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Combining Solutions of Semilinear Partial DifferentialEquations in ℝ n with Critical Sobolev ExponentMan Chun Leung a ba Department of Mathematics , National University of Singapore , Singapore , Republic ofSingaporeb Department of Mathematics , National University of Singapore , 2 Science Drive 2,Singapore , 117543 , Republic of SingaporePublished online: 14 Feb 2007.

To cite this article: Man Chun Leung (2004) Combining Solutions of Semilinear Partial Differential Equations in ℝ n

with Critical Sobolev Exponent, Communications in Partial Differential Equations, 29:5-6, 763-784, DOI: 10.1081/PDE-120037331

To link to this article: http://dx.doi.org/10.1081/PDE-120037331

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COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONSVol. 29, Nos. 5 & 6, pp. 763–784, 2004

Combining Solutions of Semilinear Partial DifferentialEquations in�n with Critical Sobolev Exponent

Man Chun Leung*

Department of Mathematics, National University of Singapore,Singapore, Republic of Singapore

ABSTRACT

We consider a positive C2-function u in �n which is equal to two differentspherical solutions respectively in two disjoint domains. u satisfies the equation�u+ n�n− 2�Ku

n+2n−2 = 0 in �n. By a result of Gidas, Ni and Nirenberg, K �≡ 1

in �n. In this paper we discuss lower bounds on sup�n �K − 1�. The estimatesare applied to study blow-up solutions and interaction of bubbles.

Key Words: Scalar curvature equation; Combining solutions; Slow decay.

AMS Classification: 35J60; 53C21.

∗Correspondence: Man Chun Leung, Department of Mathematics, National University ofSingapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore; Fax: 65-6779-5452;E-mail: matlmc@math.nus.edu.sg.

763

DOI: 10.1081/PDE-120037331 0360-5302 (Print); 1532-4133 (Online)Copyright © 2004 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

In this article we consider the variation of scalar curvature when the conformalmetric is predetermined on two regions in �n �n ≥ 3�. Specifically, let u be a positiveC2-function on �n such that u = u1 in �1 and u = u2 in �2, where �1 and �2 aredisjoint domains in �n, and

ui�x� =(

�1�2i + �x − �i�2

) n−22

for x ∈ �n� �i = 1� 2�� (1.1)

Here ��1� �1� and ��2� �2� are distinct fixed points in �+ �n. Consider theK-function of u, which is defined by

K�x� = −�u�x�

n�n− 2�un+2n−2 �x�

for x ∈ �n� (1.2)

With the function K, u satisfies the equation

�u+ n�n− 2�Kun+2n−2 = 0 in �n� (1.3)

By a result of Gidas, Ni and Nirenberg (Gidas et al., 1979, 1981; cf. Caffarelli et al.,1989), one knows that K �≡ 1. We investigate how much K can stay close to 1 bymeasuring positive lower bounds on sup�n �K − 1�.

In (1.1), ui satisfies the equation

�ui + n�n− 2�un+2n−2i = 0 in �n� �i = 1� 2�� (1.4)

They are called spherical solutions as they define conformal metrics that areisometric to Sn (rescaled and with a removable singularity). The scalar curvatureof the conformal metric u4/�n−2�ij is given by 4n�n− 1�K. Geometrically, one canvisualize the inflation at two regions, and seeks to measure the change in scalarcurvature. We refer to the articles Ambrosetti et al. (2000), Aubin (1994), Aubinand Bismuth (1997), Chen and Li (1997), Cheung and Leung (2001), Escobar andSchoen (1986), Kato (2001), Leung (1999, 2001a,b), Li (1995, 1996), Loewner andNirenberg (1974), Mazzeo and Pacard (1999), Mazzeo et al. (1996), Schoen (1988),Schoen and Zhang (1996) for relation with prescribing scalar curvature problems.

The set-up occurs in a natural manner on blow-up solutions of Eq. (1.3). Undermild assumption on K, if u a positive C2 blow-up solution of (1.3), that is, u fails tosatisfy the slow decay

u�x� ≤ C�x�− n−22 for �x� � 1� (1.5)

then bubbles appear in the form of approximate spherical solutions (see Sec. 5).At present, the intricate pattern with many bubbles is a key factor impeding furtherunderstanding of the equation.

On the other hand, once u�x��x��n−2�/2 is bounded on both sides by positiveconstants when �x� � 1, definite asymptotic property of u can be obtained.

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Semilinear Partial Differential Equations in �n 765

See Chen and Lin (1999), Taliaferro and Zhang (Preprint) and Leung (1999,2001c,d) for relation with asymptotic geometry on complete manifolds, gradientestimates and Harnack inequalities. Consequently, many authors are motivated tofind conditions on K so as to rule out the blow-up scenario. Results in this directionare obtained by a version of reflection or the moving plane/sphere method. Majoradvances (cf. Chen and Lin, 1997, 1998; Korevaar et al., 1999; Leung, 2003; Lin,2000; Taliaferro and Zhang, Preprint) purport the question: Suppose that

��K�x�� ≤ C1�x�− n2 for �x� � 1�

where C1 is a positive constant, does every positive C2-solution u of Eq. (1.3) haveslow decay (1.5)?

For n > 4, it can be deduced from Leung (2003) that there are functions Kwhich decay just slightly above the critical order −n/2 and afford solutions notsatisfying the slow decay (cf. also Leung, 2001b; Taliaferro, 1999; Taliaferro andZhang, 2003). Thus blow-up solutions form a singular but popular class. It wouldbe interesting to gain better understanding on them. Our approach is to juxtaposetwo spherical solutions and investigate the change in K. In Secs. 3 and 4, we obtainthe following results.

Theorem A. Consider a positive function u ∈ C2��n� with

u�x� = u1�x� =(

�1�21 + �x�2

) n−22

for all �x� < �

and

u�x� = u2�x� =(

�2�22 + �x�2

) n−22

for all �x� ≥ R�

Here 0 < � < R are positive numbers. Assume that either

�1�2

≤ �2

R2

(1

1+ �22/R2

)� or

�21�22

≥ 3�n+ 2�2�n− 2�

(1+ R4

�42

)� (1.6)

Then the K-function of u satisfies

supx∈B0�R�

�K�x�− 1� ≥ n+ 2n

� (1.7)

Estimate (1.7) is based on a more general inequality (3.4). We observe that byusing the Kelvin transform, there is a dual condition to (1.6) (see Remark 3.17).

