weak solutions for some quasilinear elliptic equations by the sub-supersolution method

Nonlinear Analysis 42 (2000) 995–1002www.elsevier.nl/locate/na

Weak solutions for some quasilinear ellipticequations by the sub-supersolution method

Manuel Delgado ∗, Antonio Su�arezDepartamento de Ecuaciones Diferenciales y An�alisis Num�erico, C/Tar�a s/n, Universidad de Sevilla,

41012-Sevilla, Spain

Received 6 March 1998; accepted 11 September 1998

Keywords: Quasilinear elliptic equation; Sub-supersolutions; Leray–Schauder theorem; Gagliardo–Nirenberg inequalities

1. Introduction

Let be a bounded and regular domain in RN ; N ≥ 2, with a smooth boundary@. In this paper we study the solvability of the quasi-linear elliptic boundary valueproblem

Lu= f(x; u; Du) in ;

u= 0 on @(1)

by the sub-supersolution method, where Du is the gradient of u and f : ×R×RN 7→ Ris a Carath�eodory function satisfying

|f(x; s; �)| ≤ g(x; s) + k|�|�with k ∈R+0 ; G(x) := sup|s|≤rg(x; s)∈Lp() for all r ¿ 0 and � a positive constantwith some restrictions that will be detailed below. In (1), L is an elliptic second-orderoperator of the form

Lu := −N∑

i; j=1

@@xi

(aij(x)

@u@xj

)+

N∑i=1

bi(x)@u@xi

and the precise assumptions on aij and bi will also be explained below.

∗ Corresponding author.E-mail addresses: [email protected] (M. Delgado), [email protected] (A. Su�arez)

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(99)00151 -0

996 M. Delgado, A. Su�arez / Nonlinear Analysis 42 (2000) 995–1002

The existence of a solution to a problem of this kind has been considered by sev-eral authors. Amann [2] proved the existence of a minimal and a maximal classicalsolution (notions which appear already in papers of Sato [13], Mlak [12] and Ako[1]) for (1), assuming that a classical sub-supersolution couple exists. When p¿N ,Amann and Crandall [3] proved a similar result in W 2;p(), again with the help ofa sub-supersolution couple for (1), with a more direct proof. Almost simultaneously,using the Leray–Schauder �xed point theorem, Kazdan and Kramer demonstrated in[9] that a classical solution exists. Here, the proof is considerably easier than in pre-vious papers but the important information concerning the existence of minimal andmaximal solutions is lost (see also [11], where the results are generalized including thecase of some nonlinear boundary conditions). Several years later, Dancer and Sweers[5] remarked that, under the assumption of existence of a sub-supersolution couplein W 1;∞(), there exists a minimal and a maximal solution that belong to W 1;∞().To this end, a nonconstructive argument which uses Zorn’s lemma was employed bythe authors. Moreover, Hess [8] proved the existence of a solution u∈W 1;2() for(1), and so by a bootstrap argument u ∈ W 2; m() where m=min{2=�; p}. In all thesepapers, the growth of the gradient in the nonlinear term is at most quadratic, i.e. � ≤ 2.This is a reasonable assumption, in accordance with a well known result of Serrin [14]which states, roughly speaking, that if f grows faster than quadratically in Du and is smooth, there are smooth data for which (1) possesses no solution.More recently, several authors have used the Leray–Schauder’s theorem without

any assumption concerning the existence of a sub-supersolution couple. Thus, Xavier[15] has deduced the existence of a solution in W 2;p() in the case in which L isLaplace’s operator and p¿N . It has been shown in [16] that the condition p¿Ncan be relaxed. In both papers, some conditions on the growth of f in terms of the�rst eigenvalue �1 of Laplace’s operator are imposed. In this paper, we will prove that,when p¡N , there exists at least one weak solution to (1) in W 2;p(). We will usethe sub-supersolution method for a general elliptic operator L.

