (Rn) (the set of smooth functions with compact support in Rn) and where
ξ is the Euclidean metric of Rn.
Let A0 = A0(n) be the sharp constant in the Sobolev inequality (1.10). That is, A0
is the smallest constant satisfying (1.10). We define:
1
A20
= infu∈D2
2(Rn)\0
∫Rn
(∆ξu)2 dx(∫Rn|u|2# dx
) 2
2#
, (1.11)
where D22(Rn) is is the completion of the C∞c (Rn) com a norma ‖u‖D2
2(Rn) := ‖∆ξu‖2.
Follows from the Sobolev immersion theorem that the constant A0 > 0 is well defined.
This constant was calculated by Lieb [25], Lions [26], Edmunds-Fortunato-Jannelli [14]
and Swason [37]. We have:
1
A20
=n(n2 − 4)(n− 4)w
4nn
16,
where wn is the volume of the canonical unit sphere Sn in Rn+1.
Moreover, the extremal of the sharp inequality, that is, functions in D22(Rn) that
achieve the infimum in (1.11) are known and they are of the form:
uλ,µ,x0(x) = µ
(λ
λ2 + |x− x0|2
)n−42
,
where µ 6= 0, λ > 0 and x0 ∈ Rn are arbitrary.
The following Sobolev immersion is continuous, but not compact. If (M, g) is a Rie-
mannian manifold of dimension n ≥ 5 then H2,2(M) → L2#(M) continuously, where
2# = 2nn−4
. That is, exists A > 0 such that:
‖u‖L2# (M)
≤ A‖u‖H2,2(M) .
Further, from the continuous immersion H2,2(M) → L2#(M), then there are constants
A,B > 0 such that:
‖u‖22# ≤ A‖∆gu‖2
2 +B‖u‖2H1,2(M) , ∀ u ∈ H2,2(M) , (1.12)
where
34
‖u‖2H1,2(M) = ‖∇u‖2
2 + ‖u‖22 .
We are interested in the sharp constant A,B from the inequality above. More precisely,
we are interested in taking A minimized.
Easily see that A ≥ A20. Moreover, it holds true, for all ε > 0 exists Bε > 0 such that,
for any u ∈ H2,2(M) we have:
(∫M
|u|2# dvg) 2
2#
≤ (A20 + ε)
(∫M
(∆gu)2 dvg
)+Bε
∫M
(|∇gu|2 + |u|2
)dvg . (1.13)
In particular, we define
Kn = infA ∈ R; exists B such that the above inequality (1.12) is valid ∀ u ∈ H2,2(M)
,
(1.14)
then Kn = A20 for all manifold (M, g). A question that naturally comes is the following:
• The infimum Kn is achieved?
Or, equivalently, we can take ε = 0 in (1.13)?A positive response was given in 2000
by Djadli-Hebey-Ledoux [10] with the restriction that the metric g is conformally flat.
Then in 2003, Emmanuel Hebey proved the result for any Riemannian manifold [16].
Specifically:
Theorem 6 (E. Hebey, 2003). Let (M, g) be a compact manifold of dimension n ≥ 5.
Then exists B > 0, which depends on the manifold and the metric g, such that, for all
u ∈ H2,2(M)
(∫M
|u|2# dvg) 2
2#
≤ A20
∫M
(∆gu)2 dvg +B
∫M
(|∇gu|2 + |u|2
)dvg .
In particular, the infimum is achieved in (1.12).
Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. We denote
H2,2(M) as the completion of the space C∞(M) in the norm:
‖u‖2H2,2(M) =
∫M
(∆gu)2 dvg +
∫M
(|∇gu|2 + |u|2
)dvg .
As 2# = 2nn−4
, consider two positive functions f ∈ C∞(M) and h ∈ C0(M), exist two
positive constantes A, B such that, for any u ∈ H2,2(M):
(∫M
|u|2# dvg) 2
2#
≤ A
∫M
(∆gu)2 dvg +B
∫M
(f(x)|∇gu|2 + h(x)|u|2
)dvg . (1.15)
35
The first best Sobolev constant associated to the (1.15) is defined by:
A0(f, h, g) = inf A ∈ R; exists B ∈ R such that (1.15) is valid .
The first sharp Riemannian Sobolev inequality asserts that, for any u ∈ H2,2(M),
(∫M
|u|2# dvg) 2
2#
≤ A0(f, h, g)
∫M
(∆gu)2 dvg +B
∫M
(f(x)|∇gu|2 + h(x)|u|2
)dvg ,
(1.16)
for some constant B ∈ R.
The second sharp Sobolev constant associated to (1.15) is defined by:
B0(f, h, g) = inf B ∈ R; (1.16) is valid .
The second Sobolev Riemannian sharp inequality sates that, for any u ∈ H2,2(M) we
have:
(∫M
|u|2# dvg) 2
2#
≤ A0(f, h, g)
∫M
(∆gu)2 dvg +
+ B0(f, h, g)
∫M
(f(x)|∇gu|2 + h(x)|u|2
)dvg . (1.17)
A nonzero function u0 ∈ H2,2(M) is called extremal for the inequality (1.17) if
(∫M
|u|2# dvg) 2
2#
= A0(f, h, g)
∫M
(∆gu)2 dvg+B0(f, h, g)
∫M
(f(x)|∇gu|2 + h(x)|u|2
)dvg .
The AB scalar program consists of several questions of interest involving the sharp
constant A0(f, h, g) e B0(f, h, g), and the inequalities otimas (1.17) and (1.16). In what
follows, we will divide this program into parts: A program A and B program.
The A program consists of some problems involving A0(f, h, g) and the inequality
(1.16).
• question 1A: What is the exact value, or estimates of A0(f, h, g)?
• question 2A: The inequality (1.16) is valid?
• question 3A: The validity of (1.16) implies some geometric obstruction?
• question 4A: A0(f, h, g) depends continuously of f and h in some topology?
• question 5A: A0(f, h, g) depends continuously on the metric g in some topology?
• question 6A: What is the role of geometry on these questions?
36
The B program consists of some questions involving B0(f, h, g) and the inequality
(1.17):
• question 1B: What is the exact value, or estimates of B0(f, h, g)?
• question 2B: B0(f, h, g) depends continuously on f and h in some topology?
• question 3B: B0(f, h, g) depends continuously on the metric g in some topology?
• question 4B: The inequality (1.17) has extremal function?
• question 5B: The set of the extremal functions E(f, h, g) of L2#-norm is compact
in the C0 topology?
• question 6B: What is the role of geometry on these questions?
1.6.1 Partial Answers
As shown at the beginning of the previous section, we have answers only to the questions
1A and 2A.
There are no reponses for the questions of the scalar case in the B program. That is,
no author has worked on these questions.
We will answer some of the questions of the vector case, which in turn will provide
ansswers to similar questions of the scalar program.
37
1.7 AB Vector Program
Let n ≥ 5 and k ≥ 1 be integer. We denote by D2,2k (Rn) the Euclidean Sobolev vector
space D2,2(Rn)× · · · × D2,2(Rn) with norm
‖∆U‖D2,2k (Rn) =
(∫Rn
(∆U)2 dx
) 12
,
where
U = (u1, . . . , uk)
and
∫Rn
(∆U)2 dx =k∑i=1
∫Rn
(∆ui)2 dx .
Let F : Rk → R be a positive continuous function and 2#-homogeneous. In this case,
follows directly from (1.10) the existence of a constant A > 0 such that
(∫RnF (U) dx
) 2
2#
≤ A∫Rn
(∆U)2 dx , (1.18)
for all U ∈ D2,2k (Rn).
The sharp Euclidean Sobolev constant A associated with the inequality (1.18) is:
A0(F, n) = inf A ∈ R; (1.18) is valid .
The sharp Euclidean Sobolev vector inequality states that:
(∫RnF (U) dx
) 2
2#
≤ A0(F, n)
∫Rn
(∆U)2 dx , (1.19)
for all U ∈ D2,2k (Rn).
A nonzero map U0 ∈ D2,2k (Rn) is called extremal of (1.19), if
(∫RnF (U) dx
) 2
2#
= A0(F, n)
∫Rn
(∆U)2 dx . (1.20)
two basic questions related to (1.19) are:
(a) What is the exact value of A0(F, n)?
(b) (1.19) has extremal map?
We have reponses for these two questions in the following result.
38
Proposition 7. We have that
A0(F, n) = M2
2#
F A0(n) ,
where MF = maxSk−12
F and Sk−12 =
t ∈ Rk;
∑ki=1 |ti|2 = 1
and A0(n) is given by (1.11).
Furthermore, U0 ∈ D2,2k (Rn) is a extremal map of (1.19) if and only if, U0 = t0u0 for some
t0 ∈ Sk−12 such that MF = F (t0) and some extremal function u0 ∈ D2,2(Rn) of (1.10).
Proof. From the 2#− homogeneity of F :
F (t) ≤ MF
(k∑i=1
|ti|2) 2#
2
, ∀ t ∈ Rk .
Thus, using the Minkowski inequality and by (1.10), we have for any U ∈ D2,2k (Rn):
(∫M
F (U) dx
) 2
2#
≤ M2
2#
F
∫M
(k∑i=1
|ui|2) 2#
2
dvg
2
2#
≤ M2
2#
F
k∑i=1
(∫M
|ui|2#
dvg
) 2
2#
≤ M2
2#
F A0(n)
∫Rn
k∑i=1
(∆ui)2 dx = M
2
2#
F A0(n)
∫Rn
(∆U)2 dx . (1.21)
Hence, we obtain that:
A0(F, n) ≤M2
2#
F A0(n) .
now, by choosing U0 = t0u0 where t0 ∈ Sk−12 is such that MF = F (t0) and u0 ∈
D2,2(Rn) is a extremal function of (1.10), we have2
(∫RnF (U0) dx
) 2
2#
= M2
2#
F
(∫Rn|u0|2
#
dx
) 2
2#
= M2
2#
F A0(n)
∫Rn
(∆u0)2 dx
= M2
2#
F A0(n)
∫Rn
(k∑i=1
∆(ti0u0
))2
dx
= M2
2#
F A0(n)
∫Rn
(∆U0)2 dx . (1.22)
A0(F, n) ≥M2
2#
F A0(n) .
2Note that, in this case:
(∆U0)2
=
k∑i=1
(∆(ti0u0
))2= (∆u0)
2
39
Therefore,
A0(F, n) = M2
2#
F A0(n) .
We conclude also that the k-maps U0 = t0u0, as constructed above, are extremals. We
affirm that all extremals (1.19) has this form. In fact, let U ∈ D2,2k (Rn) be a extremal
of (1.19). In this case, U satisfies (1.21) with equality instead of the three inequalities.
But note that the second equality corresponds to the Minkowski inequality . This implies
that exists t ∈ Rk in |t|2 = 1 and u ∈ D2,2(Rn) such that U = tu. And, concluding, by
the first equality it follows that F (t) = MF and, from the third equality, u is extremal
function of (1.10).
Let F : Rk → R be a positive continuous and 2#-homogeneous function, and G :
M × Rk → R a positive continuous and 2-homogeneous function on the second variable.
Follows from the continuous immersion H2,2(M) → L2#(M) that exist two positive
constants A and B such that
(∫M
F (U)dvg
) 2
2#
≤ A∫M
(∆gU)2 dvg +
+ B∫M
(A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg . (1.23)
The first Sobolev best constant associated to (1.23) is defined as:
A0 = A0(A,F,G, g) = inf A ∈ R; exists B ∈ R such that (1.23) is valid .
The first sharp Riemannian Sobolev vector inequality states that, for any U ∈ H2,2k (M),
we have:
(∫M
F (U)dvg
) 2
2#
≤ A0
∫M
(∆gU)2 dvg +
+ B∫M
(A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg , (1.24)
for some constante B ∈ R.
The second Sobolev best constant associated to (1.24) is defined as:
B0 = B0(A,F,G, g) = inf B ∈ R; (1.24) is valid .
The second sharp Sobolev Riemannian vector inequality states that, for any U ∈H2,2k (M), vale
40
(∫M
F (U)dvg
) 2
2#
≤ A0
∫M
(∆gU)2 dvg +
+ B0
∫M
((A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg . (1.25)
This inequality is sharp in relation to the first and the second sharp Sobolev constant, in
that where neither can be reduced.
A natural question arises: It is possible to achieve equality in (1.25)? A nonzero
k-function U0 ∈ H2,2k (M) is called extremal of (1.25), if
(∫M
F (U0)dvg
) 2
2#
= A0
∫M
(∆gU0)2 dvg +
+ B0
∫M
((A((∇gU0)#, (∇gU)#
)+G(x, U0)
)dvg .
The AB vector program consists of several questions of interest involving the best
constants A0(A,F,G, g) and B0(A,F,G, g) and the sharp inequalities (1.24) AND (1.25).
This program is separated into two parts: the A program and the B program.
The A program is composed of the following questions involving A0(A,F,G, g) and
(1.24):
• question 1A: What is the exact value or estimates of A0(A,F,G, g)?
• question 2A: The inequality (1.24) is valid?
• question 3A: The validity of (1.24) implies in some geometric obstruction?
• question 4A: A0(A,F,G, g) depends continuously of F and G in some topology?
• question 5A: A0(A,F,G, g) depends continuously of g and A in some topology?
• question 6A: What is the role of geometry in these questions?
The B program consists of the following questions involvingB0(A,F,G, g) and (1.25):
• question 1B: What is the exact value or estimates of B0(A,F,G, g)?
• question 2B: B0(A,F,G, g) depends continuously of F and G in some topology?
• question 3B: B0(A,F,G, g) depends continuously on the metric g and A in some
topology?
• question 4B: The inequality (1.25) has extremal?
41
• question 5B: The set of the extremal E(A,F,G, g) normalized by∫MF (U)dvg = 1
is compact in the C0-topology?
• question 6B: What is the role of geometry in these questions?
1.7.1 Partial Answers
Some questions of the A program have been answered only by some authors for the scalar
case, that is, no previous studies exist for the vector case. One goal of this work is to give
a contribution to these questions in the vectorial case. The B program does not have any
previous result, even for the scalar case. We will show below answers to some questions
of the B vector program, which will give us, as already mentioned, responses to program
B in the scalar case.
We begin answering the question 1A of the AB program
Proposition 8. We have
A0(A,F,G, g) = M2
2#
F A0 ,
where MF = maxSk−12
F and Sk−12 =
t ∈ Rk;
∑ki=1 |ti|2 = 1
. In particular the inequality
otima (1.24) is valid.
Proof. We have, from the 2#− homogeneity of F :
F (t) ≤ MF
(k∑i=1
|ti|2) 2#
2
, ∀ t ∈ Rk .
Thus, using the Minkowski inequality, we have:
(∫M
F (U) dx
) 2
2#
≤ M2
2#
F
∫M
(k∑i=1
|ui|2) 2#
2
dvg
2
2#
≤ M2
2#
F
k∑i=1
(∫M
|ui|2#
dvg
) 2
2#
.
That is,
(∫M
F (U) dx
) 2
2#
≤M2
2#
F
(∫M
|U |2# dvg) 2
2#
. (1.26)
On the other hand, by the definition of A0, we have
k∑i=1
(∫M
|ui|2#
dvg
) 2
2#
≤ A0
∫M
k∑i=1
(∆gui)2 dvg +B
∫M
k∑i=1
(|∇ui|2 + u2
i
)dvg .
That is,
42
(∫M
|U |2# dvg) 2
2#
≤ A0
∫M
(∆gU)2 dvg +B
∫M
(|∇U |2 + |U |2
)dvg . (1.27)
By the hypotheses of G (remember that G : M×Rk → R is a positive continuous function
and 2-homogeneous on the second variable), there exists a constant m > 0 such that:
G(x, t) ≥ mk∑i=1
|ti|2, ∀ x ∈ M, t ∈ Rk ,
where m = minM×Sk−12
G. Hence, by joining (1.26) and (1.3) in the inequality (1.27), we
obtain for all U ∈ H2,2k (M):
(∫M
F (U) dvg
) 2
2#
≤M2
2#
F A0
∫M
(∆gU)2dvg+BM
2
2#
F
max c, m
∫M
(A((∇U)#, (∇U)#
)+G(x, U)
)dvg ,
this implies que
A0(A,F,G, g) ≤ M2
2#
F A0 . (1.28)
Now consider constant A and B such that
(∫M
F (U) dvg
) 2
2#
≤ A∫M
(∆gU)2 dvg + B∫M
((A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg ,
for all U ∈ H2,2k (M). We use (1.3) and let us now consider the constant A and B =
min BC,B, such that
(∫M
F (U) dvg
) 2
2#
≤ A0
∫M
(∆gU)2 dvg + B∫M
(|∇U |+G(x, U)) dvg .
