In this paper, we prove, in the spirit of
[3
,
12
,
20
,
22
,
23]
, the existence of infinitely many small solutions to the following quasilinear elliptic equation −Δ
_{p(x) }
u
+
u

^{p(x)−2}
u
=
u

^{q(x)−2}
u
+
λƒ
(
x, u
) in a smooth bounded domain Ω or ℝ
^{N}
. We also assume that
where
p
^{∗}
(
x
) =
Np
(
x
)/(
N

p
(
x
)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya
[22]
, and property of these solutions are also obtained.
AMS Mathematics Subject Classification : 35J60, 35B33.
1. Introduction
In this paper we deal with quasilinear elliptic problem of the form
where Ω ⊂ ℝ
^{N}
(
N
≥ 3) is a bounded domain with smooth boundary and
p
(
x
),
q
(
x
) are two continuous functions on
where denote by
p
(
x
) ≪
q
(
x
) the fact that inf
_{x∈Ω}
(
q
(
x
) −
p
(
x
)) > 0.
λ
is a positive parameter, Δ
_{p(x)}
u
:= div(∇
u

^{p(x)−2}
∇
u
) is the
p
(
x
)Laplacia operator. On the exponent
q
(
x
) we assume that is the critical exponent in the sense that
is the critical exponent according to the Sobolev embedding. Our goal will be to obtain infinitely many small weak solutions which tend to zero for (1) in the generalized Sobolev space
for the general nonlinearities of the type
ƒ
(
x, u
).
The study of differential equations and variational problems involving variable exponent conditions has been a very interesting and important topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, image processing and so on. For example, Chen, Levin and Rao
[4]
proposed the following model in image processing
where
p
(
x
) is a function satisfies 1 ≤
p
(
x
) ≤ 2 and
ƒ
is a convex function. For more information on modelling physical phenomena by equations involving
p
(
x
)growth condition we refer to
[1
,
19
,
28
,
30]
. The appearance of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces,
L
^{p(x)}
and
W
^{1,p(x)}
, where
p
(
x
) is a realvalued function. On the variable exponent Sobolev spaces which have been used to study
p
(
x
)Laplacian problems, we refer to
[5
,
21
,
29]
. On the existence of solutions for elliptic equations with variable exponent, we refer to
[2
,
6
,
7
,
8
,
9
,
10
,
11
,
16
,
17
,
31]
.
In recent years, the existence of infinitely many solutions have been obtained by many papers. When
p
(
x
) ≡
p
= 2 (a constant) with Dirichlet boundary condition, Li and Zou
[23]
studied a class of elliptic problems with critical exponents, they obtained the existence theorem of infinitely many solutions under suitable hypotheses. He and Zou
[20]
proved that the existence infinitely many solutions under case the general nonlinearities. When
p
(
x
) ≡
p
≠ 2. Ghoussoub and Yuan
[18]
obtained the existence of infinitely many nontrivial solutions for HardySobolev subcritical case and Hardy critical case by establishing PalaisSmale type conditions around appropriate chosen dual sets in bounded domain. Li and Zhang
[24]
studied the existence of multiple solutions for the nonlinear elliptic problems of
p&q
Laplacian type involving the critical Sobolev exponent, they obtained infinitely many weak solutions by using LusternikSchnirelman’s theory for
Z
_{2}
invariant functional.
On the existence of infinitely many solutions for
p
(
x
)Laplacian problems have been studied by
[2
,
7
,
9
,
31]
, but they did not give any further information on the sequence of solutions. Moreover, these papers deal with subcritical nonlinearities. Very little is known about critical growth nonlinearities for variable exponent problems
[14
,
15]
, since one of the main techniques used in order to deal with such issues is the concentrationcompactness principle. This result was recently obtained for the variable exponent case independently in
[12
,
13]
. In both of these papers the proof are similar and both relates to that of the original proof of P.L. Lions
[25
,
26]
.
Recently, Kajikiya
[22]
established a critical point theorem related to the symmetric mountain pass lemma and applied to a sublinear elliptic equation. But there are no such results on
p
(
x
)Laplacian problem with critical growth (1).
Motivated by reasons above, the aim of this paper is to show that the existence of infinitely many solutions of problem (1), and there exists a sequence of infinitely many arbitrarily small solutions converging to zero by using a new version of the symmetric mountainpass lemma due to Kajikiya
[22]
. In order to use the symmetric mountainpass lemma, there are many difficulties. The main one in solving the problem is a lack of compactness which can be illustrated by the fact that the embedding of
W
^{1,p(x)}
(Ω) into
L
^{p*(x)}
(Ω) is no longer compact. Hence the concentrationcompactness principle is used here to overcome the difficulty. The main result of this paper is as follows.
Theorem 1.1.
Suppose that f
(
x, u
)
satisfies the following conditions:

(H1)ƒ(x, u) ∈C(Ω×R, R),ƒ(x, u) =ƒ(x, u)for all u∈R;

(H2)

(H3)
Then there exists λ
^{∗}
such that for any λ
∈ (0,
λ
^{∗}
),
problem
(1)
has a sequence of nontrivial solutions
{
u_{n}
}
and u_{n}
→ 0
as n
→ ∞.
Remark 1.1.
If without the symmetry condition (i.e.
ƒ
(
x,−u
) =
ƒ
(
x, u
)) in Theorem 1.1, we get an existence theorem of at least one nontrivial solution to problem (1) by the same method in this paper.
Remark 1.2.
In this paper, we use concentrationcompactness principle due to
[12]
which is slightly more general than those in
[13]
, since we do not require
q
(
x
) to be critical everywhere.
Remark 1.3.
There exist many functions
ƒ
(
x, t
) satisfy condition (
H
_{1}
)(
H
_{3}
), for example,
ƒ
(
x, u
) =
u
^{(p1)/3}
, where
p
^{−}
> 1.
Remark 1.4.
Theorem 1.1 is new as far as we know and it generalizes results in
[3]
for
p
(
x
)Laplacian type problem. We mainly follow the way in
[3]
to prove our main result.
Definition 1.2.
We say that
is a weak solution of problem (1) if for any
The energy functional corresponding to problem (1) is defined as follows,
then, it is easy to check that as arguments
[27]
show that
J
(
u
) is well defined on
and the weak solutions for problem (1) coincides with the critical points of
J
. We try to use a new version of the symmetric mountainpass lemma due to Kajikiya
[22]
. But since the functional
J
(
u
) is not bounded from below, we could not use the theory directly. So we follow
[3]
to consider a truncated functional of
J
(
u
). Denote
J
′ :
E
→
E
^{∗}
is the derivative operator of
J
in the weak sense. Then
Definition 1.3.
We say
J
satisfies PalaisSmale condition ((
PS
) for short) in
which satisfies that {
J
(
u_{n}
)} is bounded and ∥
J
′(
u_{n}
)∥
_{p(x)}
→ 0 as
n
→ ∞, has a convergent subsequence.
Under assumption (
H
_{2}
), we have
which means that, for all
ε
> 0, there exist
a
(
ε
),
b
(
ε
) > 0 such that
Hence, for any constants
β
we have
for some
c
(
ε
) > 0.
The remainder of the paper is organized as follows. In Section 2, we shall present some basic properties of the variable exponent Sobolev spaces. In Section 3, we will prove the corresponding energy functional satisfies the (
PS
) condition. In Section 4, we shall prove our main results.
2. Weighted variable exponent Lebesgue and Sobolev spaces
We recall some definitions and properties of the variable exponent LebesgueSobolev spaces
L
^{p(·)}
(Ω) and W
^{1,p(·)}
(Ω), where Ω is a bounded domain in ℝ
^{N}
.
Set
For any
we define
We can introduce the variable exponent Lebesgue space as follows:
L
^{p(·)}
(Ω) = {
u
:
u
is a measurable realvalued function such that
for
Equipping with the norm on
L
^{p(x)}
(Ω) by
which is a Banach space, we call it a generalized Lebesgue space.
Proposition 2.1
(
[5
,
11]
). (i)
The space
(
L
^{p(x)}
(Ω),  · 
_{p(x)}
)
is a separable, uniform convex Banach space, and its conjugate space is L
^{q(x)}
(Ω),
where
1/
q
(
x
) + 1/
p
(
x
) = 1.
For any u
∈
L
^{p(x)}
(Ω)
and v
∈
L
^{q(x)}
(Ω),
we have
(ii)
If
0 < Ω < ∞
and p
_{1}
,
p
_{2}
are variable exponents in
such that p
_{1}
≤
p
_{2}
in
Ω,
then the embedding L
^{p2(·)}
(Ω) ,→
L
^{p1(·)}
(Ω)
is continuous
.
Proposition 2.2
(
[5
,
11]
).
The mapping ρ
_{p(·)}
:
L
^{p(·)}
(Ω) → ℝ
defined by
Then the following relations hold:
Next, we define
W
^{1,p(x)}
(Ω) is defined by
and it can be equipped with the norm
Denote
under the norm
We know that if Ω ⊂ ℝ
^{N}
is a bounded domain, ∥
u
∥ and ∥
u
∥
_{1}
are equivalent norms on
Proposition 2.3
(
[5
,
11]
). (i)
W
^{1,p(x)}
(Ω)
are separable re exive Banach spaces;
(ii)
If
then the embedding W
^{1,p(x)}
(Ω) → L
^{q(x)}
(Ω)
is continuous
.
In this paper, we use the following equivalent norm on
W
^{1,p(x)}
(Ω):
Proposition 2.4
(
[21
,
6]
).
Let I
(
u
) =
ƒ
_{Ω}
∇
u