Theorem B. Let D1 and D2 be disjoint domains in �n with

B�1�r1� ⊂ D1 and B�2

�a� ⊂ D2�

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where �1� �2 ∈ �n� r1 and a are positive numbers. Consider a positive C2-function u on�n with

u�x� = u1�x� =(

�1�21 + �x − �1�2

) n−22

for all x ∈ D1

and

u�x� = u2�x� =(

�2�22 + �x − �2�2

) n−2n

for all x ∈ D2�

Assume that both u and the K-function of u have removable singularities at theorigin under the Kelvin transformation x → x/�x�2� x �= 0. Given a number ≥ 1, ifr1 ≥ �1� a ≥ �2 and

�22�21

≥ 8nn��1 − �2�4

r41� 2 + 6�� (1.8)

then we have

supx∈�n

�K�x�− 1� ≥ n+ 22n

2� (1.9)

In Sec. 5, we discuss the appearance of bubbles, the blow-up analysis whichallows us to grasp a vital property of the solutions. Naturally, one can think ofreplacing parts of the blow-up solution by the spherical solutions. Locally, u candrop rapidly, and the variation in K can be exaggerated (cf. (1.2)). In order tominimize the change in K and to preserve as large a part of the spherical solutionas possible, a suitable place is found for the cut-and-glue-in procedure to takeplace. The last section is devoted to obtaining information on blow-up solutions viatheorem B. Key findings include restrictions on the “depths” of the bubbles.

Throughout this article n is an integer bigger than two, and Bx�r� the openball in �n with center at x and radius r > 0. In conformity with the convention, wedenote by C�C ′� C�� C0� C1� � � � various positive constants, which may have differentvalues from section to section, unless it is stated otherwise. The term domain is usedin a somewhat confined way – it is a non-empty open connected subset in �n, whichis also bounded and has smooth boundary.

2. ESTIMATES

Let H�x� �� be the fundamental solution for the Laplacian on �n. It is given by

H�x� �� = 1�2− n��n

1�x − ��n−2

for x� � ∈ �n with x �= ��

Here �n is the volume of the unit sphere in �n. Note that H < 0.

Theorem 2.1. Let �1��2 and � be domains in �n with

�1 ⊂ �2 ⊂ �2 ⊂ ��

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Semilinear Partial Differential Equations in �n 767

Consider two positive smooth solutions u1 and u2 of the equation

�u+ n�n− 2�un+2n−2 = 0 in �� (2.2)

Let uc be a positive C2-function on � with uc = u1 in �1 and uc = u2 in �\�2.The K-function of uc, denoted by K�x�, is defined as in (1.2). For any � ∈ �1, we have

4n∫�2

H�x� ���K�x�− 1�dx

= u− 4

n−21 ���− u

− 4n−2

2 ���+ �n+ 2�∫�2

�H�x� ���(∣∣∣�u− 2

n−2c �x�

∣∣∣2 − ∣∣∣�u− 2n−2

2 �x�∣∣∣2)dx�(2.3)

Proof. Given a real valued function f ∈C2���, a calculation shows that

�f�u� = f ′�u��u+ f ′′�u���u�2�

Hence we obtain

�u− 4

n−2c = − 4

n− 2u− n+2

n−2c �uc +

(4

n− 2

)(n+ 2n− 2

)u− 2n

n−2c ��uc�2

= 4nK + �n+ 2�∣∣∣�u− 2

n−2c

∣∣∣2� (2.4)

Given � ∈ �1 ⊂ �2, it follows from the representation formula (Gilbarg andTrudinger, 1997, Sec. 2.4, p. 18) that

u− 4

n−2c ��� =

∫�2

H�x� ���u− 4

n−2c �x�dx

+∫��2

[u− 4

n−2c �x�

�H�x� ��

�nx−H�x� ��

�

�nx

(u− 4

n−2c �x�

)]dSx

= 4n∫�2

H�x� ��K�x�dx − �n+ 2�∫�2

�H�x� ���∣∣∣�u− 2

n−2c �x�

∣∣∣2dx+

∫��2

[u− 4

n−2c �x�

�H�x� ��

�nx−H�x� ��

�

�nx

(u− 4

n−2c �x�

)]dSx� (2.5)

as �H� = −H . Here n is the unit outward normal on ��2. Likewise, we can draw theconclusion that

u− 4

n−22 ��� = 4n

∫�2

H�x� ��dx− �n+ 2�∫�2

�H�x� ���∣∣∣�u− 2

n−22 �x�

∣∣∣2dx+∫��2

[u− 4

n−22 �x�

�H�x� ��

�nx−H�x� ��

�

�nx

(u− 4

n−22 �x�

)]dSx� (2.6)

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Using the fact uc = u2 on �\�2, (2.5) and (2.6), we have

u− 4

n−2c ���− u

− 4n−2

2 ���

= 4n∫�2

H�x� ���K�x�− 1�dx − �n+ 2�∫�2

�H�x� ���∣∣∣�u− 2

n−2c �x�

∣∣∣2dx+ �n+ 2�

∫�2

�H�x� ���∣∣∣�u− 2

n−22 �x�

∣∣∣2dx�Since uc��� = u1��� for � ∈ �1, after rearranging the above equality we obtain (2.3).

�

Lemma 2.7. Let � ⊂ Bx0�R� be a domain in �n. We have∫

��H�x� ���dx ≤ R2

2�n− 2�for all � ∈ �� (2.8)

Proof. It suffices to show the case when � = B0�R�. Let � ∈ B0�R� andW = B0�R� ∩ B��R�. Consider

I =∫B0�R�

�H�x� ���dx and II =∫B��R�

�H�x� ���dx = R2

2�n− 2��

We have �x − �� < R for x ∈ B��R�\ W , and �y − �� > R for y ∈ B0�R�\ W . From theform of H and the symmetry of B��R�\ W and B0�R�\ W , one sees that

