1 Introduction
It is well known that the (S)+-property of the mappings plays an important
role in the monotone mapping theory and the critical point theory (see e.g.
[7,16,26,39]). This paper deals with the (S)+-property of the quasilinear
elliptic operators.
Let
 be an open subset of RN and let T(u) = −div(a(x,ru)) be a
quasilinear elliptic operator on
 and possess the variational structure, that
is a(x, ) = A0(x, ), where A :
 × RN ! R and A0(x, ) denotes the
derivative of A(x, ) with respect to . For simplicity, let us consider the
Dirichlet 0-boundary value problem on a bounded domain
. It is well
known that, when T(u) = −div(|ru|p−2 ru) is the p-Laplacian with p 2
Research supported by the National Natural Science Foundation of China (10671084)
†Corresponding author. Fax:+86-931-8912481. E-mail address: fanxl@lzu.edu.cn
(X.L.Fan), guanchx05@lzu.cn (C.X.Guan).
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(1,1), the operator T, as a mapping from W1,p
0 (
) into W1,p
0 (
) defined
by
T(u)v = Z
 |ru|p−2 rurv, 8u, v 2 W1,p
0 (
),
is of type (S)+, and when T(u) = −div(|ru|p(x)−2 ru) is the p(x)-Laplacian
with 1 < inf
 p(x)  sup
 p(x) < 1, the operator T, as a mapping from
W1,p(x)
0 (
) into W1,p(x)
0 (
) is of type (S)+ (see e.g. [16]). Note that the
space W1,p(x)
0 (
) is uniformly convex (see e.g. [17,29,30]), it is a modular
space generated by the function A(x, ) = ||p(x) (see [17,24,29,30]) and the
function A(x, ) is uniformly convex (for the definition of the uniform convexity
of the function A(x, ) see Definition 2.9 below). In this paper we
shall give a sufficient condition for the (S)+-property of the general quasilinear
elliptic operator T, which is stated by the term of the uniform convexity
of the function A (see Section 3 for details).
Many authors have studied the quasilinear elliptic operators of general
form (see e.g. [3,5,9,10,12,18,19,20,21,28]). When A(x, ) is independent of
x, the theory of Orlicz spaces can be applied (see e.g. [3,9,18,19,20,28]) and
when A(x, ) is dependent of x, the theory of Musielak-Orlicz spaces can
be applied. When A is of scalar type, that is there exists  :
 × R ! R
such that A(x, ) = (x, ||), the theory of the scalar valued Musielak-Orlicz
spaces is needed essentially. When A is not of scalar type, the theory of the
vector valued Musielak-Orlicz spaces is needed. In this paper we need the
theory of the vector valued Musielak-Orlicz spaces for which we refer to
[8,22,23,25,33,34,36,38].
This paper is organized as follows. In Section 2, we recall some basic facts
about the vector valued Musielak-Orlicz spaces, particularly, the facts on the
uniform convexity of the spaces. In Section 3, we give a sufficient condition
for the (S)+-property of the subdifferential mapping of the convex modular
generated by a k-Musielak-Orlicz function A(x, ) and based on this we
obtain the (S)+-property of the corresponding quasilinear elliptic operator.
In Section 4, we study a special class of the quasilinear elliptic operators,
called of the p(x)-Laplacian type, which corresponds to a special class of the
N-Musielak-Orlicz functions, called of type with variable exponent p(x),
which includes several important special cases, for example, the case that
A(x, ) = b(x) !2(x) + ||2p(x)
2 − b(x)(!(x))p(x), 8x 2
, 8 2 RN, (1.1)
(see Examples 4.1-4.3 below for more details). In Section 5, we give several
simple results on the existence and multiplicity of solutions for the equations
of the p(x)-Laplacian type, which are a generalization of those in the p(x)-
Laplacian case.
The problems, considered in Sections 4 and 5, involve the variable exponent.
In recent years there has been an increasing interest in the study of
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various mathematical problems with variable exponent. We refer to [11,32]
for the overview and the references of this area, to [31] for the application
backgrounds, to [11,13,17,24,32] for the variable exponent Lebesgue-Sobolev
spaces and to [1,4,5,14,15,16,27,40] for the p(x)-Laplacian equations and
the corresponding variational problems. In this paper we extend the p(x)-
Laplacian to the case of the p(x)-Laplacian type.