2. The main result

We consider problem (1), where

Lu := −N∑

i; j=1

@@xi

(aij(x)

@u@xj

)+

N∑i=1

bi(x)@u@xi:

We assume aij ∈W 1;∞(); aij = aji; bi ∈W 1;∞0 () and div(b)≤ 0, with b = (b1; : : : ;

bN ); we also assume there exists a positive constant � such thatN∑

i; j=1

aij(x)�i�j ≥ �|�|2; ∀� ∈ RN ;∀x ∈ :

In (1), f : ×R×RN → R is a Carath�eodory function, i.e. measurable in x ∈ andcontinuous in (s; �) ∈ R× RN . In addition, we suppose that f satis�es

|f(x; s; �)| ≤ g(x; s) + k|�|�; (2)

M. Delgado, A. Su�arez / Nonlinear Analysis 42 (2000) 995–1002 997

where k; � ∈ R+0 ; g is also a Carath�eodory function and satisfy:(H1) For all r ¿ 0,

sup|s|≤r

|g(·; s)| ∈ Lp(); (3)

(H2) (a) If N = 2, then

1¡p¡ 2 and2

p+ 1≤ �¡ 2:

(b) If N ≥ 3, then2NN + 2

≤ p¡ N2

and2

p+ 1≤ �¡ N

N − por

N2

≤ p¡N and2

p+ 1≤ �¡ 2:

De�nition 1. A couple {u0; u0}, where the functions u0 and u0 belong to W 2;p() ∩L∞(), is a sub-supersolution couple for (1) (in W 2;p()) if:(1) u0 ≤ 0 ≤ u0 on @,(2) u0 ≤ u0 in ,(3) Lu0 − f(x; u0; Du0) ≤ 0 ≤ Lu0 − f(x; u0; Du0).

Remark 1. Notice that the last inequalities have a meaning a.e. in . Indeed; wheneverv ∈ W 2;p() ∩ L∞(); we have f(·; v; Dv) ∈ Lq() for some q¿ 1; since

|f(x; v; Dv)| ≤ |g(x; v)|+ k|Dv|� a:e: in :

In this paper, the main result is the following:

Theorem 1. Assume 1¡p¡ 2 if N = 2 and 2N=(N + 2) ≤ p¡N if N ≥ 3. Also;assume (H1) and (H2). If there exists a sub-supersolution couple for (1) in W 2;p();then (1) possesses at least one solution in W 2;p().

Proof From (H2), we know that p∗ = Np=(N − p) ≥ 2. We also know that theinterval [�p; p∗) is nonempty. For any q ∈ [�p; p∗), one has W 2;p() ,→ W 1;q()with a compact imbedding.Fixed q0 ∈ [�p; p∗), let T : W 1;q0 ()→ W 1;q0 ()∩L∞() be the truncation operator

associated to the sub-supersolution couple {u0; u0}

Tu(x) =

u0(x) if u0(x) ≤ u(x);u(x) if u0(x) ≤ u(x) ≤ u0(x);u0(x) if u(x) ≤ u0(x):

Observe that

|Tu(x)| ≤ max{|u0(x)|; |u0(x)|}= m a:e: in ;

whence Tu ∈ L∞() for any u ∈ W 1;q0 ().


On the other hand, if u ∈ W 1;q0 () then Tu ∈ W 1;q0 (). Indeed,

Tu(x) =u(x) + u0(x) + 2u0(x)− |u(x)− u0(x)|

4

+|u(x) + u0(x)− 2u0(x)− |u(x)− u0(x)||

4and, by Corollary A:6. in [10] for example, it follows that Tu ∈ W 1;q0 ().Let us introduce U 0=u0+K and U0=u0−K , where K is a positive constant chosen

in order to have

U 0 ≥ 1 and U0 ≤ −1: (4)

Observe that TU 0 = u0 and TU0 = u0. De�ne now

a(x) = max{−LU 0(x);LU0(x); 1}:Then a ∈ Lp() and a ≥ 1. For each 0 ≤ t ≤ 1, let us consider the boundary valueproblem

Lu+ (1− t)a(x)u= tf(x; Tu; D(Tu)) in ;

u= 0 on @: (5)

Then, {U 0; U0} is a sub-supersolution couple for (5). Indeed,LU 0 + (1− t)a(x)U 0 ≥ LU 0 + (1− t)a(x) ≥ LU 0 − (1− t)LU 0

= tLu0 ≥ tf(x; u0; Du0) = tf(x; TU 0; D(TU 0)):

Similarly, we can prove that U0 is a sub-solution of (5).Furthermore, if u ∈ W 2;p() is a solution of (5) for some t ∈ [0; 1], then

U0 ≤ u ≤ U 0: (6)

Indeed, let us just check that u ≤ U 0. Let us set w = u− U 0. Then w ∈ W 2;p() andwe want to prove that w+ = 0, where w+(x) = max{0; w(x)}. We know that