We choose U = ut0 =(ut10, ut
20, ..., ut
k0
), where t0 =
(t10, . . . , t
k0
)∈ Sk−1
2 is such that
F (t0) = MF .using also that G is 2-homogeneous, we obtain:
43
(∆U)2 =k∑i=1
(∆uti0
)2= (∆u)2|t0|2 = (∆u)2
|∇U |2 =k∑i=1
∣∣∇uti0∣∣2 =k∑i=1
|ti|2|∇u|2 = |t0|2|∇u|2 = |∇u|2
|U |2 =k∑i=1
(uti0)2
= u2|t0|2 = u2
G(x, U) = G(x, t0u) = |u|2G(x, t0) ≤ c|u|2 .
where u ∈ H2,2(M). We found a constant B1 > 0 such that:
(∫M
|u|2# dvg) 2
2#
≤ A0M− 2
2#
F
∫M
(∆gu)2 dvg +B1
∫M
(|∇u|2 + u2
)dvg .
thus, we obtain that
A0M− 2
2#
F ≥ A0 . (1.29)
Hence, from (1.29) and (1.28), we obtain:
A0(A,F,G, g) = M2
2#
F A0 .
If F (U) =∑k
i=1 |ui|2#
, then A0 = Kn, because MF = 1. That is, in this case, the first
best constant A0 only depends on n.
Note that the first best constant from the vector theory does not depend on the
geometry and neither G, and depends on n and F with respect to C0loc(M) topology.
The first best constant A0 depends continuously on F . Let (Fα)α be a family of
positive continuous functions, Fα : Rk → R and 2#−homogeneous for each α which
converges toF : Rk → R on the C0 topology. Thus we have Aα0 = M2
2#
FαKn is the first
best constant associated to Fα:
(∫M
Fα(U) dvg
) 2
2#
≤ Aα0∫M
(∆gU)2 dvg +B∫M
((A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg .
But note that we have the following convergence: Aα0 → A0. That is, A0 depends
continuously on F .
What was done above answer some of the questions involved in A vector program.
Let us now focus on B vector program. More precisely, we answer some questions of the
B vector program in chapter 3.
44
In chapter 2 that follows, we developed a theory that will be used to demonstrate the
existence of extremal in chapter 3.
45
Chapter
2Elliptic Systems of Fourth Order
This part that follows return attention to the system of fourth order:
−∆2gU + divg
(A(∇U)#
)+
1
2∇UG(x, U) =
1
2#∇F (U) , (2.1)
where U = (u1, . . . , uk). That is,
−∆2gui + divg
(Ai(∇ui)#
)+
1
2∂iG(x, U) =
1
2#∂iF (U) ,
for i = 1, . . . , k. For most of the results in this section, we consider F : Rk → R a
C1 positive function and 2#-homogeneous and G : M × Rk → R a C1 positive function
and 2-homogeneous on the second variable. But, some results (as the L2 concentration
and compactness) we will take the following particular function: G(x, t) =∑k
ij=1Aijtitj,
where t = (t1, . . . , tk) ∈ Rk, Ai = big (bi ∈ R and g is the metric of the manifold) and
Aij : M → R are functions such that a matrix (Aij(x)) is positive defined and symmetric,
for all x ∈M . Thus, o system (2.1) is being:
−∆2gui + bi∆gui +
1
2
k∑j=1
Aijuj =1
2#|ui|2
#−2ui . (2.2)
In the scalar version of (2.1), a particular case is the following:
∆2gu+ bα∆gu+ cαu = u2#−1 (2.3)
where bα and cα are positive continuous functions. This case has been studied by many
authors (see for exemple [10], [15] and [24]). They analyze the solutions and the energy
functional associated. One of the questions in the work of Hebey-Wen is if bα and cα
converges to b0 and c0 respectively, that is, we have
∆2gu+ b0∆gu+ c0u = u2#−1 (2.4)
and if the solutions uα converges weakly in H2,2(M), then which conditions the weak limit
is not trivial? In this section we put the same question to solutions of (2.2) and we estab-
lished a result in the same direction of Hebey-Wen, that is the theorem of compactness
46
that is the end of this section.
One of the main differences between (2.1) and (2.3) is that the second equation can
be written, in most cases, as the product of elliptic operators of second order. This helps
to use standards techniques of second order as the Moser iteration from analysis. But
in the case of (2.1) we will not have a product in general. Thus, we are not able to use
directly the techniques of second order. Thus we used a technique of Moser iteration for
operators of fourth order done by Sandeep (see [31]).
Consider the operator Pg := ∆2g − divg
(A(∇·)#
)+ a (where a ∈ C∞(M) and A
is a smooth symmetric (2,0)-tensor). This operator Pg does not satisfy the principle of
punctual comparison, even when it is coercive. This is an important difference to operators
of second order.
A question of importance is the following: the operator Pg satisfies the maximum
principle? A resposta is positive if Pg is the product of two elliptic operators, each one
coercive. Thus, Pg = (∆g + a) (∆g + a′), where a and a′ ∈ C∞(M). Also, if 4b0 ≤ c20 in
(2.4), we obtain the decomposition of Pg in elliptic coercive operators and therefore we
have also the maximum principle (see [30]).
One of the highlight of this section is the bubbles decomposition theorem, since using
this theorem, we were able to prove the main results and get our contributions. We
demonstrate that theorem proving initially that Palais-Smale sequence converges weakly
and we obtain the weak limit of the sequence. We obtain a new Palais-Smale sequence
for a new functional. Thence we use the scalar case to complete the theorem..
Our contribution here gives extension to some work already done. (for example E.
Hebey, F. Robert, Z. Djadli, M. Ledoux, V. Felli, and others) where has been studied
(2.2) in the escalar case.
This section will begin with the existence of solutions, then we will through the regu-
larity, punctual estimates and L2 concentration, ending with compactness.
2.1 Existence
Let q ∈[2, 2#
]. We define:
µq = infIq(U);U ∈ H2,2
k (M) \ 0 e Φ(U) = 1,
where Φ(U) =
∫M
F (U) dvg (see (1.7)) and Iq : H2,2k (M) \ 0 → R is the following
functional:
Iq(U) =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg(∫
M
F (U) dvg
) 2q
, (2.5)
47
defined for all U ∈ H2,2k (M) \ 0, where F : Rk → R is a q-homogeneous function. The
functional Iq is well defined, because Sobolev embeddingH2,2k (M) → Lqk(M) is continuous.
To get solutions of (2.1) in the critical case, that is, when q = 2#, we have to use
another method. This presented later. In the result which follows, we focus on the
subcritical case, that is, when q < 2#.
Proposition 9. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. Let A
be a symmetric (2, 0)−tensor; F : Rk → R is a C1 function positive and q−homogeneous
and G : M × Rk → R is a continuous function, of class C1 and 2-homogeneous on the
second variable. Assume that q ∈[2, 2#
). Then, µq is finite and is achieved. That is,
µq ∈ R and there is U ∈ H2,2k (M) \ 0 such that Iq(U) = µq.
Proof. Initially, we prove that µq > −∞. Let U ∈ H2,2k (M) \ 0. As A is smooth, then
there exists C > 0 such that∣∣∣∣∫M
A((∇U)#, (∇U)#
)dvg
∣∣∣∣ ≤ C
∫M
|∇U |2g dvg , (2.6)
for all U ∈ H2,2k (M). We note that there is a constant C ′ > 0 such that∣∣∣∣∫
M
A((∇U)#, (∇U)#
)dvg
∣∣∣∣ ≤ 1
2
∫M
(∆gU)2 dvg + C ′‖U‖22 , (2.7)
for all U = (u1, . . . , uk) ∈ H2,2k (M), where ‖U‖2
2 =∑k
i=1
∫M|ui|2 dvg.
To prove the above inequality, we prove the two following lemmas.
Lemma 10. Let (M, g) be a compact Riemannian manifold. Then, for each ε > 0, exists
C(ε) > 0 such that
‖∇u‖2 ≤ ε‖∆gu‖2 + C(ε)‖u‖2, ∀ u ∈ H2,2(M) .
Proof. Suppose, by contradiction, that there is ε > 0 such that, for each α ∈ N∗, exists
uα ∈ H2,2(M) such that
‖∇uα‖2 > ε‖∆guα‖2 + α‖uα‖2, e ‖∇uα‖2 = 1 .
Thence,
1 > ε‖∆guα‖2 e 1 > α‖uα‖2 .
That is,
‖∆guα‖2 <1
εe ‖uα‖2 <
1
α.
Thus,
48
‖∆guα‖2 + ‖∇uα‖2 + ‖uα‖2 <1
ε+ 1 +
1
α.
Therefore, we have that exists a constant C > 0 such that
‖uα‖H2,2(M) ≤ C .
Furthermore, we have ‖uα‖2 → 0. As the embedding H2,2(M) → H1,2(M) is compact,
up to a subsequence, we have uα u (u ∈ H2,2(M)) and uα → u in H1,2(M). Therefore,
‖∇u‖2 = 1 and ‖u‖2 = 0, that is a contradiction.
We will use this lemma 10 above to prove the lemma which follow. To prove (2.7),
just we check the escalar case. The vector case follows directly from the lemma below.
Lemma 11. Let A be a (2, 0)−tensor. There exists a constant C ′ > 0 such that∣∣∣∣∫M
A((∇u)#, (∇u)#
)dvg
∣∣∣∣ ≤ 1
2
∫M
(∆gu)2 dvg + C ′‖u‖22 .
Proof. Let C > 0 be a constant on the inequality (2.6). We take ε = 1√4C
in the lemma
10 above:
‖∇u‖2 ≤1√4C‖∆gu‖2 + C(ε)‖u‖2 .
Thence,
‖∇u‖22 ≤
(‖∆gu‖2√
4C+ C(ε)‖u‖2
)2
=‖∆gu‖2
2
4C+
2C(ε)√4C‖∆gu‖2‖u‖2 + C2(ε)‖u‖2
2
≤ ‖∆gu‖22
4C+‖∆gu‖2
2
4C+
4C2(ε)‖u‖22
2+ C2(ε)‖u‖2
2 .
Thus,
C
∫M
|∇u|2 dvg ≤1
2
∫M
(∆gu)2 dvg + C ′‖u‖22 .
Therefore, by (2.6) we obtain the result of the lemma.
Let us now turn to the proof of the proposition. As F is q-homogeneous and positive,
we have that there exists a constant MF,2 = M ′F > 0 such that:
F (U) ≤M ′F |U |
q2 ≤ M ′
F |U |qq , para toda U ∈ Rk , (2.8)
49
where |U |22 = |u1|2 + · · · + |uk|2 and U = (u1, . . . , uk) (see section 1.3, p. 23). And, from
the homogeneity of G, we obtain that there is a constant mG > 0 such that:
mG|U |22 ≤ G(x, U) , para toda U ∈ Rk . (2.9)
We will use now the above inequalities (2.9), (2.8), (2.7) and the Holder inequality (also
used the idea at the beginning of proof of proposition 8). Note that, for each i = 1, . . . , k
we have
∫M
|ui|2 dvg ≤(∫
M
|ui|q dvg) 2
q
vol(M)1− 2q
and as ‖ui‖2q ≤ ‖U‖2
q, we have:
∫M
|U | dvg ≤ k
(∫M
|U |qq dvg) 2
q
vol(M)1− 2q
Then we have:
Iq(U) =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg(∫
M
F (U) dvg
) 2q
≥
1
2
∫M
(∆gU)2 dvg − (C ′ −mG)‖U‖22(∫
M
F (U) dvg
) 2q
≥
∫M
(∆gU)2 dvg
2
(∫M
F (U) dvg
) 2q
− (C ′ −mG) ‖U‖22
V olg(M)2q−1(M ′
F
) 2q ‖U‖2
q
=
∫M
(∆gU)2 dvg
2
(∫M
F (U) dvg
) 2q
− k (C ′ −mG)
V olg(M)2q−1(M ′
F
) 2q
, (2.10)
for all U ∈ H2,2k (M) \ 0, where ‖U‖2
2 =∑k
i=1 ‖ui‖22. Note that we are assuming that
(C ′ −mG) > 0. Otherwise, if (C ′ −mG) < 0 we would have:
Iq(U) ≥
∫M
(∆gU)2 dvg
2 (M ′F )
2q ‖U‖2
q
This prove, for both cases, that µq > −∞ and then µq ∈ R. Now, let (Uα)α∈N ∈
50
H2,2k (M) \ 0 be a minimizing sequence for Iq, that is:
limα→∞
Iq(Uα) = µq . (2.11)
Without loss of generality, we can assume that∫M
F (Uα) dvg = 1 , (2.12)
for all α ∈ N. From (2.10) and (2.11), we have that theres exists a constant C0 > 0 such
that: ∫M
(∆gUα)2 dvg ≤ C0 ,
for all α ∈ N. From (1.4) and (2.12), we obtain:
‖Uα‖22 =
∫M
|Uα|22 ≤ C1, para uma constant C1 > 0 .
And by lemma 10, using the two limitations above, we have:
‖∇U‖22 ≤ C2 ,
where C2 > 0. Thence, exists a constant C > 0, such that:
‖Uα‖H2,2k (M) ≤ C ,
for all α ∈ N. As the unit ball of H2,2k (M) is weakly compact, we have that there exists
U ∈ H2,2k (M) that is the weak limit of (Uα)α∈N, up to the extraction of a subsequence.
That is, exists a subsequence (Uα′) ⊂ (Uα) such that
Uα′ U ,
weak convergence in(H2,2k (M)
)∗, when α → ∞. Without loss of generality, we assume
that the convergence is for the initial sequence (Uα)α∈N. As the embedding H2,2k (M) →
H1,2k (M) is compact, we have
limα→∞
Uα = U em H1,2k (M) .
For 2 ≤ q < 2#, the embedding H2,2k (M) → Lqk(M) is compact, we have also
limα→∞
Uα = U em Lqk(M) .
We also have Uα → U qtp. Thus∫MF (U) dvg = 1 and by (1.4) we conclude that U 6= 0,
where U = (u1, . . . , uk). Consider Θα = Uα − U ∈ H2,2k (M) for α ∈ N. We have:
51
∫M
((∆gUα)2 + A
((∇Uα)#, (∇Uα)#
)+G(x, Uα)
)dvg
=
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg +
∫M
(∆gΘα)2 + o(1) ,
where limα→∞ o(1) = 0, because Θα 0 in(H2,2k (M)
)∗and Uα → U in H1,2
k (M) when
α→∞. Assume that:
µq =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg +
∫M
(∆gΘα)2 + o(1) , (2.13)
when α→∞. As U 6= 0, we have Iq(U) ≥ µq. But we also have:
µq ≤∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg +
∫M
(∆gΘα)2 . (2.14)
that is µq ≤ Iq(U). Joining (2.13) and (2.14), we have:
µq =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg = Iq(U) ,
and
limα→∞
∫M
(∆gΘα)2 dvg = 0 .
In particular, the infimum µq is achieved in U ∈ H2,2k (M).
Now, consider the set Λ defined by:
Λ =U ∈ H2,2
k (M); Φ(U) = 1,
where Φ(U) =∫MF (U) dvg (see (1.7)). Consider also the number λ given by:
λ = infU∈Λ
Ψ(U) ,
where Ψ(U) = I2#(U) (see (1.8) and (2.5)).
We see easily that if (Uα)α is a minimizing sequence for λ, then (Uα)α is a limited
sequence in H2,2k (M). In fact, such statement arises from the fact that (Uα)α ⊂ Λ and
from the existence of a constant mF > 0 such that mF |U |2#
2 ≤ F (U).
In the theorem which follows, we see that the system (2.1) has a solution of minimum
energy.
52
Theorem 12. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. Let
F : Rk → R be a positive function of class C1 and 2#−homogeneous and G : M×Rk → Rbe a continuous function of class C1 and 2-homogeneous on the second variable. Let also
A be a smooth symmetric (2, 0)−tensor and assume that Ψ is coercive. If λ < 1A0
, then
(2.1) has weak solution U0 ∈ Λ such that
Ψ(U0) = λ .
Proof. As Ψ is coercive, we have that exists α > 0 such that
Ψ(U) ≥ α‖U‖22# ≥ M
− 2
2#
F,2#α
(∫M
F (U) dvg
) 2
2#
,
(see the proposition 5). We claim that the infimum λ is positive, that is, λ > 0. From
the above inequality we obtain:
Ψ(U) =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg(∫
M
F (U) dvg
) 2
2#
≥ α
M2
2#
F
,
for all U ∈ H2,2k (M). Thus λ ≥ αM
− 2
2#
F > 0. This proves the assertion.