^{p(x)}
+ 
u

^{p(x)}
dx
.
If u, u_{n}
∈
W
^{1,p(x)}
(Ω),
then the following relations hold:
3. Preliminaries and lemmas
In the following, we always use
C
and
c_{i}
(
i
= 1, 2, · · · ) to denote positive constants. We give the concentrationcompactness principle of the variable exponent due to
[12
,
15]
.
Lemma 3.1.
Let q
(
x
)
and p
(
x
)
be two continuous functions such that
Let
{
u_{j}
}
_{j∈ℕ}
be a weakly convergent sequence in
with weak limit u, and such that
∇
u_{j}

^{p(x)}
⇀
μ weakly

^{∗}
in the sense of measures;

u_{j}

^{q(x)}
⇀
v weakly

^{∗}
in the sense of measures. Assume, moreover that
Then, for some countable index set I we have
(i)
ν
= 
u

^{q(x)}
+ Σ
_{i∈I}
ν_{i}
δ
_{xi}
,
ν_{i}
> 0;
(ii)
μ
≥ ∇
u

^{p(x)}
+ Σ
_{i∈I}
μ_{i}
δ
_{xi}
,
μ_{i}
> 0;
(iii)
where
{
x_{i}
}
_{i∈I}
⊂ Γ
and S is the best constant in the GagliardoNirenbergSobolev inequality for variable exponents, namely
In order to prove the functional
J
satisfies the local (
PS
)
_{c}
condition, we take continuous function
η
(
x
) satisfies
Denote
Lemma 3.2.
Assume condition
(
H
_{2}
)
holds. Then for any λ
> 0,
there exists positive constant m
^{∗}
> 0
such that the functional J satisfies the local
(
PS
)
_{c}
condition in
in the following sense: if
and J
′(
u_{n}
) → 0
for some sequence in
Then
{
u_{n}
}
contains a subsequence converging strongly in
Proof
. First, we show that {
u_{n}
} is bounded in
If ∥
u_{n}
∥
_{p(x)}
→ ∞ as
n
→ ∞. Thus, we may assume that ∥
u_{n}
∥
_{p(x)}
> 1 for any integer
n
.
Then for
n
sufficiently large, we have
By (4), for any (
x, t
) ∈ Ω × ℝ, we have
On the other hand, noting that
p
(
x
) ≪
q
(
x
), by the Young inequality, for any
ε
_{2}
,
ε
_{3}
∈ (0, 1), we get
and
Thus, relations (13)(16) imply that
where
Thus, we choose
ε
_{2}
,
ε
_{3}
be so small that
d
_{1}
−
c
_{1}
ε
_{2}
> 0 and
It follows from (8) and (17) that {
u_{n}
} is bounded in
Therefore we can assume that
Note that if
I
= ∅ then
u_{n}
→
u
strongly in L
^{q(x)}
(Ω). If not, let
x_{i}
be a singular point of the measures
μ
and
ν
, define a function
such that
ϕ
(
x
) = 1 in
B
(
x_{i}
,
ε
),
ϕ
(
x
) = 0 in Ω \ (
x_{i}
, 2
ε
) and ∇
ϕ
 ≤ 2/
ε
in Ω. As
we obtain that
i.e.
On the other hand, by Hölder inequality and boundedness of {
u_{n}
}, we have that
From (18), (19) and (21), we get that
Combing this with Lemma 2.1 (iii), we obtain
