Positive Solutions of Quasilinear Elliptic Equations …fe.math.kobe-u.ac.jp/FE/FullPapers/49-2/49_235.pdfPositive Solutions of Quasilinear Elliptic Equations on RN 239 z 1 1 ðsÞs

Funkcialaj Ekvacioj, 49 (2006) 235–267

Positive Solutions of Quasilinear Elliptic Equations

with Critical Orlicz-Sobolev Nonlinearity on RN

By

Nobuyoshi Fukagai1�, Masayuki Ito

2 and Kimiaki Narukawa3y

(Tokushima University1,2 and Naruto University of Education3, Japan)

Abstract. A nonnegative nontrivial solution of the quasilinear elliptic di¤erential

equation (1.1) below on the entire space is obtained. The function fðtÞt in the

principal part is nonhomogeneous and bðtÞt has the critical Orlicz-Sobolev growth with

respect to f.

Key Words and Phrases. Quasilinear degenerate elliptic equation, Unbounded

domain, Orlicz-Sobolev, Critical exponents, Mountain pass, Concentration-

compactness.

2000 Mathematics Subject Classification Numbers. 35B33, 35J20, 35J25, 35J70,

46E30.

1. Introduction

Let us consider a quasilinear elliptic di¤erential equation in the divergence

form

�divðfðj‘ujÞ‘uÞ ¼ bðjujÞuþ lf ðx; uÞð1:1Þ

on the entire space RN , Nb 2, where bðjujÞu denotes a critical Sobolev growth

term with respect to the principal term, f ðx; uÞ is a subcritical term and l > 0

is a real parameter. The nonnegative solution of (1.1) can be regarded as a

critical point of the functional

IlðuÞ ¼ðRN

fFðj‘ujÞ � BðuÞ � lFðx; uÞgdx

where FðtÞ, BðtÞ and Fðx; tÞ are primitives of fðtÞt, bðtÞt and f ðx; tÞ,respectively.

The equation (1.1) with power nonlinearity FðtÞ ¼ jtjp is well known as the

p-Laplace equation involving critical Sobolev exponent p� ¼ Np=ðN � pÞ. The

boundary value problem

* Supported in part by Grant-in-Aid for Scientific Research (No. 16540197), Japan Society for the

Promotion of Science.† Supported in part by Grant-in-Aid for Scientific Research (No. 14540211), Japan Society for the

Promotion of Science.

�Dpu ¼ jujp��2

uþ lf ðx; uÞ in W

u ¼ 0 on qW

�ð1:2Þ

has been studied by many authors through variational approach. In the case of

bounded W, positive solutions of (1.2) are obtained by Brezis and Nirenberg

[10], Guedda and Veron [17], Garcıa Azorero and Peral Alonso [15], Ben-

Naoum, Troestler and Willem [7], and Silva and Xavier [21]. Further, Benci

and Cerami [6], Goncalves and Alves [16], Ambrosetti, Garcia Azorero and

Peral [2, 3], and Silva and Soares [20] have studied the case of W ¼ RN .

There often arise the equations associated by nonhomogeneous non-

linearities F in the fields of nonlinear elasticity, plasticity and non-Newtonian

fluids etc. (e.g., [13], [14]). It is meaningful to study such equations with

general f. Here, for such f, we consider the equation (1.1) with critical

Sobolev growth bðjujÞu on the entire space. It seems to be di‰cult to deal with

the functional Il on the usual Sobolev space. As an example, let FðtÞ ¼ð1þ t2Þg � 1 ðg0 1Þ. It has a di¤erent power-like behavior at 0 and at

infinity:

FðtÞ@ 2gt2 t ! 0

t2g t ! y:

�Since neither of the Sobolev spaces W 1;2ðRNÞ, W 1;2gðRNÞ includes the other,

the functional Il is not well defined on neither of them. The most natural

function space on which Il is defined is the Orlicz-Sobolev space associated with

the function F. Introduction of such space also makes us possible to deal with

non power-like nonlinearity, e.g., FðtÞ ¼ tp logð1þ tÞ. Further, the lack of

compactness of the functional occurs due to the critical growth of bðjujÞuof (1.1). Considering this we make some modification of the concentration-

compactness principle.

The purpose of the present paper is to show the existence of nonnegative

and nontrivial (weak) solutions to (1.1) with general nonlinearity f by applying

the variational methods in the Orlicz-Sobolev space.

Here we give a brief review of Orlicz spaces. For an N-function A ¼ AðtÞand an open set WHRN , the Orlicz space LAðWÞ is defined (see Adams and

Fournier [1, Chap. 8]). When A satisfies D2-condition, i.e.,

Að2tÞa kAðtÞ; tb 0;

for some constant k > 0, the space LAðWÞ coincides with the set of measurable

functions u on W such that ðW

AðjuðxÞjÞdx < y:

236 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa

Equipped with the (Luxemburg) norm defined by

kukA;W ¼ inf k > 0;

ðW

AjuðxÞjk

� �dxa 1

� �ð1:3Þ

for u A LAðWÞ, this space is a Banach space. For the case of W ¼ RN , we shall

simply denote the norm kukA;RN as kukA. The complement of A is given by the

Legendre transformation

~AAðsÞ ¼ maxtb0

ðst� AðtÞÞ for sb 0:ð1:4Þ

These A and ~AA are complementary each other. From Young’s inequality

staAðtÞ þ ~AAðsÞ;

a Holder type inequalityðW

uðxÞvðxÞdx�� a 2kukA;Wkvk ~AA;Wð1:5Þ

can be obtained for u A LAðWÞ, v A L ~AAðWÞ. The Sobolev conjugate function A�of A is defined by setting

A�1� ðtÞ ¼

ð t0

A�1ðsÞsðNþ1Þ=N ds:ð1:6Þ

Let

FðtÞ ¼ð t0

fðsÞs ds;ð1:7Þ

BðtÞ ¼ð t0

bðsÞs ds; Fðx; tÞ ¼ð t0

f ðx; sÞds;ð1:8Þ

bðtÞ ¼ bðtÞ ðt > 0Þ0 ðta 0Þ;

�f ðx; tÞ ¼ f ðx; tÞ ðt > 0Þ

0 ðta 0Þ

�ð1:9Þ

for x A RN , t A R. We make the following assumptions.

Assumptions on f:

(H1) fðtÞ A C1ðð0;yÞÞ satisfies

fðtÞ > 0; ðfðtÞtÞ0 > 0 for t > 0;

(H2) there exist l;m A ð1;NÞ, lam < l� ð¼ Nl=ðN � lÞÞ, such that

lafðtÞt2FðtÞ am for t > 0;ð1:10Þ

237Positive Solutions of Quasilinear Elliptic Equations on RN

Assumptions on b:

(H3) bðtÞ A Cðð0;yÞÞ satisfies

bðtÞt A Cð½0;yÞÞ; limt!þ0

bðtÞt ¼ 0;

l�a

bðtÞt2BðtÞ am� ð¼ Nm=ðN �mÞÞ for t > 0;

(H4) there exist b0; b1 > 0 such that

b0 aBðtÞF�ðtÞ

a b1 for t > 0:

Assumptions on f :

(H5) f ðx; tÞ A CðRN � ½0;yÞÞ satisfies

f ðx; 0Þ ¼ 0 for x A RN ;

(H6) there exist r0; r1 > 0 and a nonnegative function gðxÞ A L1ðRNÞVLyðRNÞ such that

m

l� m� < r0 < m�; m < r1 < l�

and

jF ðx; tÞja gðxÞtr0 ð0a ta 1ÞgðxÞtr1 ðtb 1Þ

�for x A RN ;

(H7) there exists an open set W0 HRN such that

F ðx; tÞ > 0 for x A W0; t > 0;

(H8) there exists C > 0 such that

j f ðx; tÞtjaCjFðx; tÞj for x 2 RN ; tb 0:

Under assumptions (H1) and (H2), F, F�, ~FF and fF�F� are N-functions and

satisfy D2-condition (see Lemma 2.7 below). Let D1;FðRNÞ be a Banach space

obtained by the completion of Cy0 ðRNÞ with norm

jujD1;FðRN Þ ¼ kukF�þ k‘ukF:ð1:11Þ

Since the Orlicz-Sobolev inequality

kukF�aS0k‘ukFð1:12Þ

holds for u A D1;FðRNÞ with a constant S0 > 0, the norm (1.11) is equivalent to

the norm


kukD1;FðRN Þ ¼ k‘ukFð1:13Þ

on D1;FðRNÞ. In this paper, we will use (1.13) as the norm of D1;FðRNÞ.Now we state

Theorem 1.1. Suppose (H1)–(H8). Then there exists l0 > 0 such that

equation (1.1) has a nonnegative nontrivial (weak) solution u ¼ ul A D1;FðRNÞfor every l > l0.