Let (Uα)α ⊂ Λ be a minimizing sequence for λ. That is,
limα→∞
Ψ (Uα) = λ .
The sequence is limited, existe U0 ∈ H2,2k (M) such that, up to a subsequence, we have:
Uα U0 weakly in H2,2k (M)
Uα → U0 strongly in H1,2k (M) and in L2
k(M) .
We consider Θα = Uα − U0 ∈ H2,2k (M) for all α ∈ N. As in the proof of the previous
proposition (proposition 9), we have:
λ = Ψ(U0) +
∫M
(∆gΘα)2 dvg + o(1) , (2.15)
where limα→∞ o(1) = 0. Without loss of generality, we can assume that:
limα→∞
Θα(x) = 0 q.t.p x ∈ M .
By the Brezis-Lieb generalized lemma (see [1]), we have:
53
limα→∞
(∫M
F (U0 + Θα) dvg −∫M
F (U0) dvg −∫M
F (Θα) dvg
)= 0 . (2.16)
With the inequality (1.24) (also from (1.3) and (1.5)) and the convergence of Θα in
H2,2k (M), we have:
(∫M
F (Θα) dvg
) 2
2#
≤ A0
∫M
(∆gΘα)2 dvg +B
∫M
(|∇Θα|2 +G(x,Θα)
)dvg
≤ A0
∫M
(∆gΘα)2 dvg + o(1) . (2.17)
By the definition, we have Ψ is 2-homogeneous, that is, Ψ(αU) = α2Ψ(U), for α ∈ R.
Thus:
λ ≤ Ψ
U0(∫M
F (U0) dvg
) 1
2#
=1(∫
M
F (U0) dvg
) 2
2#
Ψ(U0)
Therefore,
Ψ(U0) ≥ λ
(∫M
F (U0) dvg
) 2
2#
. (2.18)
Using in (2.15) the inequalities (2.16), (2.17) and (2.18), we obtain:
λ ≥ λ
(∫M
F (U0) dvg
) 2
2#
+ (A0)−1
(∫M
F (Θα) dvg
) 2
2#
+ o(1)
≥ λ
(∫M
F (U0) dvg
) 2
2#
+ (A0)−1
(1−
∫M
F (U0) dvg
) 2
2#
+ o(1) .
When α→ +∞ in this last inequality we obtain:
λA0
(1−
(∫M
F (U0) dvg
) 2
2#
)≥(
1−∫M
F (U0) dvg
) 2
2#
.
Note that 22#
= 1− 4n< 1 and
F (Uα) F (U0) em L1(M) ⇒∫M
F (U0) dvg ≤ lim inf
∫M
F (Uα) dvg = 1 .
We then conclude then:
54
∫M
F (U0) dvg ≤ 1 . (2.19)
As 1−Xp ≤ (1−X)p for all X ∈ [0, 1] and for all 0 ≤ p ≤ 1, we obtain1:
(λA0 − 1)
(1−
(∫M
F (U0) dvg
) 2
2#
)≥ 0 .
From the hypothesis λ < 1A0
, we obtain:∫M
F (U0) dvg ≥ 1 .
On the other hand, from weak convergence (see (2.19)), we obtain that:∫M
F (U0) dvg ≤ 1 .
Therefore, ∫M
F (U0) dvg = 1 .
Thus, U0 ∈ Λ and U0 6= 0 (by (1.4), because F is 2#-homogeneous). Thus as in
the proof of the previous proposition, we also have limα→∞Θα = 0 in H2,2k (M) and
limα→∞∫M
(∆Θα)2 dvg = 0. Thus, from (2.15), we have
λ = Ψ(U0) .
Therefore, U0 ∈ H2,2k (M) \ 0 is a minimizer for Ψ and U0 is solution for the problem
(2.1), that is,
Ψ′(U0)V = λΦ′(U0)V for all V ∈ H2,2k (M) ,
Lagrange multipliers theorem.
1If (X − 1) ≥ 0, then 1 = (1−X +X)p ≤ (1−X)p +Xp, because (a+ b)s ≤ as + bs if a, b > 0 ands ∈ [0, 1] .
55
2.2 Regularidade
The Trudinger strategy for second order operators, as in the case of Yamabe type op-
erators, does not apply satisfactorily to the case of fourth order operators. But we can
note that Sandeep ([31]) was able to develop the De Giorgi-Nash-Moser method for fourth
order equations, which is a technique that is very close to the technique of Trudinger.
Some techniques used to fourth order operators were developed by Van der Vorst and,
and in the Riemannian context, by Esposito-Robert (see [15], [38] and also [10]).
An important result that we obtain with coercivity is the following.
Proposition 13. Let (M, g) be a compact Riemannian manifold and let A be a sum of
smooth symmetric and positive (2, 0)−tensores. Assume that S ∈ Hr,pk (M) and G :
M × Rk → R of class C1 is a positive 2-homogeneous function on the second variable, of
class C1. Then existe U ∈ H4+r,pk (M) such that
−∆2gU + divg
(A(∇U)#
)+
1
2∇UG(x, U) = S(x) . (2.20)
Morover, we have
‖U‖H4+r,pk (M) ≤ C‖S‖Hr,p
k (M) .
for all U ∈ H2,2k (M) where C = C(M, g, k) > 0. And if G(x, t) =
∑ki,j=1 Aij(x)titj, where
(Aij(x)) is positive as bilinear form and symmetric, then we have unicity in (2.20).
Proof. Let A be a sum of smooth and symmetric (2, 0)−tensores and G a C1 and 2-
homogeneous function on the second variable. The functional Ψ is coercive, that is, exists
λ > 0 such that:
Ψ(U) =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥ λ
∫M
|U |22 dvg .
In this proof, we consider the operator
PgU = −∆2gU + divg
(A(∇U)#
)+
1
2∇UG(x, U) ,
or, considering the coordinates functions of U = (u1, . . . , uk), we have (we use the same
notation for the operator Pg):
Pgui = −∆2gui + divg
(Ai(∇ui)#
)+
1
2∂iG(x, U) ,
for i = 1, . . . , k.
Affirmation 1. Let p > 1. We affirm that there is C > 0 such that
‖U‖Lpk(M) ≤ C‖PgU‖Lpk(M) ∀ U ∈ H4,pk (M) .
56
We prove this affirmation by contradiction. We assume that, for all α ∈ N∗, exists
Uα = (u1α, . . . , u
kα) ∈ H4,p
k (M) such that
‖Uα‖p = 1 e ‖PgUα‖p ≤1
α. (2.21)
We denote Sα(x) = (S1α, . . . , S
kα). Let Siα = Pgu
iα. Thus,
−∆2gu
iα + divg
(Ai(∇uiα)#
)+
1
2∂iG(x, Uα) = Siα(x) ,
for i = 1, . . . , k. Applying the Lp theory in
−∆2gui + divg
(Ai(∇ui)#
)= Siα(x)− ∂iG(x, Uα)
e we obtain
‖uiα‖H4,pk (M) ≤ C
(‖Siα − ∂iG(x, Uα)‖p + ‖uiα‖p
)≤ C
(‖Siα‖p + ‖∂iG(x, Uα)‖p + ‖uiα‖p
)≤ C0
(‖Siα‖p + ‖uiα‖p + ‖uiα‖p
)≤ C0
(1
α+ 2
)≤ C ′ ,
where C ′ > 0 does not depend on α. Thus, the sequence (uiα) is limited in H4,p(M).
Then, up to a subsequence, we have:
uiα ui em H4,p(M) ,
for i = 1, . . . , k. And, up to another subsequence,
uiα → ui em H2,p(M) ,
for i = 1, . . . , k, because the embedding H4,pk (M) → H2,p
k (M) is compact. For any
ϕ ∈ C∞(M), we have
∫M
(∆gu
iα∆gϕ+ Ai
((∇uiα)#, (∇ϕ)#
)+
1
2∂iG(x, Uα)ϕ
)dvg =
∫M
Siα(x)ϕ dvg ,
for i = 1, . . . , k. Or, writing in another way:∫M
ϕPguiα dvg =
∫M
Siα(x)ϕ dvg .
Now, doing α → ∞ and by using (2.21) together with the Holder inequality, we have
57
Pgui = 0 in the weak sense, for i = 1, . . . , k, or PgU = 0. Form the Schauder theory, we
have ui ∈ C4(M). And, form the coercivity, we have:
0 =
∫M
UPgU dvg ≥ λ
∫M
|U |2 dvg ,
that is, U ≡ 0. That is a contradiction, because:
‖U‖p = limα→+∞
‖Uα‖p = 1 .
So, we prove the affirmation.
Affirmation 2. Let α ∈ (0, 1). We affirm that for any S ∈ C0,αk , G and A as in
the affirmation 1, then there exists U ∈ H4,2k (M) such that
PgU = S .
Now we prove the affirmation. Let us consider the functional:
I(U) =1
2
∫M
UPgU dvg −∫M
S · U dvg ,
for all U ∈ H2,2k (M), where U(x) = (u1(x), . . . , uk(x)), S(x) = (S1(x), . . . , Sk(x)) e S ·U =
s1u1 + · · ·+ uksk. From the coercivity and Holder inequality, we have:
I(U) ≥ λ‖U‖22 − ‖S‖2‖U‖2 ≥ −
‖S‖22
4λ. (2.22)
Then, µ = infI(U) ;U ∈ H2,2
k (M)> −∞ is defined. Let (Uα)α∈N ∈ H2,2
k (M) be a
minimizing sequence for µ, that is,
limα→∞
I(Uα) = µ . (2.23)
With the first inequality of (2.22), we have ‖Uα‖2 < α for all α ∈ N. From (2.23) and
the coercivity we have
‖Uα‖H2,2k (M) = O(1) ,
when α → +∞. As the unit ball of Hm,pk (M) is weakly compact and the sequence
(Uα) is limited in H2,2k (M), then exists a subsequence (Uα′) ∈ H2,2
k (M) and there exists
U ∈ H2,2k (M) such that
Uα′ U fracamente em H2,2k (M) .
The embedding H2,2k (M) → H1,2
k (M) is compact, then up to another subsequence, we
have
Uα → U fortemente em H1,2k (M) .
58
where, without loss of generality, we return to index of the original sequence. Then we
have:
I(Uα) = I(U) +1
2
∫M
(∆g(Uα − U))2 dvg + o(1) = µ+ o(1) ,
when α→ +∞. As µ is the infimum, we have µ ≤ I(U) and
limα→∞
∫M
(∆g(Uα′ − U))2 dvg = 0 .
Therefore, µ = I(U). We have then I ′(U) = 0, because I ∈ C1(H2,2k (M),R). That is,
PgU = S in the weak sense. From
−∆2gU + divg
(A(∇U)#
)= S(x)− 1
2∇UG(x, U) .
Follows from the Schauder theory that U ∈ H4,2k (M).
Affirmation 3. We affirm that, for any S ∈ Lpk(M) and G(x, t) =∑k
i,j=1Aij(x)titj,
where (Aij(x)) is positive as bilinear form and symmetric for all x ∈ M , exists a unique
U ∈ H4,pk (M) such that
−∆2gU + divg
(A(∇U)#
)= S − 1
2∇UG(x, U) .
Let (Sm)m∈N ∈ C∞k (M) be a sequence such that:
limm→∞
Sm = S
strongly in Lpk(M). For each m ∈ N, let Um ∈ C4k(M) be such that (see affirmation 2):
−∆2gUm + divg
(A(∇Um)#
)= Sm .
As Ψ is coercive and from Lp theory, we have for any m, n ∈ N:
‖Um − Un‖H4,pk (M) ≤ C ′
(‖Sm − Sn‖Lpk(M) + ‖Um − Un‖Lpk(M)
)≤ k‖Sm − Sn‖p
.
We have then that (Um) is a Cauchy sequence in H4,pk (M) and then exists U ∈ H4,p
k (M)
such that limm→∞ Um = U in H4,pk (M). We have then:
−∆2gU + divg
(A(∇U)#
)= S − 1
2∇UG(x, U) .
Now assume that V ∈ H4,pk (M) satisfies:
59
−∆2gV + divg
(A(∇V )#
)= S − 1
2∇VG(x, V ) .
Subtracting the last two equalities, we obtain that
−∆2g (U − V ) + divg
(A(∇(U − V ))#
)+
1
2∇G(x, U − V ) = 0 ,
remembering that G is as defined in the statement of Affirmation 3. Follows from coer-
civity that U ≡ V :
0 ≥ Ψ(U − V ) ≥ λ
∫M
|U − V |2 dvg ,
and this proves the affirmation.
The part of existence is proven in affirmation 3 above. The estimate a priori is a
consequence of affirmation 1 and Lp theory:
‖U‖H4+r,pk (M) ≤ C
(‖S‖Hr,p
k (M) + ‖U‖Lpk(M)
)≤ C ′‖U‖Hr,p
k (M)
For the particular cases of the functions F and G, we can improve the regularity of
the solution. See the proposition below. Thus, the question of regularity is resolved.
Proposition 14. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. Let
A be a sum of symmetric and positive (2, 0)−tensores . Let F : Rk → R be a positive C1
function and 2#-homogeneous and G : M × Rk → R a 2-homogeneous function on the
second variable of class C1, given by F (t) =∑k
i=1 |ti|2#
and G(x, t) =∑k
i=1Aij(x)titj. In
wath (Aij(x)) is symmetric and positive defined as bilinear form, for all x ∈ M . Assume
that U = (u1, . . . , uk) ∈ H2,2k (M) is a weak solution of:
−∆2gui + divg Ai((∇ui)#) +
1
2
k∑j=1
Aijuj =1
2#|ui|2
#−2ui (2.24)
i = 1, . . . , k. Then U ∈ C4k(M) and U is a solution of (2.24) in the usual sense.
Proof. Let p ≥ 1, R > 0 (defined after) and V = (v1, · · · , vk) ∈ Lpk(M). From Holder
inequality, we have
∂iF (U)χ|U |≥R vi ∈ Lr(M) where
1
r=
1
p+
4
n,
where denote here |U | = |U |2# and χ is the characterictic function. In fact, we obtain:
60
‖∂iF (U)χ|U |≥R vi‖r =
(∫M
|∂iF (U)1|U |≥R vi|r dvg
) 1r
≤
((∫M
|∂iF (U)χ|U |≥R|n4
) 4rn(∫
M
|vi|p dvg) r
p
) 1r
= ‖∂iF (U)χ|U |≥R‖n4‖vi‖p .
Follows from the regularity theory (see the previous result) that there is a unique W =
(w1, · · · , wk) ∈ H4,rk (M) such that:
Pgwi = ∂iF (U)χ|U |≥Rv
i , (2.25)
for i = 1, . . . , k. Exists C = C(p, r, n) > 0 such that
‖wi‖H4,r(M) ≤ C‖∂iF (U)χ|U |≥Rvi‖Lr(M) . (2.26)
We know that H4,r(M) → Lq(M) continuously, where 1q
= 1r− 4
n= 1
p. Then wi ∈ Lpk(M)
and exists C = C((M, g), p, r, n) > 0 such that
‖wi‖p ≤ C‖∇ F (U)χ|U |≥R‖n4‖vi‖p . (2.27)
We define the o operator Tp,R : Lp(M) → Lp(M) such that, for each v ∈ Lp(M),
Tp,R(v) = w where w is as above. From we have alread done above (2.27), Tp,r is a
continuous map. But also, note that this map is a linear map. Let vi1 and vi2 ∈ Lp(M) be
such that:
Pgwi1 = ∂iF (U)χ|U |≥Rv
i1
Pgwi2 = ∂iF (U)χ|U |≥Rv
i2
Then,
Pg(wi1) + Pg(w
i2) = Pg(w
i1 + wi2) = ∂iF (U)χ|U |≥R(vi1 + vi2)
On the other hand, assume that wi ∈ Lp(M) is the unique function such that:
Pg(wi) = ∂iF (U)χ|U |≥R(vi1 + vi2)
Thus, form the uniqueness of the solution of (2.25), we have wi = wi1 + wi2. Therefore,
the operator Tp,R is linear.
61
‖Tp,R‖Lp→Lp ≤ C(p, r, n)
(∫|U |≥R
|U |2#) 4
n
.
Thus, as U ∈ L2#
k (M) exists R0 = R((M, g), p, r, n) > 0 such that
‖Tp,r‖Lp→Lp ≤1
2.
Then we have IdLp − Tp,r : Lpk(M) → Lpk(M) is linear and continuous with the inverse
linear and continuous.