� =∫W�H�x� ���dx +

∫B��R�\ W

�H�x� ���dx

≥∫W�H�x� ���dx +

∫B0�R�\ W

�H�x� ���dx = I�

and equality holds only if � = 0. Thus we obtain the result. �

We return to Theorem 2.1. As K ≡ 1 in �1, from (2.3) we have

4n(supx∈�2

�K�x�− 1�) ∫

�2\�1

�H�x� ���dx

≥ u− 4

n−21 ���− u

− 4n−2

2 ���+ �n+ 2�∫�2

�H�x� ���(∣∣∣�u− 2

n−2c �x�

∣∣∣2 − ∣∣∣�u− 2n−2

2 �x�∣∣∣2)dx

(2.9)

for � ∈ �1. In addition, if �2 ⊂ Bx0�R�, then it follows from lemma 2.7 that

supx∈�2

�K�x�− 1� ≥ n− 22nR2

[u− 4

n−21 ���− u

− 4n−2

2 ���+ �n+ 2�∫�2

�H�x� ���

×(∣∣∣�u− 2

n−2c �x�

∣∣∣2 − ∣∣∣�u− 2n−2

2 �x�∣∣∣2)dx]� (2.10)

In Sec. 6 we generalize the above results to the case where there is an isolatedsingularity.

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3. THE CONCENTRIC CASE

In (2.9), consider the case when �1 = B0���, �2 = B0�R�, and

u1�x� =(

�1�21 + �x�2

) n−22

and u2�x� =(

�2�22 + �x�2

) n−22

for x ∈ �n� (3.1)

Here � and R are positive numbers with 0 < � < R. In this case we have

u− 4

n−21 �0�− u

− 4n−2

2 �0� = �21 − �22� (3.2)

and ∣∣∣�u− 2n−2

i �x�∣∣∣2 = 4�x�2

�2ifor x ∈ �n and i = 1� 2� (3.3)

It follows from (3.3) that

�n+ 2�∫�2

�H�x� 0��[∣∣∣�u− 2

n−2c �x�

∣∣∣2 − ∣∣∣�u− 2n−2

2 �x�∣∣∣2]dx

≥ 4�n+ 2�∫B0���

�H�x� 0��(r2

�21

)dx − 4�n+ 2�

∫B0�R�

�H�x� 0��(r2

�22

)dx

= n+ 2n− 2

[�4

�21− R4

�22

]�

where r = �x�. We have K�x� = 1 for �x� < � and∫B0�R�\B0���

�H�x� 0��dx = R2 − �2

2�n− 2��

Using (2.9) we obtain

supx∈B0�R�\B0���

�K�x�− 1� ≥ n− 22n

{�21 − �22 +

n+ 2n− 2

[�4

�21− R4

�22

]}1

R2 − �2� (3.4)

In particular, we have the following results.

Proposition 3.5. Let �1� �2� � and R be as above. If

�1�2

≤ �2

R2

(1

1+ �22/R2

)� (3.6)

then

supx∈B0�R�\B0���

�K�x�− 1� ≥ n+ 2n

� (3.7)

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Proof. Inequality (3.6) is equivalent to

�1�2

≤ �2R2

[1+

(�2R

)2]−1

�

Hence we have

�4

�21≥ R4

[1+ �22R

−2

�2

]2

= R4

�22+ 2R2 + �22�

It follows that

n− 22n

{�21 − �22 +

n+ 2n− 2

[�4

�21− R4

�22

]}1

R2 − �2≥ n+ 2

n� (3.8)

(3.7) follows from (3.4) and (3.8). �

Proposition 3.9. Let �1� �2� � and R be as above. If

�21�22

≥ 3�n+ 2�2�n− 2�

(1+ R4

�42

)� (3.10)

then

supx∈B0�R�\B0���

�K�x�− 1� ≥ n+ 2n

� (3.11)

Proof. It can be seen that

�22 +R4

�22≥ 2R2� (3.12)

On account of (3.10) and (3.12), we have

�21 ≥3�n+ 2�2�n− 2�

(�22 +

R4

�22

)≥ n+ 2

n− 2

(�22 +

R4

�22+ R2

)� (3.13)

Hence the result follows from (3.4). �

Combining Propositions 3.5 and 3.9, we obtain Theorem A. Let us introducethe proportions

k1 =�

�1and k2 =

R

�2� (3.14)

They are called depth factors, as they indicate how far the bubbles drop at theboundaries. More explicitly, from (3.1), we have[

u1�0�u1���

] 2n−2

= 1

�1�u1����2

n−2

= 1+ k21� (3.15)

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The notion helps to shed light on condition (3.6) in Proposition 3.5. We assert that(3.6) is equivalent to the condition that u2�R� ≥ �u1���, where � is the number givenby � = ��k21 + 1�/k21�

n−22 . Observe that � → 1 as k1 → �. Condition (3.6) can be

written as

�2�1

≤ k21k22 + 1

�

The condition u2�R� ≥ �u1��� is the same as[u2�R�

u1���

] 2n−2 1

�2

n−2

≥ 1� (3.16)

From (3.15) and (3.16), we obtain

k21k22 + 1

= k21 + 1

k22 + 1

1

�2

n−2

= �2�1

[u2�R�

u1���

] 2n−2 1

�2

n−2

≥ �2�1�

That is, condition (3.6) is satisfied if u2�R� ≥ �u1���, and vice versa. We note thatthe term �u1��� is independent on u2.

Remark 3.17. By applying the Kelvin transformation x → �2x/�x�2 on u, we cometo a situation where there is a positive C2-function u such that

u�x� = u2�x� =(

�2

�22 + �x�2) n−2

2

for �x� < �2/R�

u�x� = u1�x� =(

�1

�21 + �x�2) n−2

2

for �x� ≥ ��

(3.18)

Here �i = �2/�i for i = 1� 2. (See Leung, 2001c. Here u has a removable singularityat the origin because of the form of u2.) With the parameters in (3.18) fitted intheorem A, we conclude that

supx∈B0�R�\B0���

�K�x�− 1� = supx∈B0���\B0��

2/R�

�K�x�− 1� ≥ n+ 2n

�

if either

�1�2

≤ �2

R2

(1

1+ �2/�21

)� or

�21�22

≥ 3�n+ 2�2�n− 2�

(1+ �41

�4

)� (3.19)

Here K is the K-function of u and is given by K�x� = K��x/�x�2� for x �= 0.Consequently, in order to obtain (1.7), we may replace (1.6) by (3.19).