Concerning the results obtained by using the arguments in Orlicz-Sobolev

spaces, see Badiale and Citti [5], and Clement, Garcıa-Huidobro, Manasevich

and Schmitt [11]. They have treated the equations in the subcritical case.

Finally, we give some examples satisfying (H1)–(H2).

( i ) FðtÞ ¼ tp0 þ tp1 , 1 < p0 < p1 < N,

( ii ) FðtÞ ¼ ð1þ t2Þg � 1, maxf1=2;N=ðN þ 2Þg < g < minfN=2;N=ðN � 2Þg,(iii) FðtÞ ¼ tp logð1þ tÞ, 1 < p < N � 1.

Throughout this paper we assume (H1)–(H8).

2. Preliminaries

In this section we state some preliminary inequalites on F. These inequal-

ites will be used in our arguments.

Lemma 2.1. Let z0ðtÞ ¼ minftl; tmg, z1ðtÞ ¼ maxftl; tmg, tb 0. Then

z0ðrÞFðtÞaFðrtÞa z1ðrÞFðtÞ for r; tb 0;ð2:1Þ

z0ðkukFÞaðRN

FðjujÞdxa z1ðkukFÞ for u A LFðRNÞ:ð2:2Þ

Proof. Integrating (1.10) implies (2.1). From this and the definition of

the norm (1.3), we have (2.2). r

Lemma 2.2. Let z2ðtÞ ¼ minftl�; tm

�g, z3ðtÞ ¼ maxftl �; tm

�g, tb 0. Then

z2ðrÞF�ðtÞaF�ðrtÞa z3ðrÞF�ðtÞ for r; tb 0;ð2:3Þ

z2ðkukF�Þa

ðRN

F�ðjujÞdxa z3ðkukF�Þ for u A LF� ðRNÞ:ð2:4Þ

Proof. In the same way as in the preceeding lemma, (2.4) follows from

(2.3). Putting t ¼ FðtÞ, s ¼ z0ðrÞ in (2.1) implies

z�11 ðsÞF�1ðtÞaF�1ðstÞa z�1

0 ðsÞF�1ðtÞ for s; t > 0:

By the definition of the conjugate function (1.6),


z�11 ðsÞs�1=NF�1

� ðtÞaF�1� ðstÞa z�1

0 ðsÞs�1=NF�1� ðtÞ:

This implies (2.3). r

Lemma 2.3. FðtÞ increases essentially more slowly than F�ðtÞ near infinity,

i.e.,

limt!y

FðktÞF�ðtÞ

¼ 0ð2:5Þ

for every constant k > 0.

Proof. By (2.1) and (2.3),

0aFðktÞF�ðtÞ

aFðkÞz1ðtÞF�ð1Þz2ðtÞ

¼ FðkÞtmF�ð1Þtl

�

for tb 1. Since m < l�, we have (2.5). r

Lemma 2.4. (i) (2.1) is equivalent to (1.10).

(ii) (2.3) is equivalent to

l�a

F 0�ðtÞt

F�ðtÞam� for t > 0:ð2:6Þ

Proof. (i) It is su‰cient to show that (2.1) implies (1.10). By (2.1), we

easily see that

z0ðrÞ � z0ð1Þr� 1

FðtÞa FðrtÞ �FðtÞr� 1

az1ðrÞ � z1ð1Þ

r� 1FðtÞ

for r > 1, t > 0. Letting r ! 1þ 0, we have (1.10). (ii) is shown in the same

way. r

Lemma 2.5. Let ~FF be the complement of F and put

z4ðsÞ ¼ minfsl=ðl�1Þ; sm=ðm�1Þg; z5ðsÞ ¼ maxfsl=ðl�1Þ; sm=ðm�1Þg:

Then the following inequalities hold.

m

m� 1~FFðsÞa ~FF 0ðsÞsa l

l� 1~FFðsÞ for sb 0;ð2:7Þ

z4ðrÞ ~FFðsÞa ~FFðrsÞa z5ðrÞ ~FFðsÞ for r; sb 0;ð2:8Þ

z4ðkuk ~FFÞaðRN

~FFðjujÞdxa z5ðkuk ~FFÞ for u A L ~FFðRNÞ:ð2:9Þ

Proof. By ~~FF~FF ¼ F and (A.8) in Appendix (replacing F with ~FFÞ, we have

Fð ~FF 0ðsÞÞ ¼ ~~FF~FFð ~FF 0ðsÞÞ ¼ ~FF 0ðsÞs� ~FFðsÞ:ð2:10Þ


Di¤erentiating (2.10) and noting ~FF 00ðsÞ0 0, we have

F 0ð ~FF 0ðsÞÞ ¼ s:

Putting t ¼ ~FF 0ðsÞ in (1.10), we obtain

lFð ~FF 0ðsÞÞa ~FF 0ðsÞsamFð ~FF 0ðsÞÞ:ð2:11Þ

Substituting (2.10) in (2.11), we have (2.7). From (2.7) we obtain (2.8) and

(2.9). r

In the same way, we have

Lemma 2.6. Let fF�F� be the complement of F� and put

z6ðsÞ ¼ minfsl �=ðl ��1Þ; sm�=ðm ��1Þg; z7ðsÞ ¼ maxfsl�=ðl ��1Þ; sm

�=ðm ��1Þg:

Then the following inequalities hold.

m�

m� � 1fF�F�ðsÞafF�F�

0ðsÞsa l�

l� � 1fF�F�ðsÞ for sb 0;ð2:12Þ

z6ðrÞfF�F�ðsÞafF�F�ðrsÞa z7ðrÞfF�F�ðsÞ for r; sb 0;ð2:13Þ

z6ðkukeF�Þa

ðRN

fF�F�ðjujÞdxa z7ðkukeF�Þ for u A L eF�

ðRNÞ:ð2:14Þ

The inequalities (2.1), (2.3), (2.8) and (2.13) imply

Lemma 2.7. F, F�, ~FF and fF�F� satisfy D2-condition.

Remark 2.8. From (2.3) and (2.13), the inequalities

F�ðtÞbP0tm �

for 0a ta 1;ð2:15Þ

F�ðtÞbP0tl �

for tb 1:ð2:16Þ

fF�F�ðtÞaP1tm �=ðm ��1Þ for 0a ta 1;ð2:17Þ

fF�F�ðtÞaP1tl �=ðl��1Þ for tb 1:ð2:18Þ

hold with P0 ¼ F�ð1Þ, P1 ¼ fF�F�ð1Þ.

3. Mountain pass conditions and Palais-Smale sequences

Now we consider the variational problem of a functional

IlðuÞ ¼ðRN

fFðj‘ujÞ � BðuÞ � lFðx; uÞgdxð3:1Þ


on D1;FðRNÞ. This functional is well defined and Frechet di¤erentiable (see

Appendix). The derivative I 0lðuÞ is given by

hI 0lðuÞ; vi ¼ðRN

ffðj‘ujÞ‘u � ‘v� ðbðuÞuþ lf ðx; uÞÞvgdxð3:2Þ

for v A D1;FðRNÞ. Thus nonnegative critical points of Il are weak solutions of

(1.1).

Prior to giving the proof of Theorem 1.1, recall the Ambrosetti-Rabinowitz

mountain pass lemma without Palais-Smale condition.

Lemma 3.1. Let I be a C 1-function on a Banach space E. Suppose there

exist a neighborhood U of 0 in E and a constant a which satisfy the following:

( i ) IðuÞb a on the boundary of U,

( ii ) Ið0Þ < a,

(iii) there exists a w0 cU satisfying Iðw0Þ < a.

Set

G ¼ fg A Cð½0; 1�;EÞ; gð0Þ ¼ 0; gð1Þ ¼ w0gð3:3Þ

and

c ¼ infg AG

maxw A g

IðwÞ ðb aÞ:ð3:4Þ

Then exists a sequence fung in E such that IðunÞ ! c and I 0ðunÞ ! 0 in E 0.

A proof of this lemma is given in Aubin and Ekeland [4, p. 272, Theorem

5], which relies on Ekeland’s minimization principle. The sketch of the proof is

written in Brezis [8, Lemma 7].