The authors draw great inspiration from De N´apoli and Mariani [10]. In
[10] De N´apoli and Mariani have introduced the following condition for the
function A :
 × RN ! R, called the p-uniform convexity condition,
Ax,
 + 
2 
1
2A(x, )+
1
2A(x, )−d | − |p , 8x 2
, ,  2 RN, (1.2)
where d is a positive constant, and studied the corresponding equations of
the p-Laplacian type. However, as was noted in [10], for the case of the
usual p-Laplacian, the condition (1.2) is satisfied only if p  2. Thus the
results of [10] are not suitable for the case of the p-Laplacian with p 2 (1, 2).
To remedy this defect, in the present paper we use the uniform convexity
condition (UC) (see Definition 2.8 below) instead of the condition of form
(1.2). This is a main intention of the present paper.
2 Vector valued Musielak-Orlicz spaces and uniform
convexity
In this section we recall some basic facts about the vector valued Musielak-
Orlicz spaces, particularly, the facts about the uniform convexity of the
spaces. For the scalar valued Musielak-Orlicz spaces see [29,30,38]. For
the vector valued Musielak-Orlicz spaces see [8,22,23,33,34,36,38] and the
comments in [29]. For the Banach valued Musielak-Orlicz spaces see [25]
and the comments in [29].
Let
 be an open subset of RN and k a positive integer.
Definition 2.1. A function A :
 × Rk ! R is called a k-Musielak-Orlicz
function if A satisfies the following conditions:
1) A is a Carath´eodory function, that is, A(x, ·) : Rk ! R is continuous
for a.e. x 2
 and A(·, ) :
 ! R is measurable for every  2 Rk.
2) A(x, )  0 for a.e. x 2
 and all  2 Rk; A(x, ) = 0 if and only if
 = 0.
3) A(x,−) = A(x, ) for a.e. x 2
 and all  2 Rk.
4) The function A(x, ·) : Rk ! R is convex for a.e. x 2
.
Definition 2.2. A k-Musielak-Orlicz function A :
 × Rk ! R is said to
be of scalar type if there exists a 1-Musielak-Orlicz function  :
×R ! R
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such that A(x, ) = (x, ||) for a.e. x 2
 and all  2 Rk.
Denote S(
,Rk) = {u|u :
 ! Rk is measurable}.
Let A :
 × Rk ! R be a k-Musielak-Orlicz function. Define
A(v) = Z

A(x, v(x))dx, 8v 2 S(
,Rk). (2.1)
Proposition 2.1. If A is a k-Musielak-Orlicz function, then A is a convex
modular in the vector space S(
,Rk), that is, A : S(
,Rk) ! [0,+1]
satisfies the following conditions:
1) A(v) = 0 if and only if v = 0.
2) A(−v) = A(v), 8v 2 S(
,Rk).
3) A( u+ v)  A(u)+ A(v), 8u, v 2 S(
,Rk), ,  0, + = 1.
The proof of Proposition 2.1 is immediate. For the definition of the
modulars see [29].
Define
LA(
,Rk) = nv 2 S(
,Rk) : lim
!0
A( v) = 0o (2.2)
and for v 2 LA(
,Rk),
kvkA
= kvkLA = kvkA := inf n > 0 :  v
 1o. (2.3)
Then LA(
,Rk) is a vector subspace of S(
,Rk) and k·kA is a norm in
LA(
,Rk), which is called the Luxemburg norm (see [29,38]). The space
LA(
,Rk) is called a vector valued Musielak-Orlicz space, which belongs a
special class of the Musielak-Nakano’s modular spaces. When k = 1 it is a
scalar valued Musielak-Orlicz space.
Remark 2.1. In [33,38], a definition of the generalized N-function has been
given. By the definition in [38], A :
 × Rk ! R is called a generalized
N-function if A satisfies the conditions 1)-4) in Definition 2.1 (note that in
[33] the condition 3) is needless) and the following conditions:
5) For a.e. x 2
, lim||!1
A(x,)
||
= 1.
6) There exists a positive constant L such that
A(x, )  LA(x, ) for a.e. x 2
 and ,  2 Rk with ||  || .
In the present paper, we use the Definition 2.1 without the conditions 5)
and 6) because the conditions 5) and 6) are needless for some problems. We
will also give an improvement on the condition 6) (see the condition (2.4)
in Definition 2.5 below).