As ∇F (U)χ|U |≤R ∈ L∞k (M) we have, by the regularity (see the previous result), that
for all p ≥ 2# exists U = (u1, · · · , uk) ∈ H4,pk (M) such that:
−∆2gui + divg
(Ai(∇ui)#
)+ ∂iG(x, U) = ∂iF (U)χ|U |≤R ui ,
for i = 1, ..., k. Let U = (IdLp − Tp,R)−1 (U) ∈ Lpk(M). We have
−∆2gui + divg
(Ai(∇ui)#
)+ ∂iG(x, U) = ∂iF (U)χ|U |≥Rui + ∂F (U)χ|U |≤R ui
and
−∆2g (ui − ui) + divg
(Ai(ui − ui)#
)+ ∂iG(x, U − U) = ∂iF (U)χ|U |≥Ru
i ,
for i = 1, . . . , k. Thence we have U − U = T2#,R(U) and
(Id
L2# − T2#,R
)(U) = U = (IdLp − Tp,R) (U) =
(Id
L2# − T2#,R
)(U) ,
since p ≥ 2# and U, U ∈ L2#
k (M).
As the operator(Id
L2# − T2#,R
)has inverse, we have U = U ∈ Lpk(M) for all p ≥ 2#.
From the regularity theory (Lp theory), we have U ∈ H4,pk (M) for all p ≥ 2#. From
Sobolev embedding Hr,pk (M) → C0,α
k (M) for all α ∈ (0, 1) such that α < r − np
we have
∇F (U) ∈ C0,αk (M) and for the regularity theory (Schauder theory), we have U ∈ C4
k(M).
62
2.3 Bubbles Decomposition
Consider Uα =(u1α, ..., u
kα
)solutions of the system (2.1). Note that (2.1) has a energy
functional given by:
J(U) =
∫M
((∆gU)2 + A
((∇U)#, (∇U)#
))dvg +
+
∫M
G(x, U) dvg −∫M
F (U) dvg . (2.28)
or,
J(U) =
∫M
UPg(U) dvg −∫M
F (U) dvg ,
where
Pg(U) = −∆2gU + divg
(A(∇U)#
)+
1
2∇UG(x, U) .
Remember in which follows the definition of Palais Smale sequence.
Definition 2. Let (Uα)α ∈ H2,2k (M). The sequence (Uα)α is a Palais-Smale (or PS)
sequence if:
• J(Uα) is limited;
• limα→∞ J′(Uα) = 0 em
(H2,2k (M)
)∗.
Let (xα)α∈N be a converging sequence of points in M and let (µα)α∈N ∈ R be such that
µα > 0 for all α and limα→+∞ µα = 0. Let δ ∈(
0, ig(M)
2
), where ig(M) is the injectivity
radius of (M, g). Assume that ηδ, xα ∈ C∞(M) defined by ηδ, xα = ηδ exp−1xα , at which we
consider expxα : B2δ(0) → B2δ(xα) a exponencial map in x defined in B2δ(0) (Euclidean
ball of Rn) and that ηδ ∈ C∞(Rn), ηδ ≡ 1 in Bδ(0), ηδ ≡ 0 in Rn \ B2δ(0). We define in
which follows a family of functions, called scalar bubbles or 1-bubbles:
Bα(x) = βnηδ, xα(x)
(µα
µ2α + dg(x, xα)2
)n−42
,
for all x ∈ M . In this case we say that thr points xα are the centers and the numbers µα
are the weights of (Bα)α. The constant βn is βn = (n(n− 4)(n2 − 4))n−48 .
A k−bubbles is a sequence dof maps Bj,α = (B1j,α, · · · , Bk
j,α) such that one of the
coordinates is a scalar bubbles and the others are null..
Note that we have:
Bα(x) = ηδ, xα(x)µ−n−4
2α u
(exp−1
xα (x)
µα
), (2.29)
63
for all x ∈ M e
u(x) = βn
(1
1 + |x|2
)n−42
, (2.30)
for all x ∈ Rn. Note that u ∈ D22(Rn) is a extremal for:
1
A0
= infu∈D2
2(Rn)\0
∫Rn
(∆ξu)2 dx(∫Rn|u|2# dx
) 2
2#
, (2.31)
where D22(Rn)is the completamento de C∞c (Rn) com a norma ‖u‖D2
2(Rn) = ‖∆ξu‖2 .
Note that the function u in (2.30) satisfies the following equation:
∆2ξ u = u2#−1 em Rn .
The extremal for the sharp Euclidean inequality, that is, functions in D22(Rn) that
achieve the infimum in (2.31), are in the following form:
uλ, µ, x0(x) = µ
(λ
λ2 + |x− x0|2
)n−42
for all x ∈ Rn ,
where µ 6= 0, λ > 0 and x0 ∈ Rn are arbitrary.
For clarity, we are considering u ≥ 0, where the bubbles decomposition with 1-bubbles
defined as in (2.29), where u ∈ D22(Rn) ∩ C∞(Rn) is a solution of ∆2
ξu = |u|2#−2u in Rn.
The lack of strong convergence of Palais-Smale sequences for J can be described by
the 1-bubbles. The following theorem shows how fundamental they are for description of
Palais-Smale sequences. A description of Palais-Smale sequences for critical functional is
done by Struwe (see [36]) which was provided Palais-Smale sequences for critical functional
associated with an elliptical operator of second order in a subset of Rn. The initial idea
for the scalar case came from studies of Hebey and Robert which is the extension of the
associated functional to the fourth order operator, in a Riemannian manifold.
Theorem 15. [Bubbles Decomposition] Let (M, g) be a compact Riemannian manifold
of dimension n ≥ 5, Fα, F : Rk → R positive functions C1, and 2#−homogeneous and
Gα, G : M × Rk → R are continuous functions 2-homogeneous on the second variable,
and Gα is C1 such that
Fα → F em C1loc(Rk)
Gα → G em C0loc(M × Rk) .
Let Uα ∈ H2,2k (M) be a weak solution of (2.1). If the sequence (Uα)α∈N is limited in
64
H2,2k (M), then exists U0 ∈ H2,2
k (M) the weak limit of Uα and, up to a subsequence, we
have:
Uα = U0 +l∑
j=1
Bj,α +Rα (2.32)
for all α > 0, where (Bj,α)α∈N , j = 1, ..., l, k-bubbles and (Rα)α∈N ⊂ H2,2k (M) is such
that
limα→∞
Rα = 0 em H2,2k (M)
Proof. Without loss of generality, we suppose that the limit U0 ∈ H2,2k (M) is trivial, that
is, U0 ≡ 0.
From limitation of Uα, up to a subsequence, exists U0 ∈ H2,2k (M) such that:
Uα U0 em H2,2k (M) .
Note that, from the convergence above, we have:∫M
∆gUα∆gΘ dvg +
∫M
A((∇Uα)#, (∇Θ)#) = o(1), ∀ Θ ∈ H2,2k (M) .
By the dominated convergence, we have:
k∑i=1
∫M
∂iFα(Uα)Θi dvg =k∑i=1
∫M
∂iF (U)Θi dvg + o(1) ,
and ∫M
Gα(x, Uα) dvg = o(1) . (2.33)
As U0 ≡ 0, we have uiα 0 in H2,2k (M).
Thus, by using the energy functional J defined in (2.28):
J(Uα) =
∫M
((∆gUα)2 + A
((∇Uα)#, (∇Uα)#
))dvg −
∫M
F (Uα) dvg + o(1)
= Lk(Uα) + o(1) ,
where
Lk(Uα) =
∫M
((∆gUα)2 + A
((∇Uα)#, (∇Uα)#
))dvg −
∫M
F (Uα) dvg .
On the other hand, we have:
65
DJ(Uα)Θ =
∫M
(∆gUα∆gΘ + A
((∇Uα)#, (∇Θ)#
))dvg
+k∑i=1
∫M
∂iGα(x, Uα)Θi dvg −k∑i=1
∫M
∂iFα(Uα)Θi dvg .
We also have that
DLk(Uα)Θ = DJ(Uα)Θ−k∑i=1
∫M
∂iGα(x, Uα)Θi dvg
or,
DLk(Uα)Θ =
∫M
(∆gUα∆gΘ) + A((∇Uα)#, (∇Θ)#
)dvg −
k∑i=1
∫M
∂iF (Uα)Θi dvg .
Thus, by using (2.33), we have:
DJ(Uα)Θ =
∫M
(∆gUα∆gΘ + A
((∇Uα)#, (∇Θ)#
))dvg −
−k∑i=1
∫M
∂iFα(Uα)Θi dvg + o(1)
= DLk(Uα)Θ + o(1) .
Therefore, if (Uα)α∈N is a Palais-Smale sequence for the functional J , then (Uα)α∈N is also
a Palais-Smale sequence for Lk. As
k∑i=1
∫M
∂iF (Uα)Θi dvg = o(1) ,
and, ∫M
|uiα|2#−2uiαΘi dvg = o(1) ,
we obtain that, if (Uα)α∈N is a Palais-Smale sequence for J , then that sequence is also
Palais-Smale for Lk. We have then that (uiα)α∈N also is a Palais-Smale sequence for Lik,
with i = 1, ..., k, where
66
Lik(uiα) =
∫M
((∆gu
iα
)2+ Ai
((∇uiα)#, (∇uiα)#
))dvg −
1
2#
∫M
|uiα|2#
dvg .
Then:
DLik(uiα)Θi =
∫M
(∆gu
iα∆gΘ
i + Ai((∇uiα)#, (∇Θi)#
))dvg −
∫M
|uiα|2#−2uiαΘi dvg .
So, for each i exists a ki and a scalar bubbles(Bij,α
)α, where j = 1, ..., ki, such that, up
to a subsequence
uiα = u10 +
ki∑j=1
Bij,α +Ri
α ,
and,
Liα(uiα) =
ki∑i=1
E(uiα) + o(1) ,
where uiα ∈ D22(Rn) is a non-trivial solution of
−∆2ξ u = |u|2#−2u em Rn ,(
Bij,α
)α
is a scalar bubbles and
E(uiα) =1
2
∫Rn
(∆gu
iα
)2dx− 1
2#
∫Rn|uiα|2
#
dx .
We finished putting l =∑k
i=1 ki, because, for each coordinate function of the sequence
(Uα)α ∈ H2,2k (M) we have a bubbles decomposition. Thus, if Uα = (u1
α, . . . , ukα), we have:
Uα =l∑
j=1
Bj,α +Rα
where Bj,α =(B1j,α, · · · , B
kij,α
)is a k-bubbles and Rα =
(R1α, · · · , Rk
α
)is such that
Rα → 0 em H2,2k (M)
2.4 Pointwise Estimates
The bubbles decomposition enables us to add properties to the sequences of limited solu-
tions in H2,2k (M). With this result, we add the pointwise estimates for (Uα)α.
67
Theorem 16 (Pointwise Estimates). Let (M, g), Gα, Fα be as in the bubbles decomposition
theorem (theorem 15). Let Uα be a limited sequence of solutions of
−∆2gU + divg
(A(∇U)#
)+∇UGα(x, U) = ∇Fα(U) em M , (2.34)
converging to 0 in H2,2k (M). Considering the bubbles decomposition in the theorem 15,
then exists, up to a subsequence, a constant C > 0, independent of α such that
(mini,j
dg(xij,α, x
))n−42
√√√√ k∑i=1
(uiα)2 ≤ C ,
for all α and for all x, where xij,α are the centers of the bubbles Bj,α. And in particular
the |Uα| are uniformly bounded in any compact subset of M \ xj,0lj=1 and uiα → 0 in
C0loc(M \ xj,0
lj=1) where xj,0 is the limit of xij,α.
Proof. We define
Φα(x) = dg(xij,α, x
)e Ψα(x) = Φα(x)
n−42
(k∑i=1
(uiα(x)
)2
) 12
,
where the xij,α are the centers of the bubbles Bj,α.
We will do the proof by contradiction. Initially, consider the sequence (yα)αormed by
the points of maximum of Ψα and such that:
Ψα(yα) = maxM
Ψα(x) e limα→+∞
Ψα(yα) = +∞ .
Up to a subsequence we can assume that:
|ui0α (yα)| ≥ |uiα(yα)| ,
for some i0 = 1, ..., k and for all i. Let
µα = |ui0α (yα)|−2
n−4 ,
then µα → 0 when α→ +∞. Then, by the definition of yα
limα→+∞
dg(xij,α, yα
)µα
= +∞ . (2.35)
In fact, to demonstrate (2.35),develop the expression of Ψα:
68
Ψα(yα) =
(k∑i=1
|uiα(yα)|2) 1
2
Φα (yα)n−42
≤(k|ui0α (yα)|2
) 12 dg
(xij,α, yα
)n−42
=√k|ui0α (yα)|dg
(xij,α, yα
)n−42
=√k
(dg(xij,α, yα
)µα
)n−42
.
As Ψα(yα)→∞ when α→ +∞, we obtain (2.35).
Let 0 < δ < ig(M), where ig is the radius of injectivity of (M, g). For i = 1, ..., k on
the Euclidean ball B0 (δµ−1α ) of center 0 and radius δµ−1
α , we define the function:
wiα(x) = µn−42
α uiα(expyα(µαx)
), (2.36)
where expyα is the exponential map in yα. Given R > 0 and x ∈ B0(R) Euclidean ball
centered on 0 and radius R. By (2.35) and (2.36) we have:
|wiα(x)| ≤ µn−42
α
(k∑j=1
|ujα(expyα(µαx)
)|2) 1
2
≤ µn−42
α
Ψα
(expyα(µαx)
)Φα
(expyα(µαx)
)n−42
.
Thus,
|wiα(x)| ≤ µn−42
α
Ψα
(expyα(µαx)
)Φα
(expyα(µαx)
)n−42
, (2.37)
for all i and all α big enough. For all i, j and x on the Euclidean ball B0(R) of center 0
and radius R > 0, we obtain the inequalities:
69
dg(xij,α, expyα(µαx)
)≥ dg
(xij,α, yα
)− dg
(yα, expyα(µαx)
)≥ dg
(xij,α, yα
)−Rµα
=
(dg(xij,α, yα
)Φα(yα)
− RµαΦα(yα)
)Φα(yα)
≥(
1− RµαΦα(yα)
)Φα(yα) .
From the above inequality, by (2.37) and the definitions of yα and Ψα, we obtain:
|wiα(x)| ≤ µn−42
α
Ψα
(expyα(µαx)
)Φα
(expyα(µαx)
)n−42
≤ µn−42
αΨα(yα)
Φα
(expyα(µαx)
)n−42
≤ µn−42
α k12 |ui0α (yα)| Φα(yα)
n−42
Φα
(expyα(µαx)
)n−42
≤ k12
(1− Rµα
Φα(yα)
)−n−42
.
Therefore,
|wiα(x)| ≤ k12
(1− Rµα
Φα(yα)
)−n−42
, (2.38)
for all x ∈ B0(R) and for any i = 1, . . . , k when α is big enough. In particular, from
(2.35) and (2.38), up to a subsequence, we obtain the uniforme limitation of wiα in any
compact subset of Rn for all i.
Let Wα =(w1α, ..., w
kα
). The Wα are solutions of
−∆2gα Wα + µ2
α divg
(A(∇Wα)#
)+
1
2µ4α∇gα,UG(x,Wα) =
1
2#∇gα F (Wα) , (2.39)
where
G(x, U) = G(expyα(µαx), U
)e gα =
(exp∗yα g
)(µαx) ,
gα is the pull-back de g. Let ξ be a Euclidean metric. For each compact set K ⊂ Rn, as
µα → 0, follows that gα → ξ and, C2(K) when α→ +∞.
70
Then, from the elliptic theory, follows by (2.38) that the wiα are uniformly bounded
in C2,θloc (Rn), 0 < θ < 1, for all i. In particular, up to a subsequence, we can assume that
wiα → wi in C2loc(Rn) when α → +∞. Follows that the wi are limited in Rn by (2.37)
and they are such that |wi0(0)| = 1 by construction. Furthermore we consider the wi
belonging to the D22(Rn) space and wi ∈ L2#(Rn). Let W = (w1, ..., wk) 6= 0. For all i
and R > 0, we have: ∫Byα (Rµα)
|uiα|2#
dvg =
∫B0(R)
|wiα|2#
dvgα .
Follows from dominated convergence that, for all R > 0:∫Byα (Rµα)
|uiα|2#
dvg =
∫Rn|wi|2# dx+ εR(α) , (2.40)
where εR(α) is such that:
limR→+∞
limα→+∞
εR(α) = 0 . (2.41)
By bubbles decomposition, we have
∫Byα (Rµα)
|uiα|2#
dvg =
∫Byα (Rµα)
∣∣∣∣∣ui0 +
mi∑i=1
Bij,α +Rα
∣∣∣∣∣2#
dvg
≤ 22(2#−1)
mi∑i=1
∫Byα (Rµα)
|Bij,α|2
#
dvg + o(1) .