We also note that the results can be generalized into the situation when B0��� ⊂�1 and �2 ⊂ B0�R�, and similar technique can be applied to the eccentric case. It isclear that, from (3.4), if the parameters ��R� �2 are fixed, then

supx∈B0�R�\B0���

�K�x�− 1� → � as �1 → 0+�

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4. DISJOINT DOMAINS

Let us begin the argument toward theorem B by taking (possibly after atranslation) �2 = 0. Thus D1 and D2 are disjoint open domains in �n with B�1

�r1� ⊂D1 and B0�a� ⊂ D2 for positive numbers r1 and a, and for a point �1 ∈ �n. Considerspherical solutions

u1�x� =(

�1�21 + �x − �1�2

) n−22

and u2�x� =(

�2�22 + �x�2

) n−22

for x ∈ �n�

where �1 and �2 are positive numbers. Let uc be a positive C2-function on �n suchthat

uc = u1 in D1 and uc = u2 in D2�

uc satisfies the equation

�uc + n�n− 2�Kun+2n−2c = 0 in �n� (4.1)

where K is the K-function of uc (cf. (1.3)).We seek to apply a Kelvin transform to bring the setting into that of

Theorem 2.1. The Kelvin transform of uc about the sphere of radius a and centerat the origin is given by

uc�x� =(

a

�x�)n−2

uc

(a2x

�x�2)

for x ∈ �n\�0�� (4.2)

uc satisfies the equation

�uc + n�n− 2�Kun+2n−2c = 0 in �n\�0�� (4.3)

where

K�x� = K

(a2x

�x�2)

for x ∈ �n\�0��

See Leung (2001c). In what follows we consider a relation between two Kelvintransforms.

Lemma 4.4. For a function u defined on �n, let

u�x� = 1�x�n−2

u

(x

�x�2)

for x �= 0

be the Kelvin transform of u with center at the origin and radius one; and

u�2�a�x� = an−2

�x − �2�n−2u

(�2 +

a2�x − �2�

�x − �2�2)

for x �= �2�

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Semilinear Partial Differential Equations in �n 773

the Kelvin transform of u with center at �2 and radius a > 0. Then

u�2�a�x� = an−2

�x − �2�n−2

∣∣∣∣�2 + a2�x − �2�

�x − �2�2∣∣∣∣2−n

u

�2 + a2�x−�2�

�x−�2�2∣∣∣�2 + a2�x−�2�

�x−�2�2∣∣∣2 for x �= �2�

Proof. We have u�x� = 1/��x�n−2�u( x

�x�2)

for x �= 0. Substituting this into the

expression for u�2�a, we obtain the result. �

When x → �2, from Lemma 4.4 we have

u�2�a�x� → an−2

�x − �2�n−2

∣∣∣∣a2�x − �2�

�x − �2�2∣∣∣∣2−n

u

a2�x−�2�

�x−�2�2∣∣∣ a2�x−�2�

�x−�2�2∣∣∣2 = 1

an−2u

(x − �2a2

)�

By the assumption in Theorem B (mindful of the action of the translation), togetherwith Lemma 4.4, uc can be extended as a C2-function across the origin. Likewise, Kcan be extended as a continuous function, so that Eq. (4.3) is satisfied on the whole�n. (With all this, compare also with Lemma 6.13.)

The Kelvin transforms of u1 and u2 about the sphere of radius a > 0 and centerat the origin are given respectively by

u1�x� =(

�1

�21 + �x − �1�2) n−2

2

and u2�x� =(

a2�−12

a4�−22 + �x�2

) n−22

(4.5)

for x ∈ �n. Here

�1 =a2�1

�21 + ��1�2and �1 =

a2�1�21 + ��1�2

� (4.6)

See, for instance, Leung (2001c). Let �1 denote the image of D1 under the inversionx → a2x/�x�2� x �= 0. It follows that

uc = u1 in �1 and uc = u2 in �n\B0�a�� (4.7)

We need to find a suitable point � to enlarge the right hand side of (2.9). Theinversion x → a2x/�x�2 sends a ball to a ball. Consider the line passing throughthe origin and �1. It can be seen that the “inverted” ball of B�1

�r1� has diameter dgiven by

d =(

a2

��1� − r1− a2

��1� + r1

)= 2r1a

2

��1�2 − r21� (4.8)

(Note that ��1� > r1, as B0�a� ∩ B�1�r1� = ∅.) Assume that r1 ≥ �1. We have

a2���1� + r�

���1� + r1�2= a2��1�

��1�2 + r1��1�≤ a2��1�

��1�2 + �21≤ a2��1�

��1�2 − r1��1�= a2���1� − r�

���1� − r1�2�

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This is because ��1� > r1 ≥ �1. Hence there exists a point � ∈ B�1�r1� such that

a2�

���2 = a2�1��1�2 + �21

= �1� (4.9)

From (4.5) and (4.9) we obtain u1�a2�/���2� = u1��1� = �

2−n2

1 . That is,

u− 4

n−21 �a2�/���2� = �21 =

a4�21��21 + ��1�2�2

� (4.10)

Moreover,

u−4n−22 �a2�/���2� = ��−2

2 ���2 + 1�2a4�22���4 =

(a

���)4

��22 + ���2�2�22

� (4.11)

The following lemma can be verified by direct calculations.

Lemma 4.12. Given a point � ∈ �n and a positive number �, let

u�x� =(

�

�2 + �x − ��2) n−2

2

for x ∈ �n�

We have

∣∣∣�u− 2n−2 �x�

∣∣∣2 = 4�x − ��2

�2for x ∈ �n� (4.13)

Consequently,

∣∣∣�u− 2n−2

1 �x�∣∣∣2 = 4

�x − �1�2a4�21

��21 + ��1�2�2 and∣∣∣�u− 2

n−22 �x�

∣∣∣2 = 4�x�2�22a−4 (4.14)

for x ∈ �n. In this case �2 = B0�a�. It follows from Lemma 2.7 that

�n+ 2�∫�2

�H�x� a2�/���2��∣∣∣�u− 2

n−22 �x�

∣∣∣2dx≤ 4�n+ 2��22a

−2∫B0�a�

∣∣H�x� a2�/���2�∣∣dx ≤ 2(n+ 2n− 2

)�22� (4.15)

The inverted ball of B�1�r1� inside �1 contains a ball B�0

��� with radius � = d/8. Bychoosing the center �0 suitably [cf. (4.8) and (4.9)], we have