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Remark 2.2. As noted in Proposition 2.1, when A satisfies the conditions
in Definition 2.1, A is a convex modular in S(
,Rk). It is easy to give some
variants of Definition 2.1 such that A is a pseudomodular, or a semimodular,
or an s-convex modular respectively (see [29]). In the present paper we
will restrict ourselves to the case that A is a convex modular.
In what follows, it will always be assumed that A is a k-Musielak-Orlicz
function and A, LA(
,Rk), kvkA are defined by (2.1), (2.2), (2.3) respectively.
For simplicity we will write  and kvk instead of A and kvkA respectively
when there is no ambiguity. We often use “8x 2
” instead of
“for a.e. x 2
”. For E  RN, |E| denotes the Lebesgue measure of E. For
,  2 RN,  denotes the inner product of  and  in RN. For a sequence
{un} of a Banach spaces, un ! u and un * u denote {un} converges to u
strongly and weakly respectively.
It is easy to see that
LA(
,Rk) = nv 2 S(
,Rk) : 9 > 0 such that A v
< 1o.
Proposition 2.2. 1) ( v)  (v), 8v 2 S(
,Rk) and 2 (0, 1)
2) When kvk < 1, (v)  kvk ; When kvk > 1, (v)  kvk.
3) Let vn, v 2 LA(
,Rk). Then kvn − vk ! 0 as n ! 1 if and only if
( (vn − v)) ! 0 as n ! 1 for every > 0.
4) Let vn 2 LA(
,Rk) for n = 1, 2, · · · . Then {vn} is a Cauchy sequence
in the space (LA(
,Rk), k·kA) if and only if ( (vn−vm)) ! 0 as n,m ! 1 for every > 0.
5) If vn ! 0 in LA(
,Rk), then the sequence {vn} contains a subsequence
{vnj} convergent to 0 almost everywhere in
.
Proof. Statement 1) follows from the convexity of  and (0) = 0. For
statement 2) see [29, Theorem 1.5 and Lemma 2.4]. For statements 3) and
4) see [29, Theorem 1.6]. Now let us prove statement 6). Let vn ! 0 in
LA(
,Rk). Then by statement 3), ( vn) ! 0 for every > 0, which implies
A(x, vn(x)) ! 0 in measure in
, and consequently there exists a
subsequence {A(x, vnj (x))} of {A(x, vn(x)} such that A(x, vnj (x)) ! 0
for a.e. x 2
, which implies vnj (x) ! 0 for a.e. x 2
 and hence
vnj (x) ! 0 for a.e. x 2
. 
Definition 2.3. Let A :
×Rk ! R is a k-Musielak-Orlicz function. Define
functions A|_| and A|^| :
 × R ! R as follows. For t  0 define
A|_|(x, t) = sup
||=t
A(x, ), 8x 2
,
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三思论文代写网
A|^|(x, t) = inf
||=t
A(x, ), 8x 2
.
For t < 0 define A|_|(x, t) = A|_|(x,−t) and A|^|(x, t) = A|^|(x,−t). A|_|
and A|^| are called the scalar sup-function and the scalar inf-function of A
respectively. Define A_, A^ :
 × Rk ! R by
A_(x, ) = A|_|(x, ||), A^(x, ) = A|^|(x, ||), 8x 2
,  2 Rk.
A_ and A^ are called the sup-function and the inf-function of A respectively.
The following proposition is clear (see [33,38]).
Proposition 2.3. 1) A^(x, )  A(x, )  A_(x, ), 8x 2
, 8 2 Rk.
A|_|(x, 0) = A|^|(x, 0) = 0. When t 6= 0, A|_|(x, t)  A|^|(x, t) > 0. For
a.e. x 2
, the functions A|_|(x, t) and A|^|(x, t) are strictly increasing in
t  0. A^ = A = A_ if A is of scalar type.
2) A|_|, A|^|, A_ and A^ are all the Carath´eodory functions.
3) For a.e. x 2
, the functions A|_|(x, ·) : R ! R and A_(x, ·) : Rk !
R are convex.
Remark 2.3. The functions A|^|(x, t) and A^(x, ) need not be convex
with respect to the second variable. Note that in the usual references, for
example in [8,33,34,38], A|_| and A|^| are denoted by A and A respectively.