Here we verify that Lemma 3.1 is applicable in our situation, namely the

functional Il on D1;FðRNÞ satisfies the hypotheses (i), (ii), (iii). First note

Ilð0Þ ¼ 0 so (ii) is satisfied for any small a > 0.

Lemma 3.2. For any l > 0 there exists r0 ¼ r0ðlÞ > 0 such that

IlðuÞ > 0 for any u A D1;FðRNÞ with k‘ukF ¼ rð3:5Þ

for 0 < r < r0.

Proof. Let u A D1;FðRNÞ with r ¼ k‘ukF < minf1=S0; 1g. Then, by (2.2),

(H4), (A.4) and (1.12), we have


IlðuÞb z0ðk‘ukFÞ � b1z3ðkukF�Þ � lM0z3ðkukF�

Þr0=m�

� lM1z3ðkukF�Þr1=l

�

b z0ðk‘ukFÞ � b1z3ðS0k‘ukFÞ � lM0z3ðS0k‘ukFÞr0=m

�

� lM1z3ðS0k‘ukFÞr1=l

�

¼ rm � b1ðS0rÞl�� lM0ðS0rÞl

�r0=m�� lM1ðS0rÞr1 :

By assumptions (H2) and (H6),

m < l�; m <l�r0m� ; m < r1;

we see (3.5) for su‰ciently small r > 0. r

Lemma 3.3. Let W0 HRN be the open set in (H7) and u0 A Cy0 ðRNÞ be a

function satisfying

u0 b 0; u0 0 0; supp u0 HW0:

Then there exists t0 ¼ t0ðu0Þ > 0 such that

Ilðt0u0Þ < 0 for any l > 0:ð3:6Þ

Proof. By (H4), (H7), Lemma 2.1 and Lemma 2.2, we have

Ilðtu0Þ ¼ðRN

fFðtj‘u0jÞ � Bðtu0Þ � lF ðx; tu0Þgdxð3:7Þ

a z1ðtÞðRN

Fðj‘u0jÞdx� b0z2ðtÞðRN

F�ðju0jÞdx

¼ tmðRN

Fðj‘u0jÞdx� b0tl�ðRN

F�ðju0jÞdx

for tb 1. Since m < l�, we obtain (3.6) for some t0 ¼ t0ðu0Þ > 0. r

The preceeding lemmas show that, for l > 0, Il satisfies the assumptions

in Lemma 3.1. Thus there exists a Palais-Smale sequence fungHD1;FðRNÞsuch that

IlðunÞ ! cl and I 0lðunÞ ! 0 in D1;FðRNÞ0ð3:8Þ

as n ! y, where cl is the constant c defined by (3.4) for the functional Il, the

neighborhood U ¼ Ur and w0 ¼ t0u0. Let us consider the sequence fungin order to obtain a critical point of the functional Il. Now we show the

boundedness of this sequence in D1;FðRNÞ.


Lemma 3.4. For t > 0 there exist M2;M3 > 0 such thatðRN

F ðx; uÞ � 1

tf ðx; uÞu

�� dxð3:9Þ

aM2

ðRN

F�ðuþÞdx� �r0=m �

þM3

ðRN

F�ðuþÞdx� �r1=l �

for u A LF� ðRNÞ.

Proof. By (H8) and (A.7), it su‰ces to put M2 ¼ ð1þ C=tÞM0 and

M3 ¼ ð1þ C=tÞM1. r

Lemma 3.5. The sequence fungHD1;FðRNÞ satisfying (3.8) is bounded in

D1;FðRNÞ.

Proof. Take a constant t with m < t < l�. Then

IlðunÞ �1

thI 0lðunÞ; unið3:10Þ

¼ðRN

Fðj‘unjÞ �1

tfðj‘unjÞj‘unj2

� �dx

�ðRN

BðunÞ �1

tbðunÞu2n

� �dx

� l

ðRN

F ðx; unÞ �1

tf ðx; unÞun

� �dx

b 1�m

t

� � ðRN

Fðj‘unjÞdxþ b0l�

t� 1

� �ðRN

F�ððunÞþÞdx

� lM2

ðRN

F�ððunÞþÞdx� �r0=m �

� lM3

ðRN

F�ððunÞþÞdx� �r1=l �

:

Since r0=m� < 1 and r1=l

� < 1, the following function

hðtÞ1 b0l�

t� 1

� �t� lM2t

r0=m� � lM3t

r1=l�ð3:11Þ

is bounded below for tb 0, that is,

H1 inftb0

hðtÞ > �y:ð3:12Þ

By (3.10), (3.11), (3.12),

IlðunÞ �1

thI 0lðunÞ; unib 1�m

t

� �ðRN

Fðj‘unjÞdxþH:ð3:13Þ


On the other hand, by (3.8),

IlðunÞ �1

thI 0lðunÞ; unia IlðunÞ þ

1

tkI 0lðunÞkD1;FðRN Þ 0k‘unkFð3:14Þ

a c1 þ c2k‘unkF:

with some constants c1; c2 > 0. Therefore, by (3.13) and (3.14),

1�m

t

� �z0ðk‘unkFÞ þHa c1 þ c2k‘unkF:

This implies that fk‘unkFg is bounded. r

By (1.12), (2.2) and (2.4), we have

Corollary 3.6. The following sequences

fkunkF�g;

ðRN

Fðj‘unjÞdx� �

and

ðRN

F�ðjunjÞdx� �

are bounded.

4. Convergence of the Palais-Smale sequence

Let fungHD1;FðRNÞ be the Palais-Smale sequence obtained in the pre-

ceding section. Lemma 2.7 implies that the Orlicz spaces LFðRNÞ, LF� ðRNÞ,L ~FFðRNÞ, LeF�

ðRNÞ and D1;FðRNÞ are reflexive Banach spaces. In view of

Lemma 3.5 and Corollary 3.6, we may assume that there exist a function

u A D1;FðRNÞ and nonnegative n; m A MðRNÞ, the space of Radon measures,

such that

un * u weakly in LF� ðRNÞð4:1Þ

‘un * ‘u weakly in LFðRNÞð4:2Þ

F�ðjunjÞ * n weakly in MðRNÞð4:3Þ

Fðj‘unjÞ * m weakly in MðRNÞð4:4Þ

as n ! y. By Theorem 8.35 in [1, p. 284], for any N-function C increasing

essentially more slowly than F� near infinity, there is a subsequence of fung (still

denoting fung) such that

un ! u in LCðWÞð4:5Þ

for any bounded domain WHRN . Thus, by Lemma 2.3, we may also assume

that


un ! u in LFðWÞð4:6Þ

for any bounded domain WHRN and

un ! u a:e: in RN :ð4:7Þ

Lemma 4.1. Suppose that u ¼ 0 in (4.1)–(4.7). ThenðRN

Fðj‘ðjunÞjÞdx ¼ðRN

Fðjj‘unjÞdxþ oð1Þ as n ! yð4:8Þ

for any j A Cy0 ðRNÞ.

Proof. By (1.5),ðRN

Fðjj‘un þ un‘jjÞdx�ðRN

Fðjj‘unjÞdxð4:9Þ

¼ðRN

ð10

d

dtFðjj‘un þ tun‘jjÞdtdx

¼ðRN

ð10

fðjj‘un þ tun‘jjÞðj‘un þ tun‘jÞ � ðun‘jÞdtdx

a

ðRN

fðjj‘unj þ jun‘jjÞðjj‘unj þ jun‘jjÞjun‘jjdx

a 2kfðjj‘unj þ jun‘jjÞðjj‘unj þ jun‘jjÞk ~FFkun‘jkF

and, by (A.8) and (2.1),ðRN

~FFðfðjj‘unj þ jun‘jjÞðjj‘unj þ jun‘jjÞÞdxð4:10Þ

a

ðRN

Fð2ðjj‘unj þ jun‘jjÞÞdx

a z1ðk2ðjj‘unj þ jun‘jjÞkFÞ

a z1ð2ðkjkyk‘unkF þ kun‘jkFÞÞ:

By (4.6), we have kun‘jkF ! 0 as n ! y. Thus, by (4.9) and (4.10), the

equation (4.8) is obtained. r

The next Lemma is analogous to Lemma I.1 (the second concentration-

compactness lemma) of P. L. Lions [19].