Therefore,
∫Byα (Rµα)
|uiα|2#
dvg ≤ c
mi∑i=1
∫Byα(Rµα)
|Bij,α|2
#
dvg + o(1) , (2.42)
where o(1)→ 0 when α→ +∞ and c > 0 is independent of α and R.
Recall that the k-bubbles are vector functions where one of the coordinates functions
is a scalar bubbles and the others are null. We have, by (2.35) , that
limα→+∞
∫Byα(Rµα)
|Bij,α|2
#
dvg = 0 ,
for all R > 0 and i = 1, ..., k. From (2.40) and (2.42) we have∫Rn|wi|2# dx = εR(α) .
Using (2.41) and making α→ +∞ and R→ +∞, we have:
71
∫Rn|wi|2# dx = 0 ,
for all i = 1, . . . , k. This leads to a contradiction, because W 6= 0.
72
2.5 Concentracao L2
In what follows, we consider (Uα)α a sequence of solutions of
∆2gu
iα − divg(Ai(∇uiα)#) +
1
2∂iG(x, Uα) =
1
2#∂iF (Uα) . (2.43)
where G : M × Rk → R is a C1 positive function and 2-homogeneous and F : Rk → R is
a positive C1 function and 2#-homogeneous. For some of the results that follow, due to
technical details, we use particular cases of (2.43), we will comment later.
The points of blow up, or concentration of the sequence. (Uα)α has much of the
information of the sequence. This property is called L2 concentration.
Remark. We consider S the set of geometric blow-up points defined as
S =
lim
α→+∞xij,α; i = 1, . . . , l
. (2.44)
where l is as in the bubbles decomposition theorem.
Before discussing the L2 concentration, we prove some inequalities. Let A be a sum of
(2,0)-tensores. Consider also U = (u1, ..., uk) a k-map in H2,2k (M). We say that U satisfies
−∆2gui + divg(Ai(∇ui)#) + ∂iG(x, U) ≤ ∂iF (U) (2.45)
in the sense of distributions for all i = 1, ..., k and Φ = (ϕ1, ..., ϕk) in H2,2k (M)
∫M
(∆gui,∆gϕi) dvg +
∫M
Ai((∇ui)#, (∇ϕi)#
)dvg +
∫M
∂iG(x, U)ϕi ≤∫M
∂iF (U)ϕi dvg
where (∆gui,∆gϕi) is the punctual scalar product ∇ui and ∇ϕi.For the lemma below, we consider the coercivity of the operator ψ (see section 1.5).
Lemma 17. Consider Uα =(u1α, ..., u
kα
)solution of (2.43), and let G(x, U) =
∑ki,j=1Aij(x)ui(x)uj(x),
where (Aij(x)) is positive as bilinear form and symmetric and let ψ be coercive. Exists,
then, C > 0 such that, up to a subsequence,∫M
|Uα| dvg ≤ C
∫M
|Uα|2#−1 dvg ,
for all α, where |Uα| =∑k
i=1 |uiα| and |Uα|2#−1 =
∑ki=1 |uiα|2
#−1.
Proof. Let f iα = sign(uiα) be a function given by:
f iα = χuiα>0 − χuiα<0 , (2.46)
where χA is the characteristic function of A. Then,
f iαuiα = |uiα| ,
73
for all α and for all i. Note that we have |f iα| ≤ 1 for all α and for all i.
By coercivity (see section 1.5, p. 30), we have:
ψ(Uα) ≥ C‖Uα‖2H2,2k (M)
, (2.47)
where,
ψ(U) =
∫M
(∆gU)2 dvg +
∫M
A((∇U)#, (∇U)#
)dvg +
∫M
G(x, U) dvg . (2.48)
By (2.47), there exists a solution U ′α for the minimization problem consisting in finding
a minimum for ψ(U) under the constraint∫M
(fα, U) dvg = 1 where (fα, U) =∑k
i=1 fiαui
and U = (u1, ..., uk), that is, the ui are components of U and fα = (f 1α, . . . , f
kα). If λα
is the minimum of ψ(U) where U ∈ H2,2k (M) satisfies the constraint
∫M
(fα, U) dvg = 1,
follows by (2.47), that λα > 0. Let Uα = λ−1α U ′α. Then, Uα is solution of the system:
−∆2gu
iα + divg
(Ai(∇uiα)#
)+
k∑j=1
Aijujα = f iα , (2.49)
for all i and for all α, where the uiα are components of Uα and f iα are as in (2.46).
By multiplying (2.49) by uiα, integrating over M and summing on i, we obtain, together
with (2.47), the square of the norm H2,2k (M) of Uα is uniformly controlled by the L1 norm
of the |Uα|. That is, we have
‖Uα‖22 ≤ ‖Uα‖2
H2,2k (M)
≤ ‖Uα‖1
In particular, the uiα are uniformly bounded in L2. By the standard elliptic theory the uiαare in the Sobolev spaces Hq
2(M) for all q. Thence, the uiα are continuous.
Note that, if G(x, U) =∑k
i,j=1Aijui(x)uj(x), then ∂iG(x, U) =∑k
j=1Aijuj(x).
By the above discussion and by the elliptic theory, we have that exists a constant
C0 > 0 such that |uiα| ≤ C0 in M for all α and for all i. And, as F is 2#-homogeneous,
∂iF is 2# − 1 homogeneous. By (2.49) and (2.1):
74
k∑i=1
∫M
|uiα| dvg =k∑i=1
∫M
uiαfiα dvg
=k∑i=1
∫M
(−∆2
guiα + divg
(Ai(∇uiα)#
)+
k∑j=1
Aijujα(x)
)uiα dvg
=k∑i=1
∫M
(−∆2
guiα + divg
(Ai(∇uiα)#
)+
k∑j=1
Aijujα(x)
)uiα dvg
≤k∑i=1
∫M
∂iF (Uα)|uiα| dvg
≤ C0
k∑i=1
∫M
∂iF (Uα) dvg
≤ C
k∑i=1
∫M
|uiα|2#−1 dvg
for all α. Thus, ∫M
|Uα| dvg ≤ C
∫M
|Uα|2#−1 dvg ,
for all α, where C > 0 does not depend on α. This finalizes the proof.
Let us consider now Uα ∈ H2,2k (M) weak solution of (2.1). Suppose that the sequence
(Uα)α is limited in H2,2k (M).
Before we prove the L2 concentration, we need the following lemma. In dimensions
greater or equal to 9 this lemma is a consequence of Holder inequality and the fact that
‖Uα‖2 → 0. In dimension 8 we will use the bubbles decomposition of the theorem 15.
Lemma 18. Let n ≥ 8, then
∫M
|Uα|2#−1 = o(1)
(∫M
|Uα|2) 1
2
,
where o(1)→ 0 when α→∞.
In what follows, we consider Bδ as the union of the balls Bxi(δ), xi ∈ S, set defined
in (2.44), where i = 1, . . . , l and l is as in the bubbles decomposition theorem.
Proof. We divide the proof into two parts.. First we prove the result when n = 8. After,
we demonstrate the case n > 8.
Let n = 8 and consider:
75
∫M
|Uα|2#−1 dvg√∫
M
|Uα|2 dvg
=k∑i=1
∫M
|uiα|2#−1 dvg√∫
M
|Uα|2 dvg
,
≤k∑i=1
∫M
|uiα|2#−1 dvg√∫
M
|uiα|2 dvg
. (2.50)
Returning to the bubbles decomposition of the Uα in H2,2k (M), given by the theorem
15. Let xij,α and µij,α be the the centers and weights of the 1-bubbles (Bij,α)α involved in
the decomposition of each k-bubbles (Bα)α given by (2.32). Let R > 0 and l be as in the
bubbles decomposition theorem. We define Ωi,α(R) as the union (i = 1 to i = l) of the
geodesic balls centered in xij,α and radius Rµij,α:
Ωi,α(R) = ∪lj=1Bxij,α(Rµij,α)
We fix i = 1, ..., k. The 2# = 4 when n = 8, then we obtain, by Holder inequality:
∫M
(uiα)2#−1
dvg ≤∫
Ωi,α(R)
(uiα)2#−1
dvg +
√∫M\Ωi,α(R)
(uiα)2# dvg
√∫M
(uiα)2 dvg .
Thus,
∫M
(uiα)2#−1 dvg(∫M
(uiα)2 dvg
) 12
≤
∫Ωi,α(R)
(uiα)2#−1
dvg√∫M
(uiα)2 dvg
+
√∫M\Ωi,α(R)
(uiα)2# dvg , (2.51)
Let ϕ ∈ C∞0 (Rn), where C∞0 (Rn) is the set of smooth functions with compact support
in Rn. Let us consider ϕij,α a function defined by the equation:
ϕij,α(x) =(µij,α)−n−4
2 ϕ((µij,α)−1 expxij,α(x)) (2.52)
By the bubbles decomposition in H2,2k (M), we obtain by a direct calculation, for any
R > 0:
76
(i)
∫M\Ωij,α(R)
(Bij,α
)2#dvg = εR(α) (2.53)
(ii)
∫Ωij,α(R)
(Bij,α
)2#−1ϕij,α dvg =
∫B0(R)
(u)2#−1 ϕdx+ o(1) (2.54)
(iii)
∫Ωij,α(R)
(Bij,α
)2(ϕij,α)2#−2 dvg =
∫B0(R)
(u)2 ϕ2#−2 dvx + o(1) ,
where u is as in (2.30), Ωij,α(R) = Bxij,α
(Rµij,α), o(1)→ 0 when α→ +∞ and where
limR→+∞
limα→+∞
εR(α) = 0 , (2.55)
From (i), we obtain: ∫M\Ωi,α(R)
(uiα)2#
dvg = εR(α) (2.56)
In wath Ωi,α is as previously defined and εR(α) is such that (2.55) is valid. From now on,
let ϕ in (2.52) be such that ϕ = 1 in the B0(R). Thus,
∫Ωi,α(R)
(uiα)2#−1
dvg ≤l∑
j=1
(µij,α)n−4
2
∫Ωij,α(R)
(uiα)2#−1
ϕij,α dvg
By the bubbles decomposition and (ii):
∫Ωij,α(R)
(uiα)2#−1
ϕij,α dvg ≤ C
∫Ωij,α
(Bij,α
)2#−1ϕij,α dvg + o(1)
≤ C
∫B0(R)
u2#−1 dx+ o(1)
where o(1)→ 0 when α→ +∞ and C > 0 does not depend on α or R. In particular, we
have:
∫Ωi,α(R)
(uiα)2#−1
dvg ≤(C
∫B0(R)
u2#−1 dx+ o(1)
) l∑j=1
(µij,α)n−4
2 (2.57)
where o(1)→ 0 when α→ +∞. Independently we also have:
∫M
(uiα)2 dvg ≥∫
Ωij,α(R)
(uiα)2 dvg
≥(µij,α)n−4
∫Ωij,α(R)
(uiα)2(ϕij,α
)2#−2dvg
77
Here, 2# − 2 = 2, because n = 8. Together with the bubbles decomposition, we have
∫Ωij,α(R)
(uiα)2(ϕij,α
)2#−2dvg =
∫Ωij,α(R)
(l∑
m=1
Bim,α
)2 (ϕij,α
)2#−2dvg + o(1)
≥∫
Ωij,α(R)
(Bij,α
)2 (ϕij,α
)2#−2dvg ,
By (iii) we obtain that,∫Ωij,α(R)
(uiα)2(ϕij,α
)2#−2dvg ≥
∫B0(R)
u2 dx+ o(1) .
Thence, for any j, we have:∫M
(uiα)2 dvg ≥(µij,α)n−4
(∫B0(R)
u2 dx+ o(1)
).
And we can conclude that:∫M
(uiα)2dvg ≥
(maxj=1,...,l
µij,α
)n−4(∫B0(R)
u2 dx+ o(1)
), (2.58)
where o(1)→ 0 when α→ +∞. We denote:
R(α) =
∫M
|uiα|2#−1 dvg∫
M
|uiα| dvg.
Then, from (2.51), (2.50), (2.56), (2.57), (2.58), we obtain:
lim supα→+∞
R(α) ≤ εR + C
∫B0(R)
u2#−1 dx√∫B0(R)
u2 dx
. (2.59)
where εR → 0 when R→ +∞ and C > 0 does not depend on R. Note that we have:
limR→+∞
∫B0(R)
u2#−1 dx =
∫Rnu2#−1 dx < +∞ .
On the other hand, when n = 8,
limR→+∞
∫B0(R)
u2#−1 dx = +∞ .
By equation (2.59), we obtain then that R(α) → 0 when α → +∞. Thus we obtain the
result for n = 8.
78
For convenience, let Uα = ‖Uα‖−12#Uα, so that
∫M
|Uα|2#
dvg = 1.
Now we prove the result for n > 8. We will do the proof in two steps: n ≥ 12 and
9 ≤ n < 12.
Let n ≥ 12. Note that 2# − 1 = n+4n−4
= 1 + 8n−4
. Then 1 < 2# − 1 ≤ 2. Follows from
Holder inequality:
∫M
|Uα|2#−1 dvg ≤ C
(∫M
|Uα|2 dvg) 2#−1
2
, (2.60)
where C > 0 is independent of α. Thus,∫M
|Uα|2#−1 dvg(∫
M
|Uα|2 dvg) 1
2
≤ C
(∫M
|Uα|2 dvg) 2#−2
2
= o(1) . (2.61)
If 9 ≤ n < 12, then 2 < 2# − 1 < 2#. By the Holder inequality we obtain:
∫M
|Uα|2#−1 dvg ≤
(∫M
|Uα|2 dvg)n−4
8(∫
M
|Uα|2#
dvg
) 12−n8
(2.62)
≤(∫
M
|Uα|2 dvg)n−4
8
,
because ‖Uα‖2# = 1. We have then:∫M
|Uα|2#−1 dvg(∫
M
|Uα|2 dvg) 1
2
≤(∫
M
|Uα|2 dvg)n−8
8
= o(1) . (2.63)
As we have Uα → 0 in L2(M) when α→ +∞, by (2.61) and (2.63) follows the result,
that is, o(1)→ 0, when α→ +∞.
In the result which follows we consider a particular case of (2.43). We consider G :
M × Rk → R a positive C1 function 2-homogeneous, given by:
G(x, t) =k∑ij
Aijtitj
where (Aij) is symmetric and positive as bilinear form. We also consider that Ai = big,
where g is the metric and bi ∈ R (Ai is a (2,0)-tensor as in the section 1.2). Thus, in the
lemma which follows, we consider the following system:
79
−∆2gu
i + bi∆gui +
k∑j=1
Aijuj = ∂iF (U) (2.64)
for i = 1, . . . , k and F : Rk → R is a positive C1 function and 2#-homogeneous.
By the lemma which follows we got a useful estimate that we use for the L2 concen-
tration.
Lemma 19. Let Uα and U0 as in the bubbles decomposition (theorem 15). Assume that,
for each i = 1, . . . , k, Ai = big, where g is the metric and bi ∈ R. For all δ > 0 exists
C > 0 such that, up to a subsequence,
maxM\Bδ
|Uα| ≤ C
∫M
(1 + |Uα|2
#−2)|Uα| dvg ,
for all α, where Bδ = Bx0(δ) is the ball of center x0 and radius δ. |Uα| =∑k
i=1 |uiα| e
|Uα|2#−2 =
∑ki=1 |uiα|2
#−2 and x0 is the limit of the centers of the 1-bubbles of which the
k-bubbles are formed.
Proof. Let B = Bx(r) be such that Bx(2r) ⊂ M \ x0 and let (Uα)α, Uα = (u1α, . . . , u
kα),
solution of:
−∆2gu
iα + bi∆gu
iα +
k∑j=1
Aijuj = ∂iF (Uα) . (2.65)
for i = 1, . . . , k. Thus, by theorem 16 and (2.64), we obtain:
| −∆2gu
iα + bi∆gu
iα| ≤ C|Uα| .
We also obtain that, considering a ≤ b2i4
:
| −∆2gu
iα + bi∆gu
iα + auiα| ≤ C ′|Uα| , (2.66)
for all α and for all i and C does not depend on α or i. Let Uα be solution of:
−∆2gu
iα + bi∆gu
iα + auiα =
∣∣−∆2gu
iα + bi∆gu
iα + auiα
∣∣ , (2.67)
for all α and all i. As
−∆2g(u
iα ± uiα) + bi∆g(u
iα ± uiα) + a(uiα ± uiα) ≥ 0 ,
follows from maximum principle (see [30]) that uiα ≥ |uiα| on M and for all α and all i. In
particular, each uiα is not negative. Note that we have:
−∆2g|Uα|+ b0∆g|Uα|+ a|Uα| ≤ C|Uα| ,
80
on the ball B for all α and b0 = kmini bi. From above inequality, Uα = (u1α, . . . , u
kα) and
|Uα| =∑k
i=1 uiα. The constant C > 0 is independent of α and each uiα is not negative.