Theorem 2.1. Let A :
×Rk ! R be a k-Musielak-Orlicz function. Then
the space (LA(
,Rk), k·kA) is complete, i.e. it is a Banach space.
Proof. Let {vn} be a Cauchy sequence in (LA(
,Rk), k·kA). By Proposition
2.2, ( (vn − vm)) ! 0 as n,m ! 1 for every > 0.
1) First, let us suppose |
| < 1. We will prove that, for every > 0,
{ vn} is a Cauchy sequence in the space S(
,Rk) in the sense of convergence
in measure. Let any > 0, any  > 0 and any " > 0 be given. Since A|^| :

 × R ! R is a Carath´eodory function, by Scorza-Dragoni theorem, there
exists a compact set
" 
 such that |
\
"|  "
2 and A|^| is continuous on

"×R. For every Lebesgue measurable subset G of
", define a μ-measure
by
μ(G) = ZG
A|^|(x, )dx.
Obviously, μ(G) = 0 implies |G| = 0, that is, the Lebesgue measure |·| is
absolutely continuous with respect to the measure μ. Hence for given " > 0
there is  > 0 such that |G|  "
2 provided μ(G)  . We may assume   ".
Put
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Gn,m = {x 2
 : |vn(x) − vm(x)|  }.
Choose N0 > 0 such that when n,m  N0, ( (vn − vm)) < , i.e.
Z

A(x, (vn(x) − vm(x)))dx < .
In this case, of course,
ZGn,m\
"
A(x, (vn(x) − vm(x)))dx < .
Noting that for x 2 Gn,m, |vn(x) − vm(x)|   and A(x, (vn(x)−vm(x))) 
A|^|(x, ), we get
ZGn,m\
"
A|^|(x, )dx < ,
i.e. μ(Gn,m \
") < , which implies |Gn,m \
"|  "
2 . So |Gn,m| 
|Gn,m \
"| + |
\
"|  "
2 + "
2 = ". This shows { vn} is a Cauchy sequence
in the space S(
,Rk) in the sense of convergence in measure, and
consequently, there exists v 2 S(
,Rk) such that vn is convergent in measure
to v, moreover, {vn} contains a subsequence {vnj} convergent to v
almost everywhere in
. Hence, for every n,
A(x, (vn(x) − vnj (x))) ! A(x, (vn(x) − v(x))) a.e. in
 as j ! 1.
Applying Fatou lemma, we obtain
( (vn − v))  lim inf
j!1
( (vn − vnj ))    " for n  N0.
This shows that ( (vn − v)) ! 0 as n ! 1, which implies that v 2
LA(
,Rk) and by Proposition 2.2, kvn − vk ! 0, i.e. vn ! v in LA(
,Rk).
The proof in the case when |
| < 1 is complete.
2) The proof in the case when |
| = 1 is similar to the proof of Theorem
7.7 in [29] and hence is omitted here. 
Remark 2.4 . The Theorem 7.7 of [29] has asserted the completeness of the
space LA(
,Rk) in the case when k = 1 and A is locally integrable, where
a function A :
 × R ! R is called locally integrable if RE A(x, t)dx < 1 for every t 2 R and E 
 with |E| < 1. The ideas for proving Theorem
2.1 are essentially the same as those for proving Theorem 7.7 of [29] but
the local integrability of A|^| is not supposed. In [38] the completeness of
the space LA(
,R) in the case when A is an N-function has been proved by
using the Orlicz norm, and the completeness of the space LA(
,Rk) in the
case when A is a generalized N-function has been mentioned without proof.
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Definition 2.4. ([38]) We say that A satisfies the condition (
2,(x)) (or
(
)) if there exist a positive constant K and a nonnegative measurable
function  2 S(
,R) such that R
 A|_|(x, 2(x))dx < 1 and
A(x, 2)  KA(x, ) for a.e. x 2
 and every  2 Rk with ||  (x).
When A satisfies the condition (
2,(x)), we denote C = R
 A|_|(x, 2(x))dx.
The well-known condition (
2) is a special case of the condition (
2,(x))
when (x)  0.
Proposition 2.4. ([38]) Let A :
 × Rk ! R be a k-Musielak-Orlicz function.
If A satisfies the condition (
2,(x)), then the following statements
hold.