Lemma 4.2. (i) There exist an at most countable set J, a family fxjgj A J of

distinct points in RN and a family fnjgj A J of constants nj > 0 such that


n ¼ F�ðjujÞ þXj A J

nj dxjð4:11Þ

where dxj is the Dirac measure of mass 1 concentrated at xj.

(ii) In addition we have

mbFðj‘ujÞ þXj A J

mj dxjð4:12Þ

for some mj > 0 satisfying

0 < nj amaxfS l �

0 ml �=lj ;Sm �

0 mm �=lj ;S l�

0 ml �=mj ;Sm �

0 mm �=mj gð4:13Þ

for all j A J.

Proof. Let fxjgj A J HRN be the atoms of the measure m and decompose

m ¼ mfree þXj A J

mj dxjð4:14Þ

with mfree A MðRNÞ free of atoms. Here, mb 0 implies that mfree b 0 and

mj ¼ mðfxjgÞ > 0. Since

mðRNÞa lim supn!y

ðRN

Fðj‘unjÞdx < y;ð4:15Þ

J is an at most countable set.

The case u ¼ 0. Let j A Cy0 ðRNÞ be a function such that 0a ja 1. By

(2.4), (1.12), (2.2) and (4.8),ðRN

F�ðjjunjÞdxa z3ðkjunkF�Þa z3ðS0k‘ðjunÞkFÞð4:16Þ

a z3 S0z�10

ðRN

Fðj‘ðjunÞjÞdx� ��

¼ z3 S0z�10

ðRN

Fðjj‘unjÞdxþ oð1Þ� ��

:

By (2.1), ðRN

Fðjj‘unjÞdxaðRN

z1ðjÞFðj‘unjÞdx ¼ðRN

jlFðj‘unjÞdxð4:17Þ

and, by (2.3),ðRN

F�ðjjunjÞdxbðRN

z2ðjÞF�ðjunjÞdx ¼ðRN

jm �F�ðjunjÞdx:ð4:18Þ


Combining (4.16), (4.17), and (4.18), we haveðRN

jm �F�ðjunjÞdxa z3 S0z

�10

ðRN

jlFðj‘unjÞdxþ oð1Þ� ��

:

Letting n ! y and using (4.3), (4.4), we haveðRN

jm �dna z3 S0z

�10

ðRN

jl dm

� �� :

By approximation,

nðAÞa z3ðS0z�10 ðmðAÞÞÞ for any Borel set AHRN :ð4:19Þ

Thus n is absolutely continuous with respect to m and

nðAÞ ¼ðA

Dmn dmð4:20Þ

where

DmnðxÞ ¼ lime!þ0

nðBeðxÞÞmðBeðxÞÞ

for m-a:e: x A RNð4:21Þ

Here BeðxÞ is a ball with radius e centered at x. From (4.19),

nðBeðxÞÞmðBeðxÞÞ

az3ðS0z

�10 ðmðBeðxÞÞÞÞmðBeðxÞÞ

provided mðBeðxÞÞ0 0. Since z�10 ðtÞ ¼ maxft1=l; t1=mg and z3ðtÞ ¼

maxftl �; tm

�g imply

z3ðS0z�10 ðtÞÞ ¼ maxfS l �

0 tl�=l;Sm �

0 tm�=l;S l �

0 tl�=m;Sm �

0 tm�=mg;

limt!þ0

z3ðS0z�10 ðtÞÞt

¼ 0:

Thus, we have

DmnðxÞ ¼ 0 for m-a:e: x A RNnfxjgj A J :ð4:22Þ

Put nj ¼ DmnðxjÞmj . Then, by (4.20), (4.22) and (4.14),

n ¼Xj A J

nj dxj ; mbXj A J

mj dxj :

Further, from (4.19),

nðBeðxjÞÞa z3ðS0z�10 ðmðBeðxjÞÞÞÞ:


Letting e ! þ0, we have

0a nj a z3ðS0z�10 ðmjÞÞ

¼ maxfS l �

0 ml �=lj ;Sm �

0 mm �=lj ;S l�

0 ml�=mj ;Sm �

0 mm �=mj g:

Finally, excluding the index j of nj ¼ 0 from the index set J, the Lemma for the

case u ¼ 0 is proved.

The case u0 0. Let vn ¼ un � u A D1;FðRNÞ. Then

vn * 0 weakly in LF� ðRNÞð4:23Þ

‘vn * 0 weakly in LFðRNÞð4:24Þ

vn ! 0 in LFðWÞ for any bounded domain WHRNð4:25Þ

vn ! 0 a:e:ð4:26Þ

as n ! y. Since

k‘vnkF a k‘unkF þ k‘ukF; kvnkF�a kunkF�

þ kukF�;

the sequences fk‘vnkFg and fkvnkF�g are bounded. Therefore, by (2.2)

and (2.4), fÐRN Fðj‘vnjÞdxg and f

ÐRN F�ðjvnjÞdxg are bounded. By taking a

subsequence, we may assume that there exist nonnegative Radon measures nn,

mm A MðRNÞ such that

F�ðjvnjÞ * nn weakly in MðRNÞð4:27Þ

Fðj‘vnjÞ * mm weakly in MðRNÞð4:28Þ

as n ! y. From the above case u ¼ 0, there exist an at most countable set J,

a family fxjgj A J of distinct points in RN and positive weights fnjgj A J , fmjgj A Jsuch that

nn ¼Xj A J

nj dxj ; mmbXj A J

mj dxj :

Since F�ðtÞ, FðtÞ are convex functions and un ! u a.e., Theorem 2 and Lemma

3 of Brezis and Lieb [9] imply thatðRN

fF�ðjunjÞ �F�ðjun � ujÞ �F�ðjujÞgj dx ! 0

for j A C0ðRNÞ, jb 0. Hence, by (4.3), (4.4), (4.27) and (4.28), we haveðRN

j dn ¼ðRN

j d nnþðRN

F�ðjujÞj dx;


for any j ¼ jþ � j� A C0ðRNÞ. This proves (4.11). For any j A C0ðRNÞ,ðRN

Fðj‘unjÞj dx�ðRN

Fðj‘vnjÞj dx

¼ðRN

ð10

fðj‘un � t‘ujÞð‘un � t‘uÞ � ‘uj dtdx:

By (A.14), ðRN

~FFðfðj‘un � t‘ujÞj‘un � t‘ujÞdx

a z1ð2ÞðRN

Fðj‘un � t‘ujÞdx

az1ð2Þ2

2

ðRN

Fðj‘unjÞdxþðRN

Fðj‘ujÞdx� �

for any t A ð0; 1Þ. Thus fÐ 10 fðj‘un � t‘ujÞð‘un � t‘uÞdtg is bounded in

L ~FFðRN ;RNÞ, and there exists a subsequence which converges to some w in

L ~FFðRN ;RNÞ. Taking a limit, we have

ðRN

j dm�ðRN

j dmm ¼ðRN

w � ‘uj dx

for any j A C0ðRNÞ. Hence the atoms of m coincides with those of mm. By

weak lower semicontinuity, we have

mfree bFðj‘ujÞ:

Hence (4.12) holds. r

In order to show ‘un ! ‘u a.e. in RN we give a series of lemmas.

Lemma 4.3. The set fxjgj A J in Lemma 4.2 is a finite set.

Proof. Let an xj be fixed. Take c A Cy0 ðRNÞ such that

0aca 1; cðxÞ ¼ 1 ðjxja 1Þ0 ðjxjb 2Þ

�

and put ceðxÞ ¼ cððx� xjÞ=eÞ for e > 0. Then fceung is bounded in D1;FðRNÞwith respect to n. Since I 0lðunÞ ! 0, we have


ðRN

fðj‘unjÞ‘un � ‘ðceunÞdxð4:29Þ

¼ðRN

bðunÞu2nce dxþ l

ðRN

f ðx; unÞunce dxþ oð1Þ

a b1m�ðRN

F�ðjunjÞce dxþ lC

ðRN

jF ðx; unÞjce dxþ oð1Þ

as n ! y. By replacing dx in (A.7) with ce dx, the same calculation implies

that ðRN

jFðx; uÞjce dxð4:30Þ

aM4

ðRN

F�ðjujÞce dx

� �r0=m �

þM5

ðRN

F�ðjujÞce dx

� �r1=l�

for u A LF� ðRNÞ, where

M4 ¼ P�r0=m

�

0

ðRN

jgjm�=ðm ��r0Þce dx

� �ðm ��r0Þ=m �

aM0;

M5 ¼ P�r1=l

�

0

ðRN

jgjl�=ðl ��r1Þce dx

� �ðl ��r1Þ=l �

aM1:

On the other hand, by (H2),ðRN

fðj‘unjÞ‘un � ‘ðceunÞdxð4:31Þ

¼ðRN

fðj‘unjÞj‘unj2ce dxþðRN

fðj‘unjÞð‘un � ‘ceÞun dx

b l

ðRN

Fðj‘unjÞce dxþðRN

fðj‘unjÞð‘un � ‘ceÞun dx:

By (A.14) and (2.9), the sequence fkfðj‘unjÞ‘unk ~FFg is bounded. Thus, there is

a subsequence fung such that

fðj‘unjÞ‘un * w1 weakly in L ~FFðRN ;RNÞ

for some w1 A L ~FFðRN ;RNÞ. Since suppð‘ceÞHB2eðxjÞ and un ! u in

LFðB2eðxjÞÞ, ðRN

fðj‘unjÞð‘un � ‘ceÞun dx !ðRN

ðw1 � ‘ceÞu dx

as n ! y. Thus, combining (4.29), (4.30), (4.31) and letting n ! y, we have


ð4:32Þ l

ðRN

ce dmþðRN

ðw1 � ‘ceÞu dx

a b1m�ðRN

ce dnþ lCM4

ðRN

ce dn

� �r0=m �

þ lCM5

ðRN

ce dn

� �r1=l�

:

Now we show that the second term of the left-hand side converges 0 as

e ! 0. The sequence fbðunÞun þ lf ðx; unÞg is bounded in L eF�ðRNÞ by (A.15),

(A.18) and (2.14). Thus there is a subsequence fung such that

bðunÞun þ lf ðx; unÞ * w2 weakly in L eF�ðRNÞ

for some w2 A L eF�ðRNÞ. Since

hI 0lðunÞ; vi ¼ðRN

ffðj‘unjÞ‘un � ‘v� ðbðunÞun þ lf ðx; unÞÞvgdx

! 0

as n ! y for any v A D1;FðRNÞ,ðRN

ðw1 � ‘v� w2vÞdx ¼ 0

for any v A D1;FðRNÞ. Substituting v ¼ uce, we haveðRN

fw1 � ‘ðuceÞ � w2ucegdx ¼ 0:

Namely, ðRN

ðw1 � ‘ceÞu dx ¼ �ðRN

ðw1 � ‘u� w2uÞce dx:

Noting w1 � ‘u� w2u A L1ðRNÞ, we see that the right-hand side tends to 0 as

e ! 0. Hence we have ðRN

ðw1 � ‘ceÞu dx ! 0

as e ! 0.

Letting e ! 0 in (4.32), we obtain

lmj a b1m�nj þ lCM4n

r0=m�

j þ lCM5nr1=l

�

j :ð4:33Þ

Since

l�

m> 1;

l�r0mm� > 1;

r1

m> 1;


from (4.13) and (4.33), we see that inffnj ; j A Jg > 0. SinceP

j A J nj < y, the

Lemma is proved. r

Lemma 4.4. Let KHRNnfxjgj A J be a compact set. Then

un ! u strongly in LF� ðKÞð4:34Þ

as n ! y.

Proof. Put d ¼ distðK ; fxjgj A JÞ > 0. Choose R > 0 such that KHBRð0Þand let Ae ¼ fx A BRð0Þ; distðx;KÞ < eÞg for 0 < e < d. Take we A Cy

0 ðRNÞ suchthat

0a we a 1; weðxÞ ¼1 ðx A Ae=2Þ0 ðx A RNnAeÞ:

�Since

KHAe=2 HAe H ðRNnfxjgj A JÞVBRð0Þ

for 0 < e < d, we haveðK

F�ðjunjÞdxaðAe

weF�ðjunjÞdx ¼ðRN

weF�ðjunjÞdx:

Thus, from (4.11),

lim supn!y

ðK

F�ðjunjÞdxaðRN

we dn ¼ðRN

weF�ðjujÞdx

a

ðAe

F�ðjujÞdx:

Letting e ! þ0, by Lebesgue’s convergence theorem,

lim supn!y

ðK

F�ðjunjÞdxaðK

F�ðjujÞdx:

On the other hand, Fatou’s lemma impliesðK

F�ðjujÞdxa lim infn!y

ðK

F�ðjunjÞdx:

Thus we have

limn!y

ðK

F�ðjunjÞdx ¼ðK

F�ðjujÞdx:ð4:35Þ

Moreover, Lemma 3 of Brezis and Lieb [9] implies that


ðK

fF�ðjunjÞ �F�ðjun � ujÞ �F�ðjujÞgdx ! 0:ð4:36Þ

Hence, by (4.35) and (4.36),ðK

F�ðjun � ujÞdx ! 0:

This shows (4.34). r

Lemma 4.5. Let KHRNnfxjgj A J be a compact set. ThenðK

ðfðj‘unjÞ‘un � fðj‘ujÞ‘uÞ � ð‘un � ‘uÞdx ! 0ð4:37Þ

as n ! y.

Proof. Let w be a function in Cy0 ðRNÞ such that

0a wa 1; w ¼ 1 on K ; supp wV fxjgj A J ¼ q:

Put W ¼ fx A RN ; wðxÞ > 0g. Since

ðfðj‘unjÞ‘un � fðj‘ujÞ‘uÞ � ð‘un � ‘uÞb 0

on RN , we have

0a

ðK

ðfðj‘unjÞ‘un � fðj‘ujÞ‘uÞ � ð‘un � ‘uÞdxð4:38Þ

a

ðRN

ðfðj‘unjÞ‘un � fðj‘ujÞ‘uÞ � ð‘un � ‘uÞw dx

¼ðRN

fðj‘unjÞ‘un � ð‘un � ‘uÞw dx

�ðRN

fðj‘ujÞ‘u � ð‘un � ‘uÞw dx:

By Lemma 3.5, the sequence fðun � uÞwg is bounded in D1;FðRNÞ. Hence (3.8)

implies that

hI 0lðunÞ; ðun � uÞwi ! 0ð4:39Þ

as n ! y. Here,

hI 0lðunÞ; ðun � uÞwi ¼ðRN

fðj‘unjÞ‘un � ‘ððun � uÞwÞdxð4:40Þ

�ðRN

ðbðunÞun þ lf ðx; unÞÞðun � uÞw dx:


By using (A.15) and (A.18), we easily see that kbðunÞunkeF�and k f ðx; unÞkeF�

,

respectively, are bounded. Hence, by Lemma 4.4,ðRN

ðbðunÞun þ lf ðx; unÞÞðun � uÞw dx

�� ð4:41Þ

¼ðW

ðbðunÞun þ lf ðx; unÞÞðun � uÞw dx

�� a kbðunÞun þ lf ðx; unÞkeF�;W

kðun � uÞwkF�;W

a ðkbðunÞunkeF�þ lk f ðx; unÞkeF�

Þkun � ukF�;WkwkLyðRN Þ ! 0

as n ! y. Thus, by (4.39), (4.40) and (4.41), we haveðRN

fðj‘unjÞ‘un � ‘ððun � uÞwÞdx ! 0:ð4:42Þ

Moreover, by (4.6) and (A.14),ðRN

fðj‘unjÞð‘un � ‘wÞðun � uÞdx�� ð4:43Þ

a kfðj‘unjÞ‘unk ~FFkun � ukF;Wk‘wkLyðRN Þ ! 0:

Hence, (4.42) and (4.43) impliesðRN

fðj‘unjÞ‘un � ð‘un � ‘uÞw dx ! 0:ð4:44Þ

Note that (A.14) yields fðj‘ujÞ‘uw A L ~FFðRN ;RNÞ. By (4.2),ðRN

fðj‘ujÞ‘u � ð‘un � ‘uÞw dx ! 0:ð4:45Þ

Therefore, by (4.38), (4.44), and (4.45), we obtain (4.37). r

Corollary 4.6. There exists a subsequence, still denoted by fung, of the

Palais-Smale sequence fung in Section 2 such that

‘un ! ‘u a:e: in RNð4:46Þ

as n ! y.

Proof. Let K be a set in Lemma 4.5. There exists a subsequence of the

integrand in (4.37) converges almost everywhere on K . Using Lemma 6 in Dal

Maso and Murat [12], we have ‘un ! ‘u a.e. x A K . Since K is an arbitrary

compact set in RNnfxjgj A J , the conclusion follows. r


Applying the results stated above, we show the limit function u solves the

equation (1.1).