Follows from bubbles decomposition and (2.66), (2.67) that the (Uα)α are uniformly
bounded in L∞(B). We can apply the De Giorgi-Nash-Moser for the functions Uα. In
particular we have,
maxBx( r4)
|Uα| ≤ C
∫Bx( r2)
|Uα| dvg . (2.68)
where C > 0 does not depend on α. Since B is basically any ball in M \ x0, by (2.68)
we have:
maxM\Bδ
|Uα| ≤∫M
|Uα| dvg .
By (2.65), we have:
| −∆2gu
iα + bi∆gu
iα + auiα| ≤ C
(1 + |Uα|2
#−2)|Uα| . (2.69)
And by (2.66) and (2.67) we obtain that:∫M
|Uα| dvg ≤ C
∫M
|Uα| dvg .
Thus,
maxM\Bδ
|Uα| ≤ maxM\Bδ
|Uα| ≤∫M
|Uα| dvg ≤∫M
|Uα| dvg .
Therefore, we have from (2.69):
maxM\Bδ
|Uα| ≤ C
∫M
(1 + |Uα|2
#−2)|Uα| dvg .
Note that, from the above lemma, we obtain that:
maxM\Bδ
|Uα| ≤ C‖Uα‖2 .
Consider Uα = ‖Uα‖−12 Uα and B(δ) = x ∈ M ; dist(x, S) < δ. Thus we have
‖Uα‖2 = 1.
In the proof of L2 concentration which follows we will use the lemma (18). We assume
also Uα is solution of (2.64).
Theorem 20. Let Uα be solution of (2.43), where Aij(x) is positive as bilinear form and
symmetric. Then
Uα → 0 ,
81
in H2,2k (M \Bδ) for all δ > 0.
Proof. Initially show that ‖Uα‖L2k(M\Bδ) → 0.
Suppose (Uα) is solution of (2.43). By the lemma 17, we obtain:∫M
|Uα| dvg ≤ C
∫M
|Uα|2#−1 dvg . (2.70)
From the above inequality, together with the above lemmas, we have:
∫M\Bδ
(Uα)2 dvg ≤(maxM\Bδ |Uα|
) ∫M\Bδ
|Uα| dvg ≤
∫M\B δ
2
|Uα|2 dvg
12 ∫
M
|Uα|2#−1 dvg .
Thence, by the lemma 18, we have ‖Uα‖L2k(M\Bδ) = o(1).
From the proof of De Giorgi-Nash-Moser (see [31]), we have∫M\Bδ
|∇uiα|2 dvg ≤ C
∫M\B δ
2
(uiα)2 dvg ,
for some constant C independent of α, but it depends on δ. Thence∫M\Bδ
|∇Uα|2 dvg ≤ C
∫M\B δ
2
(Uα)2 dvg −→ 0 ,
by the previous step.
To prove the last part, let φ be a cut-off function such that 0 ≤ φ ≤ 1, φ = 0 on the
ball B δ2
and φ = 1 in M \Bδ. By multiplying (2.64) by φ2uiα and integrating over M , we
have:
∫M
∆guiα
(∆gφ
2uiα)dvg +
∫M
Ai(∇uiα,∇(φ2uiα)
)dvg +
1
2
∫M
∂iG(x, Uα)φ2uiα dvg =
=1
2#
∫M
∂iF (Uα)φ2uiα dvg , (2.71)
for i = 1, ..., k, or
∫M
∆gUα(∆gφ
2Uα)dvg +
∫M
A(∇gUα,∇g(φ
2Uα))dvg +
∫M
G(x, U)φ2 dvg =
=
∫M
F (x, Uα)φ2 dvg .
By a direct calculation, the first term can be written as:
82
∫M
∆gUα(∆gφ
2Uα)dvg =
∫M
(∆g(φUα))2 dvg +O
(‖Uα‖
H1,2k
(M\B δ
2
)),
where
‖Uα‖2H1,2k (Ω)
=
∫Ω
(|Uα|2 + |∇Uα|2
)dvg .
The other remaining terms in (2.71) can be estimated by O
(‖Uα‖2
H1,2k
(M\B δ
2
))
. Thus,
rewrite (2.71) as:
∫M
(∆g(φUα))2 dvg = O
(‖Uα‖2
H1,2k
(M\B δ
2
)),
and by the Bochner-Lichnerowicz-Weitzenbook formula
∫M
|∇2(φuiα)|2 dvg =
∫M
(∆g(φu
iα))2dvg −
∫M
Ricg(∇(φuiα),∇(φuiα)
)dvg
≤∫M
|∆g(φuiα)2|2 dvg + k
∫M
|∇(ϕuiα)|2 dvg
= O
(‖uiα‖
H1,2
(M\B δ
2
)).
Thus,
∫M\Bδ
|∇2Uα|2 dvg = O
(‖Uα‖
H1,2
(M\B δ
2
)).
Therefore,
∫M\Bδ
|∇2Uα|2 dvg ≤ C
∫M\B δ
2
(|Uα|2 + |∇Uα|2
)dvg
,
for each i = 1, . . . , k. Thus,
∫M\Bδ
|∇2Uα|2 dvg ≤ C
∫M\B δ
2
(|Uα|2 + |∇Uα|2
)dvg
,
and this last inequality converges to zero hanks to the previous step. This ends the proof
of the lemma.
We have in which follows a global estimate.
Lemma 21. Uα satisfies ‖Uα‖2 = o(1)‖∇Uα‖2, where o(1)→ 0 when α→∞.
83
Proof. By the Holder inequality we have:∫Bδ
|Uα|2 dvg ≤ vol (Bδ)2∗−22∗ ‖Uα‖2
2∗ ,
where 2∗ = 2nn−2
and V ol(Bδ) is the volume of Bδ. Now, by Sobolev embedding H1,2(M) →L2∗(M), we have
‖Uα‖22∗ ≤ A
(‖∇Uα‖2
2 + ‖Uα‖22
),
where A > 0 is independent of α. By separating the integral:∫M
|Uα|2 dvg =
∫Bδ
|Uα|2 dvg +
∫M\Bδ
|Uα|2 dvg ,
and by using as two above inequalities, we obtain:∫M
|Uα|2 dvg ≤ C1
∫M\Bδ
|Uα|2 dvg + C2V ol(Bδ)2∗−22∗
∫M
|∇Uα|2 dvg ,
for all δ > 0 small enough where C1 and C2 are positive constants independent of α and
δ. As ∫M\Bδ
|Uα|2 dvg −→ 0 ,
by the proof of the previous result, we have:
1 ≤ C2V ol(Bδ)2∗−22∗ lim inf
α→∞
((∫M
|Uα|2 dvg)−1 ∫
M
|∇Uα| dvg
),
for δ small enough, whence follows the result.
As an immediate consequence of the above two results (theorem 20 and lemma 21),
we have the following.
Lemma 22. Let Uα = ‖∇Uα‖−12 Uα, then
Uα −→ 0 em H2,2k (M \Bδ) ,
for all δ > 0.
Proof. Note that, by the theorem 20 we have:
‖Uα‖H2,2k (M\Bδ) = ‖Uα‖−2
2
(∫M\Bδ
((∆Uα)2 + |∇Uα|2 + |Uα|2
)dvg
)= o(1)
And by the lemma 21:
84
‖Uα‖2
‖∇Uα‖2
= o(1)
Therefore,
‖Uα‖2H2,2k (M\Bδ)
= ‖∇Uα‖−22
‖Uα‖−22
‖Uα‖−22
(∫M
((∆Uα)2 + |∇Uα|2 + |Uα|2
)dvg
)= o(1)
85
2.6 Compacidade
In the previous section we studied existence of solutions for the following system
∆2gui − divg(Ai(∇ui)#) + ∂iG(x, U) = ∂iF (U) , (2.72)
where i = 1, . . . , k and U = (u1, . . . , uk). We also studied the bubbles decomposition for
Uα solution of (2.72). F : Rk → R is a positive C1 function and 2#−homogeneous and
G : M × Rk → R is a positive C1 function 2-homogeneous on the second variable. Ai is
a smooth (2,0)-tensor symmetric.
Throughout this section we use the case where Gα(x, U) =∑k
ij=1Aαij(x)ui(x)uj(x)
and Aiα = big for each i = 1, . . . , k. (A(α))α, α ∈ N is a sequence of smooth maps,
A(α) : M → M sk(R), where A(α) = (Aαij). Remember that M s
k(R) is the vector space of
the real symmetric matrices of order k × k. Consider the following system
∆2gu
i − biα∆gui +
k∑j=1
Aαij(x)uj(x) = ∂iF (U) , (2.73)
for i = 1, . . . , k. In what follows we consider that Uα = (u1α, · · · , ukα) is solution of (2.73)
and (Uα)α is a limited sequence in H2,2k (M). We assume that A(α) satisfies that there is
a C1 map, A : M → M sk(R), A = (Aij) such that
Aαij 7→ Aij em C1(M) (2.74)
when α → +∞, for all i and j. We also consider that Aiα = biαg (in (2.72)) converges to
Ai = big. The limit system, by combining (2.73) and (2.74) is:
∆2gu
i − bi∆gui +
k∑j=1
Aij(x)uj(x) = ∂iF (U) ,
We will use in this section some of the results already been demonstrated before as
the bubbles decomposition and the L2 concentration of the previous section.
Let η be a cut-off function in Rn with η = 1 on the ball B0(δ) and η = 0 out of ball
B0(2δ), where B0(r) is the Euclidean ball with center 0 and radius r. We consider ηuiα as
a function defined in Rn and with support in B0(2δ). Then, in the below lemma, we will
use a Pohozaev type identity:∫Rnxk∂k(ηu
iα)∆2(ηuiα) dx+
n− 4
2
∫Rn
(∆(ηuiα))2
dx = 0 . (2.75)
where xk is the kth coordinate of x ∈ Rk. Note that, from the lemmas 21 and 22, for
j = 0, 1, 2, we have:∫B0(2δ)\B0(δ)
|∇jUα|2 dx = o(εα) =
∫B0(2δ)
|Uα|2 dx ,
86
where ε−1α o(εα)→ 0 when α→∞ and
εα =
∫M
|∇Uα|2 dvg .
Using these estimates above in (2.75), we obtain:
Lemma 23. Let Uα = (u1α, · · · , ukα) be a limited sequence in H2,2
k (M). Then we have the
following estimate:∫Rnη2(∆2uiα
)xk∂ku
iα dx+
n− 4
2
∫Rnη2uiα∆2uiα dx = o(εα) . (2.76)
where o(εα)εα→ 0 when α→ 0.
Proof. We start with the second term of (2.75). In order to simplify, uiα ≡ u. by expanding
the term (∆(ηu))2 easily we obtain that
∫Rn
(∆ (ηu))2 dx =
∫Rnη2 (∆u)2 dx+
∫Rnu2 (∆η)2 dx
− 4
∫Rn〈∇η,∇u〉 (∆η) u dx− 4
∫Rn〈∇η,∇u〉 (∆u) η dx
+ 2
∫Rnη (∆η)u (∆u) dx+ 4
∫Rn〈∇η,∇u〉2 dx .
where for the two functions ϕ and ψ, 〈∇ϕ,∇ψ〉 is the scalar product of ∇ϕ and ∇ψ.
Integrating by parts we have:
∫Rnη2 (∆u)2 dx =
∫Rnη2u∆2u dx−
∫Rn
(∆η2
)u (∆u) dx
+ 4
∫Rnη 〈∇η,∇u〉 (∆u) η dx .
By Holder inequality, for p = 1, 2, we have:
∫Rn
(∆ηp)2 u2 dx ≤(∫
Aδ
|∆ηp|n2 dx
) 4n(∫
Aδ
u2# dx
) (n−4)n
,
where Aδ = B0(2δ) \ B0(δ). Note that |∆pη| ≤ C where C > 0 and u ∈ L2#(Rn). Thus
we have, ∫Rn
(∆ηp)2 u2 dx = o(εα) .
Then we have:
87
∫Rn|∆η2|u|∆u| dx ≤
(∫Rn
(∆η2
)2u2 dx
) 12(∫
Rn(∆u)2 dx
) 12
∫Rnη|∆η|u|∆u| dx ≤
(∫Rn
(∆η)2 u2 dx
) 12(∫
Rn(∆u)2 dx
) 12
,
and as ∆u ∈ L2(Rn), we obtain then que:∫Rn
(∆η2)u(∆u) dx = o(εα) e
∫Rnη(∆η)u(∆u) dx = o(εα)
where o(εα) is as defined above. Due to Holder inequality, we write:
∫Rn〈∇η,∇u〉2 dx ≤
∫Rn|∇η|2|∇u|2 dx ≤
(∫Aδ
|∇η|n dx) 2
n(∫
Aδ
|∇u|2∗ dx) (n−2)
n
where 2∗ = 2nn−2
. Note that |∇η| ≤ C and |∇u| ∈ L2∗(Rn). Then we obtain:∫Rn〈∇η,∇u〉2 dx = o(εα) .
We can write that,
∣∣∣∣∫Rn〈∇η,∇u〉 u∆η dx
∣∣∣∣ ≤ ∫Rn| 〈∇η,∇u〉 |u|∆η| dx
≤
√∫Rn〈∇η,∇u〉2 dx
√∫Rn
(∆η)2 u2 dx ,
we obtain that ∫Rn〈∇η,∇u〉 u∆η dx = o(εα) .
Thus, form the above estimates, we obtain that:∫Rn
(∆ (ηu))2 dx =
∫Rnη2u∆2u dx+ o(εα) . (2.77)
Now we calculate the first term of (2.75). Note that we have:
∆2(ηu) = η∆2u+ ∆η∆u− 2 〈∇η,∇∆u〉+
+u∆2η + ∆u∆η − 2 〈∇u,∇∆η〉 − 2∆ 〈∇η,∇u〉 .
Thus,
88
∫Rn
∆2(ηu)xk∂k(ηu) dx
=
∫Rnη2(∆2u)xk∂ku dx+
∫Rnηu(∆2η)xk∂ku dx
+ 2
∫Rn
(∆η)(∆u)ηxk∂ku dx− 2
∫Rnη∆ 〈∇η,∇u〉xk∂ku dx
− 2
∫Rnη 〈∇u,∇∆η〉xk∂ku dx− 2
∫Rnη 〈∇η,∇∆u〉xk∂ku dx
+
∫Rnηu(∆2u)xk∂kη dx+
∫Rnu2∆2ηxk∂kη dx
+
∫Rnu∆η∆uxk∂kη dx− 2
∫Rnu(∆(〈∇η,∇u〉))xk∂kη dx
− 2
∫Rnu 〈∇η,∇∆u〉xk∂kη dx− 2
∫Rnu 〈∇u,∇∆η〉xk∂kη dx . (2.78)
Note that |∆2η| ≤ C and |x| ≤ 2δ in Aδ = B0(2δ) \ B0(δ). By Holder inequality we
obtain:
∫Aδ
u|∇u| dx ≤
√∫Aδ
|∇u|2 dx
√∫Aδ
u2 dx ,
and also
∫Aδ
|∇u|2 dx ≤ |Aδ|2n
(∫Aδ
|∇u|2∗ dx) 2
2∗
∫Aδ
u2 dx ≤ |Aδ|4n
(∫Aδ
|u|2# dx) 2
2#
.
Then from the inequality
∣∣∣∣∫Rnuη(∆2η)xk∂ku dx
∣∣∣∣ ≤ C
∫Aδ
u|∇u| dx ,
as |Aδ| ≤ C, u ∈ L2#(Rn) e |∇u| ∈ L2∗(Rn), we obtain that
∫Rnuη(∆2η)xk∂ku dx = o(εα) .
Similarly, we have
89
∣∣∣∣∫Rnη(∆u)(∆η)xk∂ku dx
∣∣∣∣ ≤ C
∫Aδ
|∇u||∆u| dx
≤ C
√∫Aδ
(∆u)2 dx
√∫Aδ
|∇u|2 dx
Thence ∫Rnη(∆u)(∆η)xk∂ku dx = o(εα) (2.79)
Note that ∣∣∣∣∫Rnη 〈∇u,∇∆η〉xk∂ku dx
∣∣∣∣ ≤ C
∫Aδ
|∇u|2 dx .
Thus we obtain that ∫Rnη 〈∇u,∇∆η〉xk∂ku dx = o(εα) .