1) (2v)  K(v) + C, 8v 2 S(
,Rk).
2) LA(
,Rk) = v 2 S(
,Rk) :  􀀀v
= 
< 1 for all  6= 0  v 2 S(
,Rk) :  (v) < 1  .
3) Let v 2 LA(
,Rk)\{0}. Then ( v), as a function of 2 [0,1), is
continuous and strictly increasing in 2 [0,1). Moreover ( v) = 1 if and
only if | | = 1
kvk
.
4) (v) < 1 (resp. = 1; > 1) () kvk < 1 (resp. = 1; > 1), where v 2
LA(
,Rk).
5) (vn) ! 0 (resp. 1) () kvnk ! 0 (resp. 1) as n ! 1, where
vn 2 LA(
,Rk).
6) (vn) ! 1 () kvnk ! 1.
7) The modular function  is continuous on the space (LA(
,Rk), k·kA),
that is, (vn) ! (v) if vn ! v in (LA(
,Rk), k·kA).
Proof. The proof of Proposition 2.4 is standard and can be found basically
in [38]. Here we give the proof of statements 6) and 7), which shows that
these statements are true without the hypotheses 5) and 6) mentioned in
Remark 2.1.
6). “=)”: By statement 2) of Proposition 2.2, we know that, kvnk 
C > 1 implies that (vn)  kvnk  C > 1, and kvnk  C1 < 1 implies that
(vn)  kvnk  C1 < 1. This shows that kvnk ! 1 if (vn) ! 1.
“(=”: To prove the part “(=”, it is sufficient to prove the following
two assertions:
(i). If kvnk ! 1−, then (vn) ! 1−. (ii). If kvnk ! 1+, then (vn) ! 1+.
Proof of the assertion (i): Let kvnk < 1 and kvnk ! 1. Then (vn) <
1. We assume by contradiction that (taking a subsequence if necessary)
(vn)  " < 1 for all n. Put 1
kvnk
= 1 + n, where n > 0 and n ! 0. We
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may assume n < 1. Then
1 =  vn
kvnk=  (2 nvn + (1 − n)vn)
 n(2vn) + (1 − n)(vn)
 n (K(vn) + C) + (1 − n)".
Going to the limit for n ! 1 in above inequality, yields 1  ", which is a
contradiction.
Proof of the assertion (ii): Suppose by contradiction that kvnk > 1 and
kvnk ! 1 but (vn)  1 + " > 1 for all n. Put 1
kvnk
= 1 − n, where n > 0
and n ! 0. Denote n = 1
1+ n
. Then n 2 (0, 1), n ! 1 and
n(1 − n)vn + (1 − n)2vn = vn.
We have that
1 + "  (vn) =  ( n(1 − n)vn + (1 − n)2vn)
 n((1 − n)vn) + (1 − n)(2vn)
= n vn
kvnk+ (1 − n)(2vn)
= n + (1 − n)(2vn).
Going to the limit for n ! 1 in above inequality and noting that (2vn) is
bounded, yields 1 + "  1, which is a contradiction.
6). By statement 3), (v) < 1 for all v 2 LA(
,Rk), so the function
 : LA(
,Rk) ! R is well defined. Let vn ! v in LA(
,Rk). By statement
4), (vn − v) ! 0. Suppose by contradiction (vn) 9 (v). Then
there exist a subsequence {vnj} of {vn} and a positive constant " such that (vnj ) − (v)  " for all nj . By statement 6) of Proposition 2.2, {vnj} contains a subsequence {vnjl } convergent to v almost everywhere in
. So
A(x, vnjl
(x)) ! A(x, v(x)) for a.e. x 2
. Note that
A(x, vnjl
(x)) = A(x, vnjl
(x) − v(x) + v(x))

1
2A(x, 2(vnjl
(x) − v(x))) +
1
2A(x, 2v(x))

1
2
[KA(x, vnjl
(x) − v(x)) + A|_|(x, 2(x))] +
1
2A(x, 2v(x)).
By the generalized Lebesgue’s dominated convergence theorem, we deduce
that R
 A(x, vnjl
(x))dx ! R
 A(x, v(x))dx, i.e. (vnjl
) ! (v). This completes
the proof. 