Lemma 4.7. Let j A Cy0 ðRNÞ. Thenð

RN

fðj‘unjÞ‘un � ‘j dx !ðRN

fðj‘ujÞ‘u � ‘j dxð4:47Þ

ðRN

bðunÞunj dx !ðRN

bðuÞuj dxð4:48Þ

ðRN

f ðx; unÞj dx !ðRN

f ðx; uÞj dxð4:49Þ

as n ! y.

Proof. Put vn ¼ fðj‘unjÞ‘un � ‘j. Take an R > 0 such that supp jHBRð0Þ. By (H1) and (2.1), we have

jfðtÞtj j‘jjamFðtÞt

k‘jkLyðRN Þ a c1tm�1; tb 1;

for a constant c1 ¼ mFð1Þk‘jkLyðRN Þ > 0. Hence, for ab

maxfc1; fð1Þk‘jkLyðRN Þg,

fx A RN ; jvnjb agH fx A RN ; j‘unjb ða=c1Þ1=ðm�1Þgand ð

fx ARN ; jvnjbagjvnjdxa

ðfx ARN ; j‘unjbða=c1Þ1=ðm�1Þg

jfðj‘unjÞ‘unj j‘jjdx

ac1

Fð1Þ

ðfx ARN ; j‘unjbða=c1Þ1=ðm�1Þg

Fðj‘unjÞj‘unj

dx

ac1

Fð1Þða=c1Þ1=ðm�1Þ

ðRN

Fðj‘unjÞdx:

Noting that fÐRN Fðj‘unjÞdxg is bounded, we see thatð

fx ARN ; jvnjbagjvnjdx ! 0ð4:50Þ

uniformly in n as a ! y. This means that fvng is uniformly integrable on RN

(and also on BRð0Þ). Moreover, Corollary 4.6 implies that vn ¼ fðj‘unjÞ‘un �‘j ! fðj‘ujÞ‘u � ‘j a.e on BRð0Þ as n ! y. Therefore Vitali’s convergence

theorem implies that fvng converges to fðj‘ujÞ‘u � ‘j in L1ðBRð0ÞÞ. This

shows (4.47). The convergence of (4.48) and (4.49) can be proved similarly.

r


Proposition 4.8. The limit function u is a weak solution of the equation

�divðfðj‘ujÞ‘uÞ ¼ bðuÞuþ lf ðx; uÞð4:51Þ

in D1;FðRNÞ.

Proof. Take a j A Cy0 ðRNÞ. Then

hI 0lðunÞ; ji ¼ðRN

ffðj‘unjÞ‘un � ‘j� ðbðunÞun þ lf ðx; unÞÞjgdx:

By (3.8) and Lemma 4.7, we haveðRN

ffðj‘ujÞ‘u � ‘j� ðbðuÞuþ lf ðx; uÞÞjgdx ¼ 0

for any j A Cy0 ðRNÞ. r

5. Proof of Theorem 1.1

Let u ¼ ul A D1;FðRNÞ be the limit of the Palais-Smale sequence fung in the

previous section.

Lemma 5.1. Let u0 A Cy0 ðRNÞ be as in Lemma 3.3. Then

maxtb0

Ilðtu0Þ ! 0 as l ! y:ð5:1Þ

Proof. Let l > 0. Since (3.7) implies that

Ilðtu0Þa c1tm � c2t

l�; tb 1;ð5:2Þ

for some constants c1; c2 > 0 independent of l, maxtb0 Ilðtu0Þ is attained at

some t ¼ tl A ð0;T0Þ with T0 ¼ maxf1; ðc2=c1Þ�1=ðl ��mÞg.Here we claim that tl ! 0 as l ! y. Indeed, if this is not true, then

there exists a sequence lj ! y and d0 > 0 such that d0 a tlj ðaT0Þ. Let

BH ðsupp u0Þ� be a closed ball in RN . Then

c3 1minfFðx; tu0ðxÞÞ; x A B; d0 a taT0g > 0:

Thus we have

maxtb0

Ilj ðtu0Þ ¼ Ilj ðtlj u0Þð5:3Þ

a z1ðtlj ÞðRN

Fðj‘u0jÞdx� b0z2ðtlj ÞðRN

F�ðju0jÞdx� ljc3 volðBÞ

! �y


as j ! y. On the other hand, Lemma 3.2 implies that

maxtb0

Ilðtu0Þ > 0

for any l > 0, which contradicts (5.3). Hence we have (5.1). r

Lemma 5.2. ðRN

F ðx; unÞdx !ðRN

F ðx; uÞdx;ð5:4Þ ðRN

f ðx; unÞun dx !ðRN

f ðx; uÞu dxð5:5Þ

as n ! y.

Proof. Let BRð0Þ be a ball in RN of radius R > 0. Put

A0;nR ¼ fx A RNnBRð0Þ; 0a unðxÞa 1g;ð5:6Þ

A1;nR ¼ fx A RNnBRð0Þ; unðxÞb 1g:ð5:7Þ

Then, by Remark 2.8, we haveðA

0; nR

jFðx; unÞjdxaðA

0; nR

gðxÞjunjr0dx

a

ðA

0; nR

jgðxÞjm�=ðm ��r0Þdx

!ðm ��r0Þ=m � ðA

0; nR

junjm�dx

!r0=m �

aP�r0=m

�

0

ðRNnBRð0Þ

jgðxÞjm�=ðm ��r0Þdx

!ðm ��r0Þ=m � ðRN

F�ðjunjÞdx� �r0=m �

and ðA1; n

R

jFðx; unÞjdxaðA1; n

R

gðxÞjunjr1dx

a

ðA

1; nR

jgðxÞjl�=ðl ��r1Þdx

!ðl��r1Þ=l� ðA

1; nR

junjl�dx

!r1=l �

aP�r1=l

�

0

ðRNnBRð0Þ

jgðxÞjl�=ðl ��r1Þdx

!ðl ��r1Þ=l� ðRN

F�ðjunjÞdx� �r1=l �

:

Noting that g A L1ðRNÞVLyðRNÞ and fÐRN F�ðjunjÞdxg is bounded, we see

that, for any e > 0, there exists R > 0 (independent of n) such that


ðRNnBRð0Þ

jF ðx; unÞjdx ¼ðA

0; nR

jFðx; unÞjdxþðA

1; nR

jFðx; unÞjdx < eð5:8Þ

for all n.

Next we show that fFðx; unÞg is uniformly integrable on BRð0Þ. Indeed,

by (H6), we have

jF ðx; tÞja kgkytr1 ; tb 1:

Hence, for ab kgky, we see that

fx A BRð0Þ; jFðx; unÞjb agH fx A BRð0Þ; junjb ða=kgkyÞ1=r1g:

Thus, ðfx ABRð0Þ; jFðx;unÞjbag

jF ðx; unÞjdx

a kgkyðfx ABRð0Þ; junjbða=kgkyÞ1=r1g

junjr1dx

akgkyP0

ðfx ABRð0Þ; junjbða=kgkyÞ1=r1g

F�ðjunjÞjunjl

��r1dx

akgky

P0ða=kgkyÞðl��r1Þ=r1

ðRN

F�ðjunjÞdx:

Since fÐRN F�ðjunjÞdxg is bounded, we haveð

fx ABRð0Þ; jF ðx;unÞjbagjF ðx; unÞjdx ! 0ð5:9Þ

uniformly in n as a ! y. Since fFðx; unÞg converges to Fðx; uÞ a.e. on RN , by

(5.8), (5.9) and Vitali’s convergence theorem, we have (5.4). The convergence

of (5.5) can be obtained similary. r

Proof of Theorem 1.1. It is su‰cient to prove that u ¼ ul satisfies ub 0

and u0 0. Let u� ¼ maxf�u; 0g. Then (4.51) implies that hI 0lðuÞ; u�i ¼ 0.

Observing

0 ¼ �ðRN

fðj‘ujÞ‘u � ‘u� dx ¼ðRN

fðj‘u�jÞj‘u�j2dx

b l

ðRN

Fðj‘u�jÞdxb lz0ðk‘u�kFÞ;

we have u� ¼ 0. Hence u ¼ ul b 0 and, by Proposition 4.8, ul is a nonnegative

(weak) solution of (1.1) for l > 0.