∣∣x − a2�/���2∣∣ = ∣∣x − �1∣∣ ≥ � for all x ∈ B�0

���� (4.16)

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Semilinear Partial Differential Equations in �n 775

Hence

�n+ 2�∫�1

�H�x� a2�/���2��∣∣∣�u− 2

n−2c �x�

∣∣∣2dx≥ �n+ 2�

∫B�0

����H�x� a2�/���2��

∣∣∣�u− 2n−2

1 �x�∣∣∣2dx

≥ 4�n+ 2���21 + ��1�2�2

a4�21

∫B�0

����H�x� a2�/���2��∣∣x − �1

∣∣2dx [by (4.14)]

≥ 4�n+ 2���21 + ��1�2�2

a4�21�2

∫B�0

���

∣∣H�x� �1��dx [by (4.16)]

≥ 4�n+ 2���21 + ��1�2�2

a4�21�2

(1

�n− 2��n

1dn−2

) ∫B�0

���dx

≥ 4�n+ 2���21 + ��1�2�2

a4�21�2

(1

�n− 2��n

1dn−2

)(�n

n�n

)≥ 4�n+ 2�

n�n− 2�1

8n+2

��21 + ��1�2�2a4�21

d4

≥ 4�n+ 2�n�n− 2�

18n+2

��21 + ��1�2�2a4�21

(24r41a

8

���1�2 − r21 �4

)≥ n+ 2

n�n− 2�18n

a4

��1�8��21 + ��1�2�2

�21r41 � (4.17)

It follows from (2.9), (4.10), (4.11), (4.15) and (4.17) that

sup�n\B0�a�

�K − 1�= supB0�a�

�K − 1�

≥ n− 22n

a−2

[a4�21

��21 + ��1�2�2+ n+ 2

n�n− 2�18n

a4

��1�8��21 + ��1�2�2

�21r41

−(

a

���)4

��22 + ���2�2�22

− 2(n+ 2n− 2

)�22

]� (4.18)

Let the positive numbers c� k and C be defined by the equations

r1 = c�1� a = k�2� ��1� = C�2� (4.19)

respectively. We obtain

a4�21��21 + ��1�2�2

= k4�42�21

��21 + C2�22�2= k4�21(

�21�22+ C2

)2 �

a4

��1�8��21 + ��1�2�2

�21r41 = k4�42

C8�82��21 + C2�22�

2c4�21 >c4

C4k4�21�

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From (4.9) we have

��� = ��1� +�21��1�

�⇒ ��� > ��1��

Assume that k = a/�2 ≥ 1. As ��1� > a ≥ �2, we obtain(a

���)4

��22 + ���2�2�22

= k4�22

(�22���2 + 1

)2

≤ k4�22

(�22��1�2

+ 1)2

≤ 22k4�22�

From (4.18) and the fact that a2 = k2�22, we have

supBo�a�

�K − 1� ≥ n− 22n

[k2(

�21�22+ C2

)2

�21�22

+ n+ 2n�n− 2�

k2

8n�21�22

c4

C4− 4k2 − 2

(n+ 2n− 2

) 1k2

]�

(4.20)

Observe that

k2(�21�22+ C2

)2

�21�22

≤ k2

4C2= a2

4��1�2≤ 1

4�

Given a number ≥ 1, we seek conditions for the inequality

n+ 2n�n− 2�

k2

8n�21�22

c4

C4− 4k2 − 2

(n+ 2n− 2

)1k2

≥(n+ 2n− 2

) 2 (4.21)

to hold. That is,

n+ 2n�n− 2�

18n

�21�22

c4

C4≥ 4+

(n+ 2n− 2

)[2k4

+ 2

k2

]� (4.22)

As c4/C4 = ��42/�41��r

41/��1�4�, (4.22) is equivalent to

�22�21

≥ 8n��1�4r41

[4n�n− 2�n+ 2

+(2n�42a4

+ n 2�22a2

)]� (4.23)

Because �2/a ≤ 1, (4.23) holds if

�22�21

≥ 8nn��1�4r41

�4+ 2+ 2�� (4.24)

Hence we conclude that if �22/�21 ≥ 8nn��1�4� 2 + 6�/r41 , and r1 ≥ �1 and a ≥ �2, then

sup�n

�K − 1� ≥ n+ 22n

2� (4.25)

Taking into the account of the translation, we obtain theorem B.

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Semilinear Partial Differential Equations in �n 777

5. GLUE-IN SPHERICAL SOLUTIONS

It is more convenient to discuss the blow-up process in the setting

�u+ n�n− 2�Kun+2n−2 = 0 in Bo�1�\�0�� (5.1)

where K is a smooth function on Bo�1�\�0�. Equation (5.1) is related to (1.3) bythe Kelvin transformation x → x/�x�2 (cf. (4.2) and (4.3)). We examine the isolatedsingularity of u at 0. Precisely, assume that u does not satisfy the slow decay

u�x� ≤ Co

�x��n−2�/2for �x� small and positive� (5.2)

The blow-up analysis for K ≡ 1 is discussed in Korevaar et al. (1999). It is basedon renormalization, rescaling and the result of Gidas, Ni and Nirenberg. Similarargument works if we assume that

limx→0

K�x� = 1 and �x� · ��K�x�� ≤ C1 for x ∈ Bo�5/8�\�0�� (5.3)

See, for instance, Leung (2001c). Following closely the discussion in Korevaar et al.(1999) and paying attention to Eq. (16) there, we perform a rescaling and find thatif u does not have slow decay, then for any given positive numbers (small) andR (large), there exist a point x1 near the origin and a small positive number �such that∥∥∥∥( �

�2 + �y�2) n−2

2

− u�x1 + y�

∥∥∥∥Co�Bo��R��

< �2−n2 (5.4)

and ∥∥∥∥( �

�2 + �y�2) n−2

2

− u�x1 + y�

∥∥∥∥C1�Bo��R��

< Cn

(

�

)�

2−n2 (5.5)

for �y� < �R. Here Cn is positive constant that depends on n only.We seek to cut and glue a bubble on the solution and keep track of the change

in the K-function, careful not to disturb it too much. Observe that from (5.4) and(5.5) the function decreases rapidly when � is small. This tends to sharpen the changein K (cf. (1.2)). So the key is to find a suitable place to perform the procedure, using(1.2) to compute the variation.