Lu+ (1− t)a(x)u= tf(x; Tu; D(Tu))

LU 0 + (1− t)a(x)U 0 ≥ tf(x; TU 0; D(TU 0)) = tf(x; u0; Du0):

Subtracting, we �nd

Lw + (1− t)a(x)w ≤ t[f(x; Tu; D(Tu))− f(x; u0; Du0)]: (7)

Multiplying this inequality by w+ and integrating over , we obtain:∫(Lw)w+ + (1− t)

∫a(x)(w+)2

≤ t∫[f(x; Tu; D(Tu))− f(x; u0; Du0)]w+

≤ t∫(g(x; Tu) + g(x; u0))w+ + k

∫(|D(Tu)|� + |Du0|�)w+: (8)


Since w+ = 0 on @, one has∫(Lw)w+ ≥ �

∫|Dw+|2 − 1

2

∫div(b)(w+)2: (9)

Here, we have used that p∗ ≥ 2, i.e. that p ≥ 2N=(N +2). Observe that every integralin (8) is well posed. Certainly,

g(·; Tu) + g(·; u0) ∈ Lp()by (H1), while w+ ∈ Lp′

() with p′ = p=(p− 1) (this follows in a standard mannerfrom Sobolev’s imbedding and (H2)). On the other hand, from the fact that |D(Tu)| ∈Lp

∗(), we have

|D(Tu)|� + |Du0|� ∈ Lp∗=�():

But denoting by s the conjugate of p∗=�, we also have w+ ∈ Ls() (here, we use that� ≤ (Np− N + 2p)=(N − p), which is also implied by (H2)). Let us introduce

A= {x ∈ : u(x) ≤ U 0(x)};B= {x ∈ : u(x)¿U 0(x)}:

We can write the last two integrals in (8) as the sum of integrals over A and B. Sincew+ =0 in A and Tu=Tu0 = u0 in B, we �nd from (8) and (9) that w+ =0. The proofthat U0 ≤ u is similar.Let us introduce the mapping S : [0; 1]×W 1;q0 () 7→ W 1;q0 (), where

S(t; u) = v

and v is the solution ofLv+ (1− t)a(x)v= tf(x; Tu; D(Tu)) in ;

v= 0 on @:

From the assumptions (H1) and (H2), it follows that f(·; Tu; D(Tu)) ∈ Lp() for anyu ∈ W 1;q0 (). According to the Lp theory of elliptic equations (see [7,9]),S(t; u)∈W 2;p() for all t ∈ [0; 1] and for all u ∈ W 1;q0 (). Furthermore, from theSobolev’s imbedding theorem, it is not di�cult to prove that S is continuous andcompact. Let us show that

‖u‖1; q0 ≤ C; (10)

whenever u∈W 1;q0 () and satis�es S(t; u) = u for some t ∈ [0; 1]. Then, from theLeray–Schauder’s �xed point theorem (see for example [7], Theorem 11:6) we will beable to a�rm there exists u ∈ W 1;q0 () such that S(1; u) = u, i.e. satisfying

Lu= f(x; Tu; D(Tu)) in ;

u= 0 on @:(11)

Since this u belongs to W 1;q0 (), we will have f(·; Tu; D(Tu))∈Lp() and u∈W 2;p().Arguing as we did in the proof of (6), we will also have

u0 ≤ u ≤ u0;whence Tu= u, that is to say u solves (1).


Hence, the proof of Theorem 1 will be achieved if we are able to establish (10).First, we observe that

‖u‖1; q0 ≤ C‖u‖2;p ≤ C(‖a‖p‖u‖∞ + ‖g(·; Tu)‖p + ‖|D(Tu)|�‖p):Thus, one also has

‖u‖2;p ≤ C(‖a‖p‖u‖∞ + ‖g(·; Tu)‖p + ‖|D(Tu)|�‖p): (12)

We know that u ∈ W 2;p()∩Lr(); ∀r ≥ 1. From the Gagliardo–Nirenberg inequalities(see [6], Theorem 10:1), we have

‖Du‖�p ≤ C‖u‖�2;p‖u‖1−�r (13)

with

�=Nr − �pr − �pN�(Nr − 2pr − pN ) (� ∈ [1=2; 1]): (14)

This gives

‖u‖2;p ≤ C(1 + ‖g(·; Tu)‖p + ‖u‖��2;p‖u‖(1−�)�r ): (15)

