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
238 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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),
239Positive Solutions of Quasilinear Elliptic Equations on RN
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
240 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
241Positive Solutions of Quasilinear Elliptic Equations on RN
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
242 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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.
243Positive Solutions of Quasilinear Elliptic Equations on RN
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
244 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
245Positive Solutions of Quasilinear Elliptic Equations on RN
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
246 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
247Positive Solutions of Quasilinear Elliptic Equations on RN
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:
248 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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;
249Positive Solutions of Quasilinear Elliptic Equations on RN
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
250 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
251Positive Solutions of Quasilinear Elliptic Equations on RN
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;
252 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
253Positive Solutions of Quasilinear Elliptic Equations on RN
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:
254 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
255Positive Solutions of Quasilinear Elliptic Equations on RN
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
256 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
257Positive Solutions of Quasilinear Elliptic Equations on RN
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
258 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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.
259Positive Solutions of Quasilinear Elliptic Equations on RN
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:
260 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
261Positive Solutions of Quasilinear Elliptic Equations on RN
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
262 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
263Positive Solutions of Quasilinear Elliptic Equations on RN
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;
264 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
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
265Positive Solutions of Quasilinear Elliptic Equations on RN
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.
266 Nobuyoshi Fukagai, Masayuki Ito and Kimiaki Narukawa
[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
E-mail: [email protected]
Kimiaki Narukawa
Department of Mathematics
Naruto University of Education
Takashima, Naruto 772-8502
Japan
E-mail: [email protected]
(Ricevita la 18-an de aprilo, 2005)
(Reviziita la 11-an de novembro, 2005)
267Positive Solutions of Quasilinear Elliptic Equations on RN