This result implies that
If the second case
ν_{i}
≥
S^{N}
holds, for some
i
∈
I
, then by using Lemma 2.1 and selecting
ε
_{2}
,
ε
_{3}
in (17) such that
we have
where
This is impossible. Consequently,
ν_{i}
= 0 for all
i
∈
I
and hence
Since {
u_{n}
} is bounded in
we deduce that there exists a subsequence, again denoted by {
u_{n}
}, and
such that {
u_{n}
} converges weakly to
Note that
On the other hand, we have
Using the fact that {
u_{n}
} converges strongly to
u
_{0}
in
L
^{q(x)}
(Ω) and inequality (5), we have
where
c
_{1}
c
_{2}
and
c
_{3}
are positive constants. Using 
u_{n}
−
u
_{0}

_{q(x)}
→ 0 as
n
→ ∞, we deduce that
By (23) and (24), we obtain
It is known that
where (· , ·) is the standard scalar product in ℝ
^{N}
. Relations (25) and (26) yield
This fact and relation (10) imply ∥
u_{n}
−
u
_{0}
∥
_{p(x)}
→ 0 as
n
→ ∞. The proof is complete.
4. Existence of a sequence of arbitrarily small solutions
In this section, we prove the existence of infinitely many solutions of (1) which tend to zero. Let
X
be a Banach space and denote
Σ := {
A
⊂
X
\ {0} :
A
is closed in
X
and symmetric with respect to the orgin}. For
A
∈ Σ, we define genus
γ
(
A
) as
If there is no mapping
φ
as above for any
m
∈
N
, then
γ
(
A
) = +∞. Let Σ
_{k}
denote the family of closed symmetric subsets
A
of
X
such that 0 ∉
A
and
γ
(
A
) ≥
k
. We list some properties of the genus (see
[22]
).
Proposition 4.1.
Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold
.

(1)If there exists an odd continuous mapping from A to B, then γ(A) ≤γ(B);

(2)If there exists an odd homeomorphism from A to B, then γ(A) =γ(B);

(3)If γ(B) < ∞,then

(4)Then ndimensional sphere Snhas a genus of n+1by the BorsukUlam Theorem;

(5)If A is compact, then γ(A) < +∞and there exists δ> 0such that Uδ(A) ∈ Σand γ(Uδ(A)) =γ(A),where Uδ(A) = {x ∈ X : ∥x − A∥ ≤ δ}
The following version of the symmetric mountainpass lemma is due to Kajikiya
[22]
.
Lemma 4.2.
Let E be an infinitedimensional space and J
∈
C
^{1}
(
E,R
)
and suppose the following conditions hold
.

(C1)J(u)is even, bounded from below, J(0) = 0and J(u)satisfies the PalaisSmale condition;

(C2)For each k∈N,there exists an Ak∈ Σksuch thatsupu∈AkJ(u) < 0.Then either(R1)or(R2)below holds.

(R1)There exists a sequence{uk}such that J′(uk) = 0,J(uk) < 0and{uk}converges to zero.