Let = 2/�n−2�. We may assume that < 1. Given a positive number � suchthat 2�1+ �� < n, let �M be a positive number such that

2�n−2��n−2−2��

2�n+2� ≥(

1

1+ �2M

) n−22

≥ �n−2��n−2−2��

2�n+2� + n−22 � (5.6)

In particular, 1/�1+ �2M� ≥ . It follows from (5.6) that[(

1

1+ �2M

) n−22

− n−22

] n+2n−2

≥ n−22 −�� (5.7)

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For a point y with �y� = ��M , we have

(�

�2 + �y�2) n−2

2

=(

�

�2 + �2�2M

) n−22

=(

1

1+ �2M

) n−22 1

�n−22

� (5.8)

Here � is the parameter in (5.4) and (5.5).We may also assume that R is large enough such that �M < R, and that is

small enough so that, say �M > 100. Let �m = �M − 10. Take a cut-off function� ∈ C���0���� so that � ≥ 0 in �0���� � = 1 in �0� ��m�� � = 0 in ���M������′� ≤ C��

−1 and ��′′� ≤ C��−2 in ���m� ��M� for some positive constant C�. Let

w�x� = ���x��u��o�x�+ �1− ���x���u�x1 + x� for �x� < �R� (5.9)

We have

�w�x� = �u��o�x�− u�x1 + x�����x�+ 2��u��o�x�− �xu�x1 + x�� · ���x�+��x��u��o�x�+ �1− ��x���xu�x1 + x�

= �u��o�x�− u�x1 + x�����x�+ 2��u��o�x�− �xu�x1 + x�� · ���x�− n�n− 2����x�u

n+2n−2��o �x�+ �1− ��x��K�x1 + x�u

n+2n−2 �x1 + x��

for �x� < �R. By (5.3), there is a positive constant �p such that

�K − 1� ≤ �p in Bx1��R�� (5.10)

Furthermore, we may choose �p → 0+ as x1 → 0 and �R → 0+. As w is a non-negative function in Bo��R�, there is a function Kg ∈ C��Bo��R�� (the K-functionof w) such that

�w + n�n− 2�Kgwn+2n−2 = 0 in Bo��R�� (5.11)

It follows that

�Kg�x�− 1�

=∣∣∣∣ �w�x�

n�n+ 2�wn+2n−2 �x�

+ 1

∣∣∣∣≤ ����x���u��o�x�− u�x1 + x��

n�n+ 2�wn+2n−2 �x�

+ 2����x����u��o�x�− �xu�x1 + x��n�n+ 2�w

n+2n−2 �x�

+∣∣∣∣∣∣u

n+2n−2��o �x�+ �1− ��x��

[K�x1 + x�u

n+2n−2 �x1 + x�− u

n+2n−2��o �x�

]�u��o�x�+ �1− ��x���u�x1 + x�− u��o�x���

n+2n−2

− 1

∣∣∣∣∣∣ (5.12)

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Semilinear Partial Differential Equations in �n 779

for x ∈ Bo���M�\Bo���m�. We have

wn+2n−2 �x� = �u��o�x�+ �1− ��x���u�x1 + x�− u��o�x���

n+2n−2

≥ �u��o�x�− �u�x1 + x�− u��o�x��� n+2n−2

≥[(

1

1+ �2M

) n−22

− n−22

] n+2n−2 1

�n+22

≥ n−22 −�

�n+22

(5.13)

for x ∈ Bo���M�\Bo���m�. As � is a radial function, so

�� = d2�

dr2+ n− 1

r

d�

dr�

where r = �x�. Together with the bounds on �′ and �′′, the fact that ��m ≤ �x� <��M , (5.4), (5.5) and (5.13), we obtain

����x���u��o�x�− u�x1 + x��n�n− 2�w

n+2n−2 �x�

≤ C� (5.14)

and

2����x����u��o�x�− �xu�x1 + x��n�n− 2�w

n+2n−2 �x�

≤ C� (5.15)

for x ∈ Bo���M�\Bo���m�. As for the last term in (5.12), we have∣∣∣∣∣∣u

n+2n−2��o �x�+ �1− ��x��

[K�x1 + x�u

n+2n−2 �x1 + x�− u

n+2n−2��o �x�

]�u��o�x�+ �1− ��x���u�x1 + x�− u��o�x���

n+2n−2

− 1

∣∣∣∣∣∣≤

∣∣∣∣un+2n−2��o �x�− �u��o�x�+ �1− ��x���u�x1 + x�− u��o�x���

n+2n−2

∣∣∣∣w

n+2n−2 �x�

+�K�x + x1��

∣∣∣un+2n−2 �x1 + x�− u

n+2n−2��o �x�

∣∣∣w

n+2n−2 �x�

+ �K�x + x1�− 1� un+2n−2��o �x�

wn+2n−2 �x�

(5.16)

for x ∈ Bo���M�\Bo���m�. We note that

xn+2n−2 − y

n+2n−2 = n+ 2

n− 2

∫ x

yt

4n−2dt ≤ n+ 2

n− 2x

4n−2 �x − y� for x ≥ y ≥ 0� (5.17)

Also,

supBo��R�

u4

n−2�o = 1

�2and sup

Bo��R�

u4

n−2 ≤ 2�2

�

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as � is small. Hence we obtain

�K�x + x1��∣∣∣un+2

n−2 �x1 + x�− un+2n−2��o �x�

∣∣∣w

n+2n−2 �x�

≤ C�−2 �u�x1 + x�− u��o�x��w

n+2n−2 �x�

≤ C ′� (5.18)

for x ∈ Bo���M�\Bo���m�. Using the left-hand-side inequality in (5.6), we have

�K�x + x1�− 1� un+2n−2��o �x�

wn+2n−2 �x�

≤ �p

(1

1+�2m

) n+22 �−

n+22

n−22 −��−

n+22

≤ 2�p

(1

1+�2M

) n+22

n−22 −�

≤ 22nn−2 �p (5.19)

for x ∈ Bo���M�\Bo���m�. Here we make use of the fact that �M ≥ 100 and�m = �M − 10. Likewise, using (5.17) and (5.14), we obtain∣∣∣un+2

n−2��o �x�− �u��o�x�+ �1− ��x���u�x1 + x�− u��o�x���

n+2n−2

∣∣∣w

n+2n−2 �x�

≤ C�−2 �u�x1 + x�− u��o�x��w

n+2n−2 �x�

≤ C ′� (5.20)

for x ∈ Bo���M�\Bo���m�. We glue in the spherical solution and find that

u�x1 + x� = u��o�x� for �x� ≤ ���M − 10��

�Kg − 1� ≤ Cmax��� �p� in Bo���M��(5.21)

Here we use (5.10), (5.12), (5.14), (5.15), (5.16), (5.18), (5.19) and (5.20), and observethat C depends on n, 1+ �p and C� only. For applications in Sec. 6, we note that �M

can be chosen to be large when is small and R is large. In (5.21), let us recall that� is less than �n− 2�/2, the same critical index appeared in Chen and Lin (1998),and in Leung (2001b, 2003).