Note that ∣∣∣∣∫Rnu2(∆2η)xk∂kη dx
∣∣∣∣ ≤ C
∫Aδ
u2 dx .
Thus we obtain ∫Rnu2(∆2η)xk∂kη dx = o(εα) .
Similarly, we write that
∣∣∣∣∫Rnu(∆η)(∆u)xk∂kη dx
∣∣∣∣ ≤ C
∫Aδ
u (∆u)
≤(∫
Aδ
(∆u)2, dx
) 12(∫
Aδ
u2 dx
) 12
.
As previously, we have ∫Rnu(∆η)(∆u)xk∂kη dx = o(εα) . (2.80)
Noting that ∣∣∣∣∫Rnu 〈∇u,∇∆η〉xk∂kη dx
∣∣∣∣ ≤ C
∫Aδ
u|∇u| dx ,
we also obtain that
90
∫Rnu 〈∇u,∇∆η〉xk∂kη dx = o(εα) .
Note that from the identity
∫M
〈∇u,∇v〉 dvg = −∫M
u∆v dvg ,
by doing u = u1u2, we obtain that
∫M
(u1 〈∇u2,∇v〉+ u2 〈∇u1,∇v〉 dvg) = −∫M
u1u2∆v dvg .
Independently, integrating by parts (by doing v = η, u1 =(xk∂ku
)η and u2 = ∆u in the
above equality and then we developed the term ∇(ηxk∂ku)), we have
∫Rnη 〈∇η,∇∆u〉 xk∂ku dx =
=
∫Rnη (∆η) (∆u) xk∂ku dx−
∫Rn
(∆u)⟨∇η,∇
(η xk∂ku
)⟩dx
=
∫Rnη (∆η) (∆u) xk∂ku dx−
∫Rn|∇η|2 (∆u) xk∂ku dx
−∫Rnη (∆u) 〈∇η,∇u〉 dx−
∫Rnη (∆u) 〈x,∇η〉∇2u dx .
But note that
∣∣∣∣∫Rn|∇η|2 (∆u) xk∂ku dx
∣∣∣∣ ≤ C
∫Aδ
|∇u||∆u| dx∣∣∣∣∫Rnη (∆u) 〈∇η,∇u〉 dx
∣∣∣∣ ≤ C
∫Aδ
|∇u||∆u| .
Then, by (2.79) we obtain that
∫Rnη 〈∇η,∇∆u〉 xk∂ku dx = o(εα)−
∫Rnη (∆u) 〈x,∇η〉∇2u dx .
Noting that |∆u| ≤√n|∇2u|, we have
∣∣∣∣∫Rnη (∆u) 〈x,∇η〉∇2u dx
∣∣∣∣ ≤ C
∫Aδ
|∇2u|2 dx .
Using the Bochner-Lichnerowicz-Weitzenbock formula, we have
91
∫Rnη|∇2u|2 dx =
∫Rnη|∆u|2 dx−
∫Rnη Ric 〈∇u,∇u〉 dx
≤∫Rnη|∆u|2 dx+ k
∫Rnη|∇u|2 dx .
Thence we obtain that |∇2u|2 ∈ L2(Rn) and∫Aδ
|∇2u|2 dx = o(εα) .
Thus we obtain that ∫Rnη 〈∇η,∇∆u〉 xk ∂ku dx = o(εα) . (2.81)
Similarly,
∫Rnη (∆ 〈∇η,∇u〉) xk ∂ku dx =
∫Rn〈∇∆η,∇u〉 η xk ∂ku dx
+
∫Rnη 〈∇η,∇∆u〉 xk ∂ku dx− 2
∫Rn
⟨∇2u,∇2η
⟩η xk ∂ku dx .
Noting that ∫Rn| 〈∇∆η,∇u〉 η xk ∂ku dx| ≤ C
∫Aδ
|∇u|2 dx ,
and that
∣∣∣∣∫Rn
⟨∇2u,∇2η
⟩η xk ∂ku dx
∣∣∣∣ ≤ C
∫Aδ
|∇u||∇2u| dx
≤ C
√∫Aδ
|∇2u|2 dx
√∫Aδ
|∇u|2 dx .
By using (2.81) we obtain that∫Rnη (∆ 〈∇η,∇u〉) xk ∂ku dx = o(εα) .
Doing similar calculations, we obtain that
∫Rn
(∆ 〈∇η,∇u〉)uxk∂kη dx =
∫Rn〈∇∆η,∇u〉uxk∂kη dx
+
∫Rn〈∇η,∇∆u〉uxk∂kη dx− 2
∫Rn
⟨∇2η,∇2u
⟩uxk∂kη dx .
92
But note that we have∣∣∣∣∫Rn〈∇∆η,∇u〉uxk∂kη dx
∣∣∣∣ ≤ C
∫Aδ
u|∇u| dx = o(εα) ,
and also
∣∣∣∣∫Rn
⟨∇2η,∇2u
⟩uxk∂kη dx
∣∣∣∣ ≤ C
∫Aδ
u|∇2u| dx
≤ C
√∫Aδ
|∇2u|2 dx
√∫Aδ
u2 dx = o(εα) .
Integrating by parts,
∫Rn〈∇η,∇∆u〉uxk∂kη dx
=
∫Rn
(∆η) (∆u)uxk∂kη dx−∫Rn
(∆u)⟨∇η,∇
(uxk∂kη
)⟩dx
=
∫Rn
(∆η) (∆u)uxk∂kη dx−∫Rn
(∆u) 〈∇η,∇u〉 xk∂kη dx
−∫Rnu (∆u) |∇η|2 dx−
∫Rnu (∆u) ∇2η 〈x,∇η〉 dx . (2.82)
Note that we have,∣∣∣∣∫Rn
(∆u) 〈∇η,∇u〉 xk∂kη dx∣∣∣∣ ≤ C
∫Aδ
|∇u||∆u| dx = o(εα) , (2.83)
and
∣∣∣∣∫Rnu (∆u) |∇η|2 dx
∣∣∣∣+
∣∣∣∣∫Rnu (∆u)∇2η 〈x,∇η〉 dx
∣∣∣∣ ≤ C
∫Aδ
u|∇u| dx . (2.84)
By using the above estimates (2.83) and (2.84) in the equality (2.82) and by using (2.80),
we obtain that ∫Rn
(∆ 〈∇η,∇u〉)uxk∂kη dx = o(εα) ,
and ∫Rn〈∇η,∇∆u〉uxk∂kη dx = o(εα) .
Finally, note that we have,
93
∫Rnη u(∆2u
)xk∂kη dx =
∫Rn
(∆u) ∆(uη xk∂kη
)dx
=
∫Rnη(xk∂kη
)(∆u)2 dx+
∫Rnu (∆u) ∆
(η xk∂kη
)dx
− 2
∫Rn
⟨∇(η xk∂kη
),∇u
⟩(∆u) dx .
Note that we have
|∆(ηxk∂kη
)| ≤ C e |∇
(ηxk∂kη
)| ≤ C .
Thence we have
∣∣∣∣∫Rnηu(∆2u
)xk∂kη dx
∣∣∣∣ ≤ C1
∫Aδ
(∆u)2 dx
+ C2
∫Aδ
u|∆u| dx+ C3
∫Aδ
|∇u||∆u| dx .
Thus we obtain that ∫Rnηu(∆2u
)xk∂kη dx = o(εα) .
Using these above estimates in (2.78), we obtain that∫Rn
∆2 (ηu)xk∂k (ηu) dx =
∫Rnη2(∆2u
)xk∂ku dx+ o(εα) . (2.85)
Therefore, by (2.77) and (2.85) the Pohozaeh identity (2.75) it becomes∫Rnη2(∆2u
)xk∂ku dx+
n− 4
2
∫Rnη2u∆2u dx = o(εα) . (2.86)
The above estimate (2.86) we use at the following theorem.
Theorem 24. Let (M, g) be a compact Riemannian manifold locally conformally flat of
dimension n ≥ 8. Assume that the tensor Aiα = biαg converges, for all i = 1, ..., k, to the
smooth symmetric tensor Ai = big.
Consider the system in (2.73) where (A(α))α is the sequence of smooth maps A(α) :
M → M sk(R), satisfying (2.74). Let Uα be solution of (2.73) that convereges weakly to U0
in H2,2k (M). Then U0 in non-trivial if big−Ag is positive or negativr definite, for some i.
Proof. We want to show, under the conditions of theorem, that the weak limit U0 is
non-trivial. Let us assume that U0 = 0 and this leads to a contradiction.
94
The Uα forms a Palais-Smale sequence for the following energy functional (see the
proof the bubbles decomposition theorem, 15):
J(U) =1
2
∫M
((∆gU)2 +G(x, U)
)dvg −
∫M
F (U) dvg .
that is defined in H2,2k (M). As already seen, up to a subsequence, the Uα have a bubbles
decomposition.
Let x ∈ S, where S is the set of the blow-up points. The (M, g) is locally conformally
flat, then we can choose δ > 0 such that g is conformal to the flat metric g = φ−4
n−4 g,
where φ is smooth and positive. Even more, we can choose δ small enough such that
S∩B(x, 4δ) = x. Note that in the Euclidean metric g = ξ, we have |∇u|2g = φ4
n−4 |∇u|2g.Let Uα = φUα, that is, uiα = φuiα, for i = 1, . . . , k. We use the conformal property of
the Paneitz-Branson operator PBg
PBg(ϕu) = ϕ2#−1PBg(u) ,
for any smooth function u and g = φ−4
n−4 g and n ≥ 5. The geometric Paneitz-Branson
operator PBg is defined by
PBgu = ∆2gu− divg(Ag du) +
(n− 4
2
)Qgu ,
where Ag is the following smooth symmetric (2,0)-tensor
Ag =(n− 2)2 + 4
2(n− 1)(n− 2)Rgg −
4
n− 2Ricg ,
Rg and Ricg denotes the scalar and Ricci curvatures respectively and Qg is the Q-curvature
Qg =1
2(n− 1)∆gRg +
n3 − 4n2 + 16n− 16
8(n− 1)2(n− 2)2R2g −
2
n− 2|Ricg |2 .
Then, by a direct calculation we have
∆2uiα − φ8
n−4 divg
((biαg − Ag)duiα
)+ 2φ
12−nn−4 (biαg − Ag)(∇uiα,∇φ) +
+ hiαuiα + φ
8n−4
k∑j=2
Aαij(x)uiα(x) = ∂iF (Uα) , (2.87)
where
hiα = −(n− 4
2
)φ
8n−4Qn
g − φn+4n−4 divg
((biαg − Ag
)dφ−1
).
Let η be a cut-off function in Rn with η = 1 on the ball B0(δ) and η = 0 out of the
ball B0(2δ), where B0(r) is the Euclidean ball with center 0 and radius r. We consider
95
ηuiα as a function defined in Rn and with support in B0(2δ). We will use the Pohozaev
type identity:
∫Rnx · ∇(ηuiα)∆2(ηuiα) dx+
n− 4
2
∫Rn
(∆(ηuiα)
)2dx = 0 . (2.88)
We prove the theorem by calculating the terms in the above identity. We calculate
the parcels in terms of εα, where:
εα =
∫M
|∇Uα|2 dvg . (2.89)
Note that, from lemmas 21 and 22, for j = 0, 1, 2, we have:
∫B0(2δ)\B0(δ)
|∇jUα|2 dx = o(εα) =
∫B0(2δ)
|Uα|2 dx , (2.90)
where ε−1α o(εα)→ 0 when α→∞. Using this estimate in (2.88) (as we did previously at
the beginning of section, in the lemma 23), we have:
∫Rnη2(x · ∇uiα
)uiα∆2uiα dx+
n− 4
2
∫Rnη2uiα∆2uiα dx = o(εα) . (2.91)
Now, we multiply the equation (2.87) by η2uiα, we integrate over Rn and sum over i =
1, . . . , k:
k∑i=1
∫Rnη2uiα∆2uiα dx−
k∑i=1
∫Rnφ
8n−4η2uiα divg
((Aiα − Ag
)duiα)dx
+ 2k∑i=1
∫Rnφ
12−nn−4 η2uiα
(Aiα − Ag
)(∇uiα,∇φ) dx+
k∑i=1
∫Rnhiα(ηuiα)2
dx+
+k∑
i,j=1
∫Rnη2φ
8n−4Aαij
(uiα)2
= 2#
∫Rnη2F (Uα) dx . (2.92)
Writing φ8
n−4η2 divg ((Aiα − Ag) duiα) as aij∂ijuα+ bk∂kuα, where aij and bk are smooth
functions with support in B0(2δ). Integrating by parts we have:
∫Rn
φ8
n−4 η2uiα divg
((biαg − Ag
)duiα)dx
= −∫Rnφ
8n−4η2
(biαg − Ag
) (∇uiα,∇uiα
)dx+ o(εα) . (2.93)
We also have, by Holder:
96
∣∣∣∣∫Rnhiα(ηuiα)2dx
∣∣∣∣ ≤ C
∫B0(2δ)
uiα|∇uiα| dx
≤ C
2
∫B0(2δ)
(|∇uiα|2 + uiα
)dx
Using (2.90):
∫Rnhiα(ηuiα)2dx = o(εα) =
∫Rnφ
12−nn−4 η2uiα
(biαg − Ag
) (∇uiα,∇φ
)dx
=
∫Rnφ
8n−4η2
k∑j=1
Aαij(x)(uiα)2dx . (2.94)
Replacing (2.93) and (2.94) in (2.92), we obtain:
k∑i=1
∫Rnη2uiα∆2uiα dx +
k∑i=1
∫Rnφ
8n−4η2
(biαg − Ag
) (∇uiα,∇uiα
)dx
= 2#
∫Rnη2F (Uα) dx+ o(εα) . (2.95)
In view of the first term in (2.91), we multiply (2.87) by η2 (∇uiα · x), we integrate in
Rn and we sum on i = 1, . . . , k:
k∑i=1
∫Rnη2(x · ∇uiα
)∆2uiα dx−
k∑i=1
∫Rnφ
8n−4η2
(x · ∇uiα
)divg
((biαg − Ag
)duiα)dx
+ 2k∑i=1
∫Rnφ
12−nn−4 η2
(∇uiα · x
) (Aiα − Ag
) (∇uiα,∇φ
)dx+
k∑i=1
∫Rnhiαη
2(x · ∇uiα
)uiα dx+
+k∑
i,j=1
∫Rnφ
8n−4η2
(x · ∇uiα
)Aαij(x)uiα dx =
k∑i=1
∫Rnη2(x · ∇uiα
)∂iF (Uα) dx . (2.96)
Note that, integrating by parts and by using (2.90), we have:∫Rnη2(x · ∇uiα
)(uiα)2#−1 dx = − n
2#
∫Rnη2(uiα)2# dx+ o(εα)
Thus, ∫Rnη2(x · ∇uiα
)∂iF (Uα) dx = −nk
2#
∫Rnη2(uiα)2#
dx+ o(εα) . (2.97)
Again by (2.90) we have:
97
∫Rnhiαη
2(x · ∇uiα
)uiαdx = o(εα) .
Also,
∫Rnφ
8n−4η2
(x · ∇uiα
) k∑j=1
Aαij(x)uiαdx = o(εα) .
And, ∫Rnφ
12−nn−4 η2
(x · ∇uiα
) (biαg − Ag
) (∇uiα,∇φ
)dx = δO(εα) ,
when |x| ≤ 2δ in the support of the integral and |ε−1α O(εα)| ≤ C, independent of α and
δ. Proceeding as in the case (2.93) and by using the fact that aij = aji:
∫Rn
φ8
n−4 η2((biαg − Ag
)duiα)dx
=n− 2
2
∫Rnφ
8n−4η2
(biαg − Ag
) (∇uiα,∇uiα
)dx+ δO(εα) . (2.98)
Replacing (2.97) and (2.98) in (2.96), we obtain:
k∑i=1
∫Rn
η2(x · ∇uiα
)∆2uiα dx−
n− 2
2
k∑i=1
∫Rnφ
8n−4 η2
(biαg − Ag
) (∇uiα,∇uiα
)dx
= −nk∫Rnη2F (Uα) dx+ o(εα) + δO(εα) . (2.99)
Replacing (2.99) and (2.95) in (2.91), we obtain:
k∑i=1
∫Rnφ
8n−4η2
(biαg − Ag
) (∇uiα,∇uiα
)dνg = o(εα) + δO(εα) . (2.100)
Returning to the manifold, we consider η defined in M . We have:∫M
φ8
n−4η2 (bαg − Ag) (∇Uα,∇Uα) dνg = o(εα) + δO(εα) .