(R2)There exist two sequences{uk}and{vk}such that J′(uk) = 0,J(uk) < 0,uk≠ 0, limk→∞uk= 0,J′(vk) = 0,J(vk) < 0, limk→∞vk= 0,and{vk}converges to a nonzero limit.
Remark 4.1.
From Lemma 4.2 we have a sequence {
u_{k}
} of critical points such that
J
(
u_{k}
) ≤ 0,
u_{k}
≠ 0 and lim
_{k→∞}
u_{k}
= 0.
In order to get infinitely many solutions we need some lemmas. We first point out that we have
Proposition 2.3 (ii) imply that
where
c
_{4}
> 0.
Next, we focus our attention on the case when
For such a
u
by relation (9) we obtain
Using (3) and (27)(29), we deduce that
where
with ∥
u
∥
_{p(x)}
< 1. If we define
Then
From the definition of
Q
(
s
) and the fact that
p
^{+}
<
q
^{+}
, we konw that there exists
λ
^{∗}
such that for
λ
∈ (0,
λ
^{∗}
),
Q
(
t
) attains its positive maximum, that is, there exists
such that
Therefore, for
e
_{0}
∈ (0,
e
_{1}
), we may find
R
_{0}
<
R
_{1}
such that
Q
(
R
_{0}
) =
e
_{0}
. Now we define
Then it is easy to see
χ
(
t
) ∈ [0, 1] and
χ
(
t
) is
C
^{∞}
. Let
φ
(
u
) =
χ
(∥
u
∥
_{p(x)}
) and consider the perturbation of
J
(
u
):
Then
where
and
From the above arguments, we have the following:
Lemma 4.3.
Let G
(
u
)
be defined as in
(31).
Then

(i)G∈C1(E,R)and G is even and bounded from below;

(ii)If G(u)