6. ESTIMATES AT THE SINGULARITY

We consider a blow-up solution v ∈ C2+��

n� of the equation

�v+ n�n− 2�Kvn+2n−2 = 0 in �n� (6.1)

The Kelvin transform of v is given by

u�x� = �x�2−nv�x/�x�2� for x �= 0� (6.2)

We know that u satisfies the equation

�u+ n�n− 2�Kun+2n−2 = 0 in �n\�0�� (6.3)

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Semilinear Partial Differential Equations in �n 781

where K�x� = K�x/�x�2� for x �= 0. Assume that

limx→0

K�x� = 1 and �x� · ��K�x�� ≤ C for x ∈ Bo�1�\�0�� (6.4)

From the fact that K is bounded near infinity and the limit in (6.4), we have

�K�x�− 1� ≤ �2 for x ∈ �n\�0�� (6.5)

where � is a positive constant. In terms of K, condition (6.4) is equivalent to

limy→� K�y� = 1 and �y� · ��K�y�� ≤ C for y ≥ 1� (6.6)

As discussed in Sec. 5, bubbles develop and they are described by (5.4) and (5.5).We choose a small and large R as in Sec. 5. There is a bubble (with center ata point �2) which satisfies (5.4) and (5.5). Applying the cut-and glue-in process inSec. 5, we may replace u by another positive smooth function which differs fromu only on small neighbourhood of �2, and because of this we still denote the newfunction by u, such that

u�x� =(

�2�22 + �x − �2�2

) n−22

in B�2�a�� (6.7)

Here

a = ��M − 10��2�

where �M is the constant in (5.6). Furthermore, by the choice of and R, we havea ≥ �2. (Actually we can have a/�2 � 1.)

We choose another set of numbers ′ (small) and R′ (large). There exists anotherbubble with center at �1. Similar to the above discussion, we may assume that

u�x� =(

�1�21 + �x − �1�2

) n−22

in B�1�r1�� (6.8)

where

r = ��′M − 10��1�

and �′M is the constant in (5.6) defined by ′. We also have r ≥ �1. By choosing �1

close to the origin, we may assume that B�2�a� ∩ B�1

�r1� = ∅. We may also assumethat r1/�1 and a/�2 are large. It follows from (5.21) and (6.4) that (6.5) remainsvalid.

Under the Kelvin transform with center at the origin and radius 1, the origin,which is a regular point for both v and K, is sent to infinity. When we apply theKelvin transform with center at �2 and radius a, the infinity is sent to �2. It canbe seen that �2 is a regular point for the latter Kelvin transforms of u and K. Anargument toward this observation is similar to the proof of Lemma 4.4.

We seek to apply the conclusion in Theorem B. As the origin in Eq. (6.3) isa singularity for u, after the Kelvin transformation with center at �2 and radiusa, there is an isolated singularity at ��2 + a2�2/��22�� ∈ B�2

�a�. Thus Theorem 2.1may not apply. However, examples of blow-up solutions exist such that the blow-up points are not too crowded together (Leung, 2003; Taliaferro, 1999). It seemsreasonable to assume the following.

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782 Leung

(H) There is a sequence of closed hypersurfaces in �n\�0�, parameterized by∑�i, where �i = max��x� � x ∈ ∑

�i� → 0 as i → �, such that they enclose domains R�i

containing the origin, respectively, and

limi→�

∫∑

�i

1 · dS = 0� limi→�

∫∑

�i

��u�dS = 0 (6.9)

One can think of condition (H) as requiring ��u� to behave regularly along∑

�i.

We have the following result (see, for example, Leung, 2001c).

Lemma 6.10. Let F be a smooth positive superharmonic function (i.e., �F ≤ 0) on�n\B0�r0�. There exist positive numbers c and r1 such that

F�y� ≥ c2�y�2−n for �y� ≥ r1�

Since K�y� → 1 as �y� → �, we can apply (6.1) and the above lemma toconclude that

v�y� ≥ c2�y�2−n for �y� ≥ r1 �⇒ u�x� ≥ c2 for �x� = �y�−1 < r−11 �

It follows that

limi→�

∫��i

u− 4n−2 dS = 0� (6.11)

Moreover,

limi→�

∫��i

��u− 4n−2 �dS ≤ C lim

i→�

∫��i

��u�dS = 0 �by �6�9��� (6.12)

Using (6.11) and (6.12) together with (2.4), one verifies the following generalizationof the representation formula in (2.5), now with a point singularity.

Lemma 6.13. Let u ∈ C2+� �\�0�� and U = u− 4

n−2 , where � is a domain whichcontains the origin. Assuming (H) for u and that the K-function of u is positive andbounded in B0��\�0� ⊂ � for some > 0. Then for any � ∈ �\�0�, we have

U��� = limi→�

∫�\ Rei

H�x� ���U�x�dx+∫�

[U�x�

�H�x� ��

�nx−H�x� ��

�U�x�

�nx

]dSx

= 4n∫�\�0�

H�x� ��K�x�dx − �n+ 2�∫�\�0�

�H�x� �����u− −2n−2 �x��2 dx

+∫d�

[U�x�

�H�x� ��

�nx−H�x� ��

�U�x�

�nx

]dSx�

As the Kelvin transform with center at �2 and radius a does not changethe essential features of the singularity at the origin, we may apply Lemma 6.13.The arguments in Secs. 2 and 4, together with (1.8), lead to

�22�21

≤ C�n� ����1 − �2�4

r41� (6.14)

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Semilinear Partial Differential Equations in �n 783

That is,

r41�21

≤ C�n� ����1 − �2�4�−22 � (6.15)

Here C�n� �� is a positive constant that depends on � and n only. In order to obtain(6.15), the conditions are (6.4) for K, and (H) for u. Fix a bubble and let the otherone be chosen closer to the origin. That is, �2 in (6.15) is fixed. We conclude that,roughly speaking, there are no “deep rooted” bubbles close to the origin. That is, r21cannot be relatively large comparing to �1.