Definition 2.5. A k-Musielak-Orlicz function A :
×Rk ! R is said to be
of quasi-scalar type if there exist a positive constant L and a nonnegative
function g 2 L1(
) such that
A|_|(x, t)  LA|^|(x, t) + g(x) for a.e. x 2
 and all  2 Rk. (2.4)
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Remark 2.5. It is clear that, if A satisfies the condition 6) in Remark 2.2,
then A is of quasi-scalar type with g(x)  0. The condition (2.4) weakened
the condition 6) in Remark 2.2. It is easy to see that, when A is of quasiscalar
type, LA(
,Rk) = LA_(
,Rk).
Definition 2.6. ([29,38]) A normed linear space X is called uniformly
convex if for every " > 0 there exists a number (") > 0 such that for all
u, v 2 X satisfying kuk = kvk = 1, the inequality ku − vk > " implies

 u+v
2

 < 1 − (").
Definition 2.7. ([29]) A convex modular  in a vector space Y is called
uniformly convex if for every " > 0 there exists a number
(") > 0 such
that for all u, v 2 Y satisfying (u) = (v) = 1, the inequality (u − v) > "
implies  􀀀u+v
2
< 1 −
(").
By Proposition 2.4 and Definitions 2.6 and 2.7, the following proposition
is clear.
Proposition 2.5. Let A :
 × Rk ! R be a k-Musielak-Orlicz function
and satisfy the condition (
2,(x)). Then the space LA(
,Rk) is uniformly
convex if and only if the modular A is uniformly convex.
To study the uniform convexity of the space LA(
,Rk) the following
definition is useful.
Definition 2.8. ([38]) Let A :
×Rk ! R be a k-Musielak-Orlicz function.
For every ", b 2 (0, 1) and x 2
, define
E",b(x) = {(, ) 2 Rk × Rk : Ax,
 − 
2 
1
2
max{A(x, "),A(x, ")},
Ax,
 + 
2  > (1 − b)A(x, ) + A(x, )
2 }, (2.5)
and
q",b(x) = sup| − |
2
: (, ) 2 E",b(x). (2.6)
We say that A satisfies the condition (UC), if
lim
b!0 Z

A|_| (x, q",b(x)) dx = 0 for every " 2 (0, 1). (2.7)
Here we adopt the agreement that q",b(x) = 0 when E",b(x) is empty.
In [38] the following theorem has been proved:
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Theorem 2.2. (Theorem 5.8 of [38]). Let A :
 × Rk ! R be a generalized
N-function, i.e. a k-Musielak-Orlicz function satisfying the additional
hypotheses 5) and 6) in Remark 2.2. Then the space LA(
,Rk) is uniform
convex if and only if A satisfies the conditions (
2,(x)) and (UC).
It can be seen from the proof of Theorem 5.8 of [38] that, in fact, in the
proof of Theorem 5.8 of [38], the hypothesis 5) in Remark 2.2 is needless,
and the hypothesis 6) in Remark 2.2 is used only in the proof of the part
“only if” and it can be replaced by the more relaxative hypothesis that A is
of quasi-scalar type. Thus we have the following:
Theorem 2.3. Let A :
 × Rk ! R be a k-Musielak-Orlicz function. If
A satisfies the conditions (
2,(x)) and (UC), then the space LA(
,Rk) is
uniform convex. Inversely, in the case when A is of quasi-scalar type, the
conditions (
2,(x)) and (UC) are also necessary in order that the space
LA(
,Rk) is uniform convex.
Because of Theorem 2.2, the following definition is reasonable.
Definition 2.9. Let A :
×Rk ! R be a k-Musielak-Orlicz function. The
function A is called uniformly convex if A satisfies the condition (UC).
The Theorem 11.6 of [29] is a corollary of Theorem 2.3, which is stated
as follows.
Corollary 2.1. (Theorem 11.6 of [29]) Let A :
×R ! R be a 1-Musielak-
Orlicz function. If
i). A satisfies the condition (
2);
ii). there exists a function  mapping the interval (0, 1) into itself such
that for every s > 0, 0 < < 1 and 0   , there holds the inequality
Ax,
s + s
2  (1 − ( ))A(x, s) + A(x, s)
2
for a.e. x 2
, (2.8)
then the space LA(
,R) is uniform convex.
It is easy to verify that the condition ii) in Corollary 2.1 implies the
condition (UC).