(iii)There exist m∗> 0such that SN−m∗> 0,and λ∗such that for λ∈ (0,λ∗),G satisfies a local(PS)ccondition for
Proof
. It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 3.2.
Lemma 4.4.
Assume that
(
H
_{3}
)
of Theorem 1.1 holds. Then for any k
∈
N
,
there exists
Proof
. First, by (
H
_{3}
) of Theorem 1.1, for any fixed
we have
Next, given any
k
∈
N
, let
E_{k}
be a
k
dimensional subspace of
We take
u
∈
E_{k}
with norm ∥
u
∥
_{p(x)}
= 1, for 0 <
ρ
< min{
R
_{0}
, 1}, we get
Since
E_{k}
is a space of finite dimension, all the norms in
E_{k}
are equivalent. If we define
It follows from (32)that
since lim
_{ρ→0}
M
(
ρ
) = +∞. That is,
This completes the proof.
Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1
Recall that
and define
By Lemmas 4.3 (i) and 4.4, we know that −∞ <
c_{k}
< 0. Therefore, assumptions (
C
_{1}
) and (
C
_{2}
) of Lemma 4.2 are satisfied. This means that
G
has a sequence of solutions {
u_{n}
} converging to zero. Hence, Theorem 1.1 follows by Lemma 4.3 (ii).
BIO
Chenxing Zhou received master’s degree from Jilin University. His research interests boundary value problems and variational problems.
College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, PR China.
email: mathfhmiao@163.com
Sihua Liang received Ph.D at Nanjing Normal University. He is currently a Postdoctor at Jilin University since 2011. His research interests boundary value problems and variational problems.
College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, PR China.
email: liangsihua@126.com
Acerbi E.
,
Mingione G.
(2001)
Regularity results for a class of functionals with nonstandard growth
Arch. Rational Mech. Anal.
156
121 
140
DOI : 10.1007/s002050100117
Andrei I.
(2009)
Existence of solutions for a p(x)Laplacian nonhomogeneous equations
E. J. Differential Equations
2009
(72)
1 
12
Azorero Garcia J
,
Aloson Peral I.
(1991)
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
Trans. Amer. Math. Soc.
323
877 
895
DOI : 10.2307/2001562
Chen Y.
,
Levine S.
,
Rao R.
(2006)
Functionals with p(x) growth in image processing
SIAM J. Appl. Math.
66
1383 
1406
DOI : 10.1137/050624522
Fan X.
,
Shen J.
,
Zhao D.
(2001)
Sobolev embedding theorems for spaces Wk,p(x)(Ω)
J. Math. Anal. Appl.
262
749 
760
DOI : 10.1006/jmaa.2001.7618
Fan X. L.
,
Han X. Y.
(2004)
Existence and multiplicity of solutions for p(x)Laplacian equations in ℝN
Nonlinear Anal.
59
173 
188
Fan X. L.
(2007)
Global C1, αregularity for variable exponent elliptic equations in divergence form
J. Differential Equations
235
397 
417
DOI : 10.1016/j.jde.2007.01.008
Fan X. L.
,
Zhao D.
(1998)
On the generalized OrliczSobolev space Wk,p(x)(Ω)
J. Gansu Educ. College
12
(1)
1 
6
Fernández Bonder J.
,
Silva A.
The concentrationcompactness principle for variable exponent spaces and applications, arXiv: 0906. 1992v2 [Math.AP].
Fu Y.
(2009)
The principle of concentration compactness in Lp(x)(Ω) spaces and its application
Nonlinear Anal.
71
1876 
1892
DOI : 10.1016/j.na.2009.01.023
Zhang X.
,
Fu Y. Q.
(2010)
Multiple solutions for a class of p(x)Laplacian equations involving the critical exponent
Ann. Polon. Math.
98
91 
102
DOI : 10.4064/ap9816
Fu Y. Q.
,
Zhang X.
(2010)
Multiple solutions for a class of p(x)Laplacian equations in ℝninvolving the critical exponent
Proc. R. Soc. Lond. Ser. A
466
1667 
1686
DOI : 10.1098/rspa.2009.0463
Chabrowski J.
,
Fu Y. Q.
(2005)
Existence of Solutions for p(x)Laplacian problems on a bounded domain
J. Math. Anal. Appl.
306
604 
618
DOI : 10.1016/j.jmaa.2004.10.028
Fu Y. Q.
(2007)
Existence of solutions for p(x)Laplacian problem on an unbounded domain
Topol. Methods Nonlinear Anal.
30
235 
249
Ghoussoub N.
,
Yuan C.
(2002)
Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents
Trans. Am. Math. Soc.
352
5703 
5743
DOI : 10.1090/S0002994700025605
He X. M.
,
Zou W. M.
(2009)
Infinitely many arbitrarily small solutions for sigular elliptic problems with critical SobolevHardy exponents
Proc. Edinburgh Math. Society
52
97 
108
DOI : 10.1017/S0013091506001568
Kováčik O.
,
Rákosnik J.
(1991)
On spaces Lp(x)and W1,p(x)
Czech. Math. J.
41
592 
618
Kajikiya R.
(2005)
A criticalpoint theorem related to the symmetric mountainpass lemma and its applications to elliptic equations
J. Funct. Analysis
225
352 
370
DOI : 10.1016/j.jfa.2005.04.005
Li G.B.
,
Zhang G.
(2009)
Multiple solutions for the p&qLaplacian problem with critical exponent
Acta Math. Scientia
29B
903 
918
Lions P. L.
(1985)
The concentrationcompactness principle in the caculus of variation: the limit case, I
Rev. Mat. Ibero.
1
45 
120
DOI : 10.4171/RMI/12
Lions P. L.
(1985)
The concentrationcompactness principle in the caculus of variation: the limit case, II
Rev. Mat. Ibero.
1
145 
201
DOI : 10.4171/RMI/6
Rabinowitz P. H.
1986
Minimax methods in criticalpoint theory with applications to differential equations
CBME Regional Conference Series in Mathematics
American Mathematical Society
Providence, RI
65
Ruzicka M.
2002
Electrorheological Fluids Modeling and Mathematical Theory
SpringerVerlag
Berlin
Sharapudinov I.
(1978)
On the topology of the space Lp(t)([0; 1])
Matem. Zametki
26
613 
632