REFERENCES

Ambrosetti, A., Garcia Azorero, J., Peral, I. (2000). Elliptic variational problems inRN with critical growth. J. Differential Equations 168:10–32.

Aubin, T. (1994). Sur le probléme de la courbure scalaire prescrite. Bull. Sci. Math.118(5):465–474.

Aubin, T., Bismuth, S. (1997). Courbure scalaire prescuite sur les variétésriemanniennes compactes dans lé cas négatif. J. Funct. Anal. 143:529–541.

Caffarelli, L., Gidas, B., Spruck, J. (1989). Asymptotic symmetry and local behaviorof semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl.Math. 42:271–297.

Chen, C.-C., Lin, C.-S. (1997). Estimates of the conformal scalar curvature equationvia the method of moving planes. Comm. Pure Appl. Math. 50:971–1019.

Chen, C.-C., Lin, C.-S. (1998). Estimates of the conformal scalar curvatureequations via the method of moving planes, II. J. Differential Geom. 49:115–178.

Chen, C.-C., Lin, C.-S. (1999). On the asymptotic symmetry of singular solutions ofthe scalar curvature equations. Math. Ann. 313:229–245.

Chen, W.-X., Li, C.-M. (1997). A priori estimates for prescribing scalar curvatureequations. Ann. Math. 145:547–564.

Cheung, K.-L., Leung, M.-C. (2001). Asymptotic behavior of positive solutions ofthe equations �u+ Ku

n+2n−2 = 0 in �n and positive scalar curvature. Discrete

Contin. Dynam. Systems, Added Volume (Proceedings of the InternationalConference on Dynamical Systems and Differential Equations, 2001; Du, J.,Hu, S., Eds.): 109–120.

Escobar, J., Schoen, R. (1986). Conformal metrics with prescribed scalar curvature.Invent. Math. 86:243–254.

Gidas, B., Ni, W.-M., Nirenberg, L. (1979). Symmetry and related properties via themaximum principle. Comm. Math. Phys. 68:209–243.

Gidas, B., Ni, W.-M., Nirenburg, L. (1981). Symmetry of positive solutions ofnonlinear elliptic equations in �n. In: Mathematical Analysis and Applications,Part A, Adv. Math. Suppl. Stud., 7a. New York-London: Academic Press,pp. 369–402.

Gilbarg, D., Trudinger, N. (1997). Elliptic Partial Differential Equations of SecondOrder. 2nd ed. Berlin-Heidelberg-New York: Springer-Verlag.

Dow

nloa

ded

by [

Fond

ren

Lib

rary

, Ric

e U

nive

rsity

] a

t 11:

28 2

2 N

ovem

ber

2014

ORDER REPRINTS

784 Leung

Kato, S. (2001). The scalar curvature equation on open Riemannian manifolds.Sugaku Expositions 14:219–236.

Korevaar, N., Mazzeo, R., Pacard, F., Schoen, R. (1999). Refined asymptoticsfor constant scalar curvature metrics with isolated singularities. Invent Math.135:233–272.

Leung, M.-C. (1999). Asymptotic behavior of positive solutions of the equation�gu+ Kup = 0 in a complete Riemannian manifold and positive scalarcurvature. Comm. Partial Differential Equations 24:425–462.

Leung, M.-C. (2001a). Boundedness of positive solutions of the conformal scalarcurvature equation and positive scalar curvature. Comm. Partial DifferentialEquations 26:285–293.

Leung, M.-C. (2001b). Exotic solutions of the conformal scalar curvature equationin Rn. Ann. Inst. H. Poincarè Anal. Non Linèaire 18:297–307.

Leung, M.-C. (2001c). Conformal scalar curvature equations in open spaces.Cubo Matemática Educational 3:415–443.

Leung, M.-C. (2001d). Growth estimates on positive solutions of the equation�u+ Ku

n+2n−2 = 0 in Rn. Canad. Math. Bull. 44:210–222.

Leung, M.-C. (2003). Blow up solutions of nonlinear elliptic equations in �n withcritical exponent. Math. Ann. 327:723–744.

Li, Y.-Y. (1995). Prescribing scalar curvature on Sn and related problems, part I.J. Differential Equations 120:319–410.

Li, Y.-Y. (1996). Prescribing scalar curvature on Sn and related problems, part II:Existence and compactness. Comm. Pure Appl. Math. 49:541–597.

Lin, C.-S. (2000). Estimates of the conformal scalar curvature equation via themethod of moving planes, III. Comm. Pure Appl. Math. 53:611–646.

Loewner, C., Nirenberg, L. (1974). Partial differential equations invariantunder conformal or projective transformations. In: Contributions to Analysis(A Collection of Papers Dedicated to Lipman Bers). New York: Academic Press,pp. 245–272.

Mazzeo, R., Pacard, F. (1999). Constant scalar curvature metrics with isolatedsingularities. Duke Math. J. 99:353–418.

Mazzeo, R., Pollack, D., Uhlenbeck, K. (1996). Moduli spaces of singular Yamabemetrics. J. Amer. Math. Soc. 9:303–344.

Schoen, R. (1988). The existence of weak solutions with prescribed singular behaviorfor a conformally invariant scalar equation. Comm. Pure Appl. Math. 41:317–392.

Schoen, R., Zhang, D. (1996). Prescribed scalar curvature on the n-sphere. Calc. Var.Partial Differential Equations 4:1–25.

Taliaferro, S. (1999). On the growth of superharmonic functions near an isolatedsingularity, I. J. Differential Equations 158:28–47.

Taliaferro, S., Zhang, L. (2003). Arbitrarily large solutions of the conformal scalarcurvature problem at an isolated singularity. Proc. Amer. Math. Soc. 131:2895–2902.

Taliaferro, S., Zhang, L. Asymptotic symmetries for conformal scalar curvatureequations with singularity (Preprint).

Received March 2003Revised February 2004

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