As big−Ag has sign for some i, biαg−Ag has sign for α big enough. Thus there exists
t > 0 such that: ∫Bx(tx)
|∇Uα|2 dνg = o(εα) + δO(εα) , (2.101)
for δ > 0 small enough and for α big enough. Now we sum (2.101) over all x ∈ S and, by
using the lemma 22, we obtain:
98
εα = o(εα) + δO(εα) .
By dividing by εα and taking the limit when α → ∞, we obtain 1 ≤ Cδ, where C is
independent of δ. This is a contradiction when δ is small enough.
99
Chapter
3Sharp Sobolev Vetorial Inequality of
Second Order
3.1 Extremal
The second vector inequality L2− Riemannian Sobolev states that, for any U ∈ H2,2k (M):
(∫M
F (U)dvg
) 2
2#
≤ A0
∫M
(∆gU)2 dvg +
+ B0
∫M
((A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg . (3.1)
is sharp concerning the first and the second Sobolev best constant, in the sense where
neither can be reduced.
As the end of the previous chapter, we assume that in this chapterG : M ×Rk → R is
a function given by∑k
i,j=1Aijuiuj, where (Aij) is positive as linear symmetric form. We
also assume that Ai = big, bi ∈ R, for i = 1, . . . , k.
Let E(A,F,G, g) be the set of the extremal maps normalized by∫MF (U) dvg = 1
associated with (3.1).
We present the following results about compactness of extremal maps for a fixed
metric.
Theorem 25 (Existence and Compactness of Extremal Maps). Let (M, g) be a compact
Riemannian manifold and locally conformally flat. Suppose that n ≥ 8 and, for some
i = 1, . . . , k,
big − Ag > 0 ou big − Ag < 0 ,
where Ag = (n−2)2+42(n−1)(n−2)
Rg g − 4n−2
Ricg. Then, the inequality (3.1) has extremal. In addi-
tion, the set E(A,F,G, g) is compact in the C0 topology.
Proof. Remember that we are assuming that G(x, U) =∑k
i,j=1 Aijuiuj so we can use the
result of the compactness and the (2, 0)-tensor Ai is given by big, bi ∈ R. However, note
100
that the demonstration that follows is valid for A being a sum of (2, 0)-tensor Ai and
G : M × Rk → R a C1 positive function and 2-homogeneous. Consider U ∈ H2,2k (M),
then:
(∫M
F (U) dvg
) 2
2#
≤ Ao∫M
(∆gU)2 dvg + B0
∫M
(A((∇U)#, (∇U)#
)+G(x, U)
)dvg .
Let (α) be sequences such that 0 < α < B0 and α→ B0.
Now consider the following functional defined in Λ:
Jα(U) =
∫M
(∆gU)2 dvg + αA−10
∫M
(A((∇U)#, (∇U)#
)+G(x, U)
)dvg ,
where
Λ =
U ∈ H2,2
k (M) :
∫M
F (U) dvg = 1
.
Consider
λα = infU∈Λ
Jα(U) .
We have that λα <1A0
. Otherwise:
Jα(U) ≥ λα ≥1
A0
, (3.2)
where U ∈ H2,2k (M). If F (U) 6= 1, just normalize it, taking 1
(∫M F (U) dvg)
12#U . By (3.2) we
have:
A0
∫M
(∆gU)2 dvg + α
∫M
(A((∇U)#, (∇U)#
)+G(x, U)
)dvg ≥
(∫M
F (U) dvg
) 2
2#
,
what is one contradiction, because 0 < α < B0. Thus, we have
λα <1
A0
.
We define
Aα = αA−10 Aiα e Gα = αA−1
0 G .
Thus
Aα → BoA−10 Ai e Gα → BoA−1
0 G .
101
Note that, from hypotheses of the theorem, Aiα = biαg converges to Ai = big. The
condition λα < A−10 implies that exists a minimizer Uα ∈ Λα for λ. Moreover, Uα satisfies
the system
∆2uiα + αA−10 divg
(Aiα(∇uiα)#
)+
1
2αA−1
0 ∂iGα(x, Uα) =λα2#∂iF (Uα) .
Then we have that Uα U0 in H2,2k (M), where U0 = (u1
0, . . . , uk0). Therefore
∆2ui0 +1
2B0A−1
0 divg
(Ai(ui0)#
)+ B0A−1
0 ∂iGα(x, U0) =λ
2#∂iF (U0) . (3.3)
Note that λα → λ = A−10 . Using Uα ∈ Λα, we obtain
limα→+∞
∫M
F (Uα) dvg = 1 .
In the inequality
(∫M
F (Uα) dvg
) 2
2#
≤ Ao∫M
(∆gUα)2 dvg+B0
∫M
(A((∇Uα)#, (∇Uα)#
)+G(x, Uα)
)dvg .
we take the limit when α→ +∞. Thus
lim infα→+∞
∫M
(∆gUα)2 dvg ≥ A−10 .
then the affirmation follows noting
lim supα→+∞
∫M
(∆gUα)2 dvg ≤ lim supα→+∞
λα ≤ A−10 .
Now multiply the equation (3.3) above by ui0 and we sum from i = 1 to i = k and
then we integrate. By theorem 24 note that U0 is a nontrivial extremal and the equality
is valid in (3.1) for U0
(∫M
F (U0) dvg
) 2
2#
= A0
∫M
(∆gU0)2 dvg+B0
∫M
(A((∇U0)#, (∇U0)
)#+G(x, U0)
)dvg .
By the above, note that the set E(A,F,G, g) of the extremal with norm equal 1 is
compact.
As a direct consequence of the previous theorem, we have the following results.
Corollary 26. Let (M, g) be a compact Riemannian manifold and locally conformally
flat. Suppose that n ≥ 8 and is true one of the statements for some i = 1, . . . , k,
i. bi >(n−2)2+4
2(n−1)(n−2)Rg and Ricg > 0 in (2.73).
102
ii. bi <(n−2)2+4
2(n−1)(n−2)Rg and Ricg < 0 in (2.73).
then (3.1) has extremal.
Another consequence of the above theorem is the following corollary.
Corollary 27. Let (M, g) be a Einstein compact Riemannian manifold and locally con-
formally flat and, for some i, bi <(n−2)2+4
2(n−1)(n−2). Then (3.1) has extremal.
103
Chapter
4Final Considerations
We will estimate the second best constant from vector theory B0(A,F,G, g) depending
on the corresponding constant in the scalar theory B0(g). In what follows, the potential
functions F and G are taken only positive continuous and homogeneous.
In these final considerations we are assuming here that there exists a Riemannian
manifold (M, g) does not have extremal for the sharp Sobolev scalar inequality.
(∫M
|u|2# dvg) 2
2#
≤ A0(g)
∫M
(∆gu)2 dvg + B0(g)
∫M
(|∇u|2g + u2
)dvg (4.1)
where A0(g) is a constante A0 defined in (9) and B0 is defined in (11). We are assum-
ingthen that there exists a nonzero function u0 ∈ H2,2(M) such that
(∫M
|u0|2#
dvg
) 2
2#
= A0(g)
∫M
(∆gu0)2 dvg + B0(g)
∫M
(|∇u0|2g + u2
)dvg
We start this section with a proposition about the estimates for the second best con-
stant in the vector theory B0(A,F,G, g) in depending on the corresponding constant in
the scalar theory B0(g). Our examples and counterexamples will be motivated by these
estimates.
Proposition 28. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. For
each t0 ∈ Sk−12 such that F (t0) = MF , we have
M2
2#
F B0(g)
min maxx∈M G(x, t0), CA≤ B0(A,F,G, g) ≤ M
2
2#
F B0(g)
max cA,mG
where mG = minM×Sk−12
G and
cA|∇gu|2 ≤ Ai((∇gu)#, (∇gu)#
)≤ CA|∇gu|2 . (4.2)
In particular, if exists t0 ∈ Sk−12 such that F (t0) = MF e MG = maxx∈M G(x, t0) and
the condition below (4.4) is satisfied, then
104
B0(A,F,G, g) =M
2
2#
F B0(g)
mG
and, furthermore, if
(∫M
|u|2# dvg) 2
2#
≤ A0(f, h, g)
∫M
(∆gu)2 dvg +B0(f, h, g)
∫M
f(x)|∇gu|2 + h(x)|u|2 dvg(4.3)
has extremal, then (1.25) has extremal (with B0(f, h, g) = B0(1, 1, g) = B0(g)).
The condition (4.4) is the following:
MG = maxx∈M
G(x, t0) ≤ CA e cA ≤ mG (4.4)
Proof. Consider U ∈ H2,2k (M), we have
(∫M
F (U) dvg
) 2
2#
≤ M2
2#
F A0
∫M
(∆gU)2 dvg +
+ B0(A,F,G, g)
∫M
(A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg
thus, taking U = ut0, with u ∈ H2,2(M), we obtain:
(∫M
|u|2# dvg) 2
2#
≤ A0
∫M
(∆gu)2 dvg +M− 2
2#
F B0(A,F,G, g) maxx∈M
G(x, t0)
∫M
|u|2 dvg
+ B0(A,F,G, g)M− 2
2#
F CA
∫M
|∇gu|2 dvg
≤ A0
∫M
(∆gu)2 dvg +
+ B0(A,F,G, g)M− 2
2#
F min
maxx∈M
G(x, t0), CA
∫M
(|∇gu|2 + |u|2
)dvg
By the definition of B0(g), then we have:
B0(A,F,G, g) ≥ M2
2#
F B0(g)
min maxx∈M G(x, t0), CA
On the other hand, we have from the proof of the proposition 8:
105
(∫M
F (U) dvg
) 2
2#
≤ M2
2#
F A0
∫M
(∆gU)2 dvg +B0(g)M
2
2#
F
mG
∫M
G(x, U) dvg
+B0(g)M
2
2#
F
CA
∫M
A((∇gU)#, (∇gU)#
)dvg
≤ M2
2#
F A0
∫M
(∆gU)2 dvg +
+B0(g)M
2
2#
F
max mG, CA
∫M
(A((∇gU)#, (∇gU)#
)+G(x, U)
)dvg
for all U ∈ H2,2k (M). Then, by the definition of B0(A,F,G, g), we have:
B0 ≤M
2
2#
F B0(g)
max cA,mG
Example 1. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 8
such that
B0 =n2 − 2n− 4
2n(n− 1)A0 Rg(x)
and (4.3) has extremal.
Let G : M × Rk → R, G(x, t) =∑k
i,j=1 Aij(x)|ti||tj| where Aij are nonnegative
continuous functions such that Ai0i0 > 0 does not depend on x and Aii ≥ Ai0i0 for some
i0. We have:
Ai0i0|t|2 ≤k∑i=1
Aii(x)|ti|2 ≤∑i,j
Aij(x)|ti||tj|
thus, mG = Ai0i0 .
Let F : Rk → R be a positive continuous function and 2#-homogeneous such that
F (ei0) = MF where ei0 is the i0-nth element of canonical basis de Rk. Then, by the
proposition 28:
B0(A,F,G, g) =M
2
2#
F B0(g)
Ai0i0
Let u0 ∈ H2,2(M) be a extremal function of (4.3). Then U = u0ei0 is a extremal map
of (1.25). Note that the regularity of F was not required.
Example 2. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 8
such that
106
B0(g) =n2 − 2n− 4
2n(n− 1)A0 Rg(x)
and (4.3) has no extremal.
Consider G : M × Rk → R, G(x, t) =∑k
i,j=1 Aij(x)|ti||tj| where Aij are nonnegative
continuous functions such that Ai0i0 > 0 does not depend on x and Aii ≥ Ai0i0 for some
i0. We have:
Ai0i0 |t|2 ≤k∑i=1
Aii(x)|ti|2 ≤∑i,j
Aij(x)|ti||tj|
thus, mG = Ai0i0 .
Let F : Rk → R be a positive continuous function and 2#-homogeneous such that
F (ei0) = MF where ei0 is the i0-nth element da canonical basis of Rk. Then, by the
proposition 28:
B0(A,F,G, g) =M
2
2#
F B0(g)
Ai0i0
Assume, by contradiction, there exists a extremal map of U0 in (1.25). The (2, 0)-
tensor Ai satisfies eqrefcCA. Then
B0(A,F,G, g)
∫M
(A((∇gU0)#, (∇gU0)#
)+
k∑i,j=1
Ai,j|uio||uj0|
)dvg =
=
(∫M
F (U0) dvg
) 2
2#
−M2
2#
F A0
∫M
(∆gU0)2 dvg
≤ M2
2#
F
k∑i=1
(∫M
|uio|2 dvg) 2
2#
−M2
2#
F A0
∫M
(∆gU0)2 dvg
≤ M2
2#
F A0(g)k∑i=1
∫M
(∆gu
io
)dvg +M
2
2#
F B0(g)k∑i=1
∫M
(|∇gu
i0|2 + |ui0|2
)2dvg
− M2
2#
F A0
∫M
(∆gU0) dvg
≤ M2
2#
F B0(g)k∑i=1
∫M
(|∇gu
i0|2 + |ui0|2
)2dvg
≤ M2
2#
F B0(g)
Ai0i0
∫M
k∑i,j=1
Ai,j|ui0||uj0| dvg +
M2
2#
F B0(g)
cA
k∑i=1
∫M
k∑i=1
Ai((∇gu
i0)#, (∇gu
i0)#)dvg
≤ M2
2#
F B0(g)
max Ai0i0 , cA
∫M
(k∑
i,j=1
Aij|ui0||uj0|+
k∑i=1
Ai((∇gu
i0)#, (∇gu
i0)#))
dvg
This implies
107
k∑i=1
(∫M
|ui0|2#
dvg
) 2
2#
= A0(g)k∑i=1
∫M
(∆gu
i0
)2dvg +
+ B0(g)
∫M
(k∑i=1
|∇guio|2 +
k∑i=1
|uio|2)dvg (4.5)
Independently, follows from the scalar sharp Sobolev inequality (4.1), that
(∫M
|ui0|2 dvg) 2
2#
≤ A0(g)
∫M
(∆gu
i0
)2dvg +B0(g)
∫M
(|∇gu
i0|2 + |ui0|2
)dvg (4.6)
for each i = 1, ..., k. then, by (4.6) and (4.5), there exists j ∈ 1, ..., k such that uj0 6= 0
and,
(∫M
|uj0|2 dvg) 2
2#
= A0(g)
∫M
(∆gu
j0
)2dvg +B0(g)
∫M
(|∇gu
j0|2 + |uj0|2
)dvg
this contradicts the initial hypothesis that (4.4) has no extremal.
4.1 Final Comments
Several studies have been devoted to the study of sharp Sobolev inequalities over the
last 30 years. Ithis is mainly due to its connection with some geometric and analytic
problems. Actually, these inequalities are related, for example, with the Yamabe problem,
isoperimetric inequalities and some properties of the Ricci flow. A scalar theory of best
constants associated to the classical Sobolev inequality was then developed in parallel
to the study of some geometric problems. Important results on the validity of sharp
Sobolev inequalities, existence or non-existence of extremal maps, characterization and
compactness of extremal maps were obtained in the last decades. Initially, the focus was
to equations of order 2 mostly because of the Yamabe problem. But in recent years several
authors have studied fourth order equations and Paneitz Branson type operators.
Every scalar theory of best constants, including our results of compactness of extremal
maps, naturally arises in the vector context.Part of this thesis was then dedicated to the
best constants vector theory. We observed that some known facts of the scalar theoryare
easily extended to the vector case. On the other hand, other facts were more complex and
required the development of some ideas that are available only in scalar context. We also
provide sufficient conditions for the existence of extremal maps. A result of compactness
of extremal maps were also established.
Although the best constants vector theory developed here, now extends much of what’s
108
in the scalar theory, some questions arise in this new context.. For example, we provide a
sufficient condition for the existence of extremal maps, with F and G of C1 class. From
the examples and counter-examples presented in the previous section, we expect some
results are valid for F and G only continuous. We also believe in more general results of
non-existence of extremal aplications that depend only on geometry and not the functions
F and G, as shown in Example 2 in the previous section. For this, the introduction of
the notion of critical maps provide a path in this direction (similarly to the scalar case of
second order). Another interesting question is the following:
Given functions F and G, we can guarantee that (3.1) has extremal map if and only
if, exists t0 ∈ Sk−12 , where F (t0) = MF , such that exists extremal?
The example 2 of the previous section shows that in fact it occurs in a particular case.
However, it is not clear that this occurs in general.
Finally, we list other issues that arise from the results of this thesis.
(a) Exists a Riemannian manifold such that does not exists extremal map for (1.17)?
(b) For any functions F and G, there exists a metric h conforme the given metric g such
that (3.1) has extremal map?
(c) How to guarantee the existence of extremal for any Riemannian manifold?
109
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