Put

M ¼ minflb=ðb�aÞS�ab=ðb�aÞ0 ðb1m�Þ�a=ðb�aÞ;ð5:10Þ

a ¼ l or m; b ¼ l� or m�g:

By Lemma 5.1, the cl in (3.8) satisfies

cl ¼ infg AG

maxw A g

IlðwÞ ! 0

as l ! y. Choose a constant l0 > 0 satisfying

0 < cl <1

m� 1

l�

� �Mð5:11Þ

for l > l0.

We show that ul 0 0 for l > l0 by contradiction. Let us assume that

ul ¼ 0. Since each of the three integrals in

hI 0lðunÞ; uni ¼ðRN

fðj‘unjÞj‘unj2dx�ðRN

bðunÞu2n dxð5:12Þ

� l

ðRN

f ðx; unÞun dx:

is bounded, we can assume that they converge as n ! y. Put

kl ¼ limn!y

ðRN

fðj‘unjÞj‘unj2dx:ð5:13Þ

Since ul ¼ 0, (5.5) implies

limn!y

ðRN

f ðx; unÞun dx ¼ 0:ð5:14Þ

Noting that hI 0lðunÞ; uni ! 0 as n ! y, by (5.12)–(5.14) we have

limn!y

ðRN

bðunÞu2n dx ¼ limn!y

ðRN

fðj‘unjÞj‘unj2dx ¼ kl:

Now we claim that kl > 0. Indeed, if we suppose that kl ¼ 0 then, by

(H2) and BðtÞb 0 for tb 0, we have

IlðunÞ ¼ðRN

fFðj‘unjÞ � BðunÞ � lF ðx; unÞgdxð5:15Þ

a1

l

ðRN

fðj‘unjÞj‘unj2dx� l

ðRN

Fðx; unÞdx:


Letting n ! y in (5.15), we have

0 < cl ¼ limn!y

IlðunÞakl

l� l

ðRN

Fðx; ulÞdx ¼ 0;

a contradiction. Thus kl > 0.

Next we show that kl bM > 0. In fact, by (H2) and Lemma 2.1, we have

z0ðk‘unkFÞaðRN

Fðj‘unjÞdxa1

l

ðRN

fðj‘unjÞj‘unj2dx:ð5:16Þ

Similarly, by (H3), (H4) and (2.4), we have

1

b1m�

ðRN

bðunÞu2n dxa

ðRN

F�ððunÞþÞdxa z3ðkunkF�Þ:ð5:17Þ

Combining (1.12), (5.16) and (5.17), we obtain

z�13

1

b1m�

ðRN

bðunÞu2n dxÞaS0z�10

1

l

ðRN

fðj‘unjÞj‘unj2dx� �

:

�Letting n ! y, we get

z�13

kl

b1m�

� �aS0z

�10

kl

l

� �:

From this inequality, we easily see kl bM with M > 0 given by (5.10).

Then, observe that

IlðunÞb1

m

ðRN

fðj‘unjÞj‘unj2dx� 1

l�

ðRN

bðunÞu2n dx� l

ðRN

F ðx; unÞdx:

Letting n ! y, we have

cl b1

m� 1

l�

� �kl b

1

m� 1

l�

� �M;

which contradicts (5.11). Thus ul 0 0 for l > l0. r

A. Appendix

For the functions FðtÞ, BðtÞ, F ðx; tÞ in Section 1, put

J1ðuÞ ¼ðRN

Fðj‘ujÞdx;ðA:1Þ

J2ðuÞ ¼ðRN

BðuÞdx;ðA:2Þ

J3ðuÞ ¼ðRN

F ðx; uÞdxðA:3Þ


for u A D1;FðRNÞ. Here we show that the functionals are well defined and

Frechet di¤erentiable on D1;FðRNÞ.

Lemma A.1. There exist M0;M1 > 0 such that

ðRN

jF ðx; uÞjdxaM0z3ðkuþkF�Þr0=m

�þM1z3ðkuþkF�

Þr1=l�

ðA:4Þ

for u A LF� ðRNÞ where uþ ¼ maxfu; 0g.

Proof. Let u A LF� ðRNÞ. Then, by (H6), Holder’s inequality, (2.3) and

Remark 2.8,ðfx ARN ;0aua1g

jFðx; uÞjdxaðfx ARN ;0aua1g

gðxÞur0 dxðA:5Þ

a kgkm �=ðm ��r0Þ

ðfx ARN ;0aua1g

um �dx

!r0=m �

aP�r0=m

�

0 kgkm �=ðm ��r0Þ

ðfx ARN ;0aua1g

F�ðuÞdx !r0=m �

aM0

ðRN

F�ðuþÞdx� �r0=m

�

and ðfx ARN ;ub1g

jF ðx; uÞjdxaðfx ARN ;ub1g

gðxÞur1 dxðA:6Þ

a kgkl �=ðl ��r1Þ

ðfx ARN ;ub1g

ul �dx

!r1=l �

aP�r1=l

�

0 kgkl�=ðl ��r1Þ

ðfx ARN ;ub1g

F�ðuÞdx !r1=l �

aM1

ðRN

F�ðuþÞdx� �r1=l

�

;

where we put M0 ¼ P�r0=m

�

0 kgkm �=ðm ��r0Þ, M1 ¼ P�r1=l

�

0 kgkl �=ðl ��r1Þ. By (A.5)

and (A.6), we have


ðRN

jFðx; uÞjdx ¼ðfx ARN ;ub0g

jFðx; uÞjdxðA:7Þ

aM0

ðRN

F�ðuþÞdx� �r0=m �

þM1

ðRN

F�ðuþÞdx� �r1=l�

for u A LF� ðRNÞ. By (2.4), inequality (A.4) is proved. r

Lemma A.2.

~FFðfðsÞsÞ ¼ fðsÞs2 �FðsÞaFð2sÞ for sb 0;ðA:8Þ

~FFFðsÞs

� �aFðsÞ for s > 0;ðA:9Þ

fF�F�F�ðsÞs

� �aF�ðsÞ for s > 0:ðA:10Þ

Proof. The convexity of FðtÞ implies

FðsÞ þF 0ðsÞðt� sÞaFðtÞ for s; tb 0

and, by using F 0ðsÞ ¼ fðsÞs,

fðsÞst�FðtÞa fðsÞs2 �FðsÞ for s; tb 0:

Hence,

~FFðfðsÞsÞ ¼ maxtb0

ðfðsÞst�FðtÞÞ

¼ fðsÞs2 �FðsÞa fðsÞs2 að2ss

fðtÞt dtaFð2sÞ

for sb 0. This shows (A.8). Since the convexity of FðtÞ and Fð0Þ ¼ 0 implies

that FðtÞ=t is increasing for t > 0, we have

FðsÞs

t�FðtÞa 0 for tb s > 0:

Therefore,

~FFFðsÞs

� �¼ max

tb0

FðsÞs

t�FðtÞ� �

¼ maxsbtb0

FðsÞs

t�FðtÞ� �

aFðsÞs

s ¼ FðsÞ

for s > 0. This shows (A.9). By the convexity of F�ðtÞ and F�ð0Þ ¼ 0, the

inequality (A.10) is proved similarly. r


Lemma A.3. J1ðuÞ, J2ðuÞ, J3ðuÞ are continuously Frechet di¤erentiable on

D1;FðRNÞ. The derivatives J 01ðuÞ, J 0

2ðuÞ, J 03ðuÞ are given by

hJ 01ðuÞ; vi ¼

ðRN

fðj‘ujÞ‘u � ‘v dx;ðA:11Þ

hJ 02ðuÞ; vi ¼

ðRN

bðuÞuv dx;ðA:12Þ

hJ 03ðuÞ; vi ¼

ðRN

f ðx; uÞv dxðA:13Þ

for u; v A D1;FðRNÞ.

Proof. Let u; v A D1;FðRNÞ. Then, by (1.5) and (1.12),ðRN

fðj‘ujÞ‘u � ‘v dx�� a 2kfðj‘ujÞ‘uk ~FFk‘vkF;ð

RN

bðuÞuv dx�� a 2kbðuÞukeF�

kvkF�a 2S0kbðuÞukeF�

k‘vkF;ðRN

f ðx; uÞv dx�� a 2k f ðx; uÞkeF�

kvkF�a 2S0k f ðx; uÞkeF�

k‘vkF:

By (A.8) and (2.1),ðRN

~FFðjfðj‘ujÞ‘ujÞdxaðRN

Fð2j‘ujÞdxðA:14Þ

a z1ð2ÞðRN

Fðj‘ujÞdx < y

and, by (2.9), kfðj‘ujÞ‘uk ~FF < y. It is easy to see that the Gateax di¤erential

of J1ðuÞ on D1;FðRNÞ is given by the right-hand side of (A.11). Hence, by

Theorem 17.2 and Theorem 17.3 in Krasnosel’skiı and Rutickiı [18, pp. 169–

170] (see also Lemma 18.2 in [18, p. 186]), J1ðuÞ is Frechet di¤erentiable and

J 01ðuÞ is continuous with respect to u A D1;FðRNÞ.