Notice, from (6) that ‖u‖r ≤ C where C is independent of t and r ≥ 1. From (H2),we see that r can be chosen such that r(N − 2p)¡pN and r ≥ 1; thus

� ∈ [1=2; 1]⇔ 2rp+ r

≤ � ≤ NN − p: (16)

On the other hand, we have

��¡ 1⇔ �¡N + 2rN + r

: (17)

Thus, for any � and r satisfying

1 ≤ r; r(N − 2p)¡pN and2rp+ r

≤ �¡ N + 2rN + r

; (18)

we have ��¡ 1 and, consequently, (10).We can consider the functions h1 and h2, with

h1(r) ≡ 2rp+ r

; h2(r) ≡ N + 2rN + r

:

They both are increasing and satisfy

limr→+∞ hi(r) = 2:

Moreover, h1(1)¡h2(1) and their graphs only intersect when r(N − 2p) = pN and,at this value, hi(r) = N=(N − p).Thus, if 2p¡N and for any � with 2=(p+ 1) ≤ �¡N=(N − p), we can �nd one

value of r which satis�es (18).If 2p¿N , the functions can only intersect at a nonpositive value, so that for any

� ∈ [2=(p+ 1); 2) we can �nd a value for r which satis�es (18).In the case 2p = N the functions never intersect, so we have the same conclusion

as the case 2p¿N .The proof is thus �nished.


Remark 2. Notice that our solution satis�es u∈W 2;p() and also u∈L∞()(remember (6)).

Remark 3. We can use a similar argument when p=N . We have W 2;p() ,→ W 1;q()for any q ∈ [1;∞), so every integral of (8) is well posed. We can de�ne S : [0; 1]×W 1;q() 7→ W 1;q() for any q ∈ [1;∞) and we need

‖u‖1; q ≤ Cwith C being a constant independent of t and u. From the Sobolev imbedding theorem,we have again

‖u‖1; q ≤ C‖u‖2; p:The Gagliardo–Nirenberg inequalities can be also applied. In this situation, we obtainthe restrictions

2N + 1

≤ �¡ 2:

Moreover, the solution belongs to W 1;q() for any q ∈ [1;∞).

Remark 4. In the case 2N=(N +2) ≤ p¡N=2, we obtain in Theorem 1 that the upperbound � is N=(N − p). This function is increasing in p, so its in�mum is achievedfor p= 2N=(N + 2). For this value, N=(N − p) = (N + 2)=N which is just the boundin [15,16].

Remark 5. As we say in the Introduction, in [9] the author proved that there exists asolution u ∈ W 2;m() for (1) where m=min{2=�; p}, and so, if � ≤ 2=p u ∈ W 2;p().However, if �¿ 2=p then u belongs to W 2;2=�() ,→ W 1;q() where

q=2N

N�− 2 :If �¿ (N + 2)=N then q¡ 2, so following [8] it cannot go further to obtain moreregularity of the solution. In particular, it cannot prove the solution belongs to W 2;p()by this way.

3. An example

In this section, we analyze a simple (academic) example to which Theorem 1 canbe applied. Let us consider the particular case N = 3, and the quasilinear problem

−�u= n(x) + u( − m(x)u) + (b(x) · ∇u)� in ;

u= 0 on @;(19)

where ∈ R; 23 ≤ �¡ 2 and

(H) m ∈ L2(); n ∈ L∞(); b ∈ (L∞())3; n ≥ 0; m ≥ m0¿ 0:

In the context of population dynamics, any positive solution of (19) can be viewed asa steady-state population density. The coe�cient m=m(x) is associated to the limiting


e�ect of crowding in the population, while the transport e�ect and the in uence of thesurronding media are responsible for the presence of b=b(x) and n=n(x), respectively.With these assumptions, it is easy to prove that {u0; u0} = {0; K} is a sub-super-

solution couple for (19), provided K ¿ 0 is su�ciently large. So, we can deduce fromtheorem 1 that there exists a positive solution of (19).When p¿N , the existence of a sub-supersolution couple for (1) is ensured by

others conditions that can be found in [15,4]

Acknowledgements

The authors would like to ackowledge support under grant DGICYT PB95-1242.They also thank the referee for helpful suggestions.

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weak solutions for some quasilinear elliptic equations by the sub-supersolution method

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