The continuous di¤erentiability of J2ðuÞ and J3ðuÞ on D1;FðRNÞ is shown

in a similar way. In fact, it is su‰cient to show that kbðuÞukeF�< y and

k f ðx; uÞkeF�< y. By (H3), (H4), (2.13) and (A.10),ð

RN

fF�F�ðjbðuÞujÞdxaðRN

fF�F� b1m� F�ðjujÞ

juj

� �dxðA:15Þ

a z7ðb1m�ÞðRN

F�ðjujÞdx < y;


which implies kbðuÞukeF�< y. Moreover, by (H6), (H8), (2.15), (2.17) and

Holder’s inequality,ðfx ARN ;0aua1g

fF�F�ðj f ðx; uÞjÞdxaðfx ARN ;0aua1g

fF�F�ðCgðxÞur0�1ÞdxðA:16Þ

a

ðfx ARN ;0aua1g

z7ðCgðxÞÞfF�F�ður0�1Þdx

a

ðfx ARN ;0aua1g

z7ðCgðxÞÞP1um �ðr0�1Þ=ðm ��1Þdx

aP1

ðfx ARN ;0aua1g

z7ðCgðxÞÞðm��1Þ=ðm ��r0Þ

!ðm ��r0Þ=ðm ��1Þ

�ðfx ARN ;0aua1g

um �dx

!ðr0�1Þ=ðm ��1Þ

a c1

ðRN

F�ðjujÞdx� �ðr0�1Þ=ðm ��1Þ

where

c1 ¼ P1

ðRN

z7ðCgðxÞÞðm��1Þ=ðm ��r0Þ

� �ðm ��r0Þ=ðm ��1ÞP�ðr0�1Þ=ðm ��1Þ0 < y:

Similarly, by (H6), (H8), (2.16), (2.18) and Holder’s inequality,ðfx ARN ;ub1g

fF�F�ðj f ðx; uÞjÞdxaðfx ARN ;ub1g

fF�F�ðCgðxÞur1�1ÞdxðA:17Þ

a

ðfx ARN ;ub1g

z7ðCgðxÞÞfF�F�ður1�1Þdx

a

ðfx ARN ;ub1g

z7ðCgðxÞÞP1ul�ðr1�1Þ=ðl ��1Þ dx

aP1

ðfx ARN ;ub1g

z7ðCgðxÞÞðl��1Þ=ðl ��r1Þ

!ðl ��r1Þ=ðl ��1Þ

�ðfx ARN ;ub1g

ul �dx

!ðr1�1Þ=ðl ��1Þ

a c2

ðRN

F�ðjujÞdx� �ðr1�1Þ=ðl ��1Þ


where

c2 ¼ P1

ðRN

z7ðCgðxÞÞðl��1Þ=ðl ��r1Þ

� �ðl��r1Þ=ðl ��1ÞP�ðr1�1Þ=ðl��1Þ0 < y:

Hence, by (1.9), (A.16) and (A.17), we haveðRN

fF�F�ðj f ðx; uÞjÞdx ¼ðfx ARN ;ub0g

fF�F�ðj f ðx; uÞjÞdxðA:18Þ

a c1

ðRN

F�ðjujÞdx� �ðr0�1Þ=ðm ��1Þ

þ c2

ðRN

F�ðjujÞdx� �ðr1�1Þ=ðl ��1Þ

< y:

This implies k f ðx; uÞkeF�< y. r

References

[ 1 ] Adams, A. and Fournier, J. F., Sobolev Spaces, 2nd ed., Academic Press, 2003.

[ 2 ] Ambrosetti, A., Garcia Azorero, J. and Peral, I., Perturbation of Duþ uðNþ2Þ=ðN�2Þ ¼ 0, the

scalar curvature problem in RN , and related topics, J. Funct. Anal. 165 (1999), 117–149.

[ 3 ] Ambrosetti, A., Garcia Azorero, J. and Peral, I., Elliptic variational problems in RN with

critical growth, Special issue in celebration of Jack K. Hale’s 70th birthday, Part 1 (Atlanta,

GA/Lisbon, 1998), J. Di¤erential Equations 168 (2000), 10–32.

[ 4 ] Aubin, J.-P. and Ekeland, I., Applied nonlinear analysis, Pure and Applied Mathematics, A

Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.

[ 5 ] Badiale, M. and Citti, G., Concentration compactness principle and quasilinear elliptic

equations in RN , Comm. Partial Di¤erential Equations 16 (1991), 1795–1818.

[ 6 ] Benci, V. and Cerami, G., Existence of positive solutions of the equation �Duþ aðxÞu ¼uðNþ2Þ=ðN�2Þ in RN , J. Funct. Anal. 88 (1990), 90–117.

[ 7 ] Ben-Naoum, A. K., Troestler, C. and Willem, M., Extrema problems with critical Sobolev

exponents on unbounded domains, Nonlinear Anal. 26 (1996), 823–833.

[ 8 ] Brezis, H., Some variational problems with lack of compactness, Nonlinear functional

analysis and its applications, Part 1 (Berkeley, Calif., 1983), 165–201, Proc. Sympos. Pure

Math., 45, Part 1, Amer. Math. Soc., Providence, RI, 1986.

[ 9 ] Brezis, H. and Lieb, E., A relation between pointwise convergence of functions and

convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

[10] Brezis, H. and Nirenberg, L., Positive solutions of nonlinear elliptic equations involving

critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.

[11] Clement, Ph., Garcıa-Huidobro, M., Manasevich, R. and Schmitt, K., Mountain pass type

solutions for quasilinear elliptic equations, Calc. Var. Partial Di¤erential Equations 11 (2000),

33–62.

[12] Dal Maso, G. and Murat, F., Almost everywhere convergence of gradients of solutions to

nonlinear elliptic systems, Nonlinear Anal. 31 (1998), 405–412.

[13] Fuchs, M. and Li, G., Variational inequalities for energy functionals with nonstandard

growth conditions, Abstr. Appl. Anal. 3 (1998), 41–64.

[14] Fuchs, M. and Osmolovski, V., Variational integrals on Orlicz-Sobolev spaces, Z. Anal.

Anwendungen 17 (1998), 393–415.


[15] Garcıa Azorero, J. and Peral Alonso, I., Multiplicity of solutions for elliptic problems with

critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877–

895.

[16] Goncalves, J. V. and Alves, C. O., Existence of positive solutions for m-Laplacian equations

in RN involving critical Sobolev exponents, Nonlinear Anal. 32 (1998), 53–70.

[17] Guedda, M. and Veron, L., Quasilinear elliptic equations involving critical Sobolev expo-

nents, Nonlinear Anal. 13 (1989), 879–902.

[18] Krasnosel’skiı, M. A. and Rutickiı, Ja. B., Convex Functions and Orlicz Spaces, Translated

from the first Russian edition by L. F. Boron, P. Noordho¤ Ltd., Groningen, 1961.

[19] Lions, P. L., The concentration-compactness principle in the calculus of variations, The limit

case, I, Rev. Mat. Iberoamericana 1 (1985), 145–201.

[20] Silva, E. A. de B. and Soares, S. H. M., Quasilinear Dirichlet problems in RN with critical

growth, Nonlinear Anal. 43 (2001), 1–20.

[21] Silva, E. A. B. and Xavier, M. S., Multiplicity of solutions for quasilinear elliptic problems

involving critical Sobolev exponents, Ann. Inst. H. Poincare Anal. Non Lineaire 20 (2003),

341–358.

nuna adreso:

Nobuyoshi Fukagai

Department of Mathematics

Faculty of Engineering

Tokushima University

Tokushima 770-8506

Japan

E-mail: [email protected]

Masayuki Ito

Department of Mathematics and Computer

Sciences

Tokushima University

Tokushima 770-8502

Japan


Kimiaki Narukawa

Department of Mathematics

Naruto University of Education

Takashima, Naruto 772-8502

Japan


(Ricevita la 18-an de aprilo, 2005)

(Reviziita la 11-an de novembro, 2005)


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