The following theorem is a special case of Theorem 5.14 of [38] when
X = Rk, see also [22].
Theorem 2.4. Suppose that A :
 × Rk ! R is a k-Musielak-Orlicz
function of scalar type, that is there exists a 1-Musielak-Orlicz function
 :
 × R ! R such that A(x, ) = (x, ||) for a.e. x 2
 and all
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三思论文代写网
 2 Rk. If  satisfies the conditions (
2,(x)) and (UC), then A satisfies
the conditions (
2,(x)) and (UC), and consequently the space LA(
,Rk) is
uniform convex.
In the end of this section we give the following theorem which is useful
in Section 3 and is also of independent interest.
Theorem 2.5. Let A :
×Rk ! R be a uniformly convex k-Musielak-Orlicz
function satisfying the condition (
2,(x)). If {vn}  LA(
,Rk), vn * v in
LA(
,Rk) and (vn) ! (v) as n ! 1, then vn ! v in LA(
,Rk) as
n ! 1.
Proof. By Theorem 2.3, the space LA(
,Rk) is uniformly convex. Let
{vn}  LA(
,Rk), vn * v in LA(
,Rk) and (vn) ! (v) = c. When c = 0,
by 5) of Proposition 2.4, vn ! 0 = v in LA(
,Rk). When c = 1, by 6) of
Proposition 2.4, kvnk ! 1 = kvk. Since LA(
,Rk) is uniformly convex, it
follows from vn * v and kvnk ! kvk that vn ! v in LA(
,Rk). When c > 0
and c 6= 1, consider the function Ac = 1
cA and the corresponding modular
Ac = 1
c A. It is easy to see that Ac is also a uniformly convex k-Musielak-
Orlicz function satisfying the condition (
2,(x)), LAc(
,Rk) = LA(
,Rk)
and the norms k·kAc
and k·kA are equivalent. Thus we have that vn * v in
(LAc(
,Rk), k·kAc) and Ac(vn) ! Ac(v) = 1, which implies that vn ! v
in LA(
,Rk). 
3 (S)+-condition
The aim of this section is to give a sufficient condition in order that a quasilinear
elliptic operator is of type (S)+. For the mappings of type (S)+ see
e.g. [7,16,26,39].
Definition 3.1. Let X be a normed linear space and let X be its dual
space. Suppose T : X ! 2X is a set-valued mapping with nonempty values,
where 2X = {E : E  X}. T is said to be a mapping of type (S)+ if it
satisfies the following (S)+- condition:
(S)+: For any sequence {un}  X for which un * u in X and
limn!1(T(un), un − u)  0, (3.1)
un must converges strongly to u in X.
In this paper we adopt the usual agreement about the notation involving
the set-valued mappings. For example, the inequality (3.1) means that the
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三思论文代写网
inequality holds when T(un) is replaced by every y 2 T(un).
Remark 3.1. When T : X ! 2X is a monotone mapping, the condition
(3.1) implies that limn!1(T(un), un − u) = 0.
For a convex functional  : X ! R, @(u) denotes the subdifferential of
 at u 2 X, that is,
@(u) = {x 2 X : (x, v − u)  (v) − (u), 8v 2 X} .
For the subdifferentials of the convex functionals see e.g. [6].
Theorem 3.1. Let A :
 × Rk ! R be a uniformly convex k-Musielak-
Orlicz function satisfying the condition (
2,(x)) and let the modular  = A
and the space (LA(
,Rk), k·kA) be as in Section 2. Denote X = LA(
,Rk).
Then @ : X ! 2X is of type (S)+.
Proof. We know that, under the hypotheses of Theorem 3.1,  : X ! R
is a continuous convex functional. Hence @(u) 6= ? for all u 2 X and
@ : X ! 2X is a (maximal) monotone mapping (see e.g. [6]). Now let
{un}  X, un * u in X and
limn!1(@(un), un − u)  0. (3.2)
The boundedness of {un} implies the boundedness of {(un)}. We may
assume, taking a subsequence if necessary, that (un) ! c. Noting that
 : X ! R is weakly lower-semicontinuous, we have (u)  c. By the
definition of @(un),
(u)  (un) + (@(un), u − un),
which and (3.2) imply (u)  c. Thus (u) = c = limn!1 (un). By Theorem
2.5, un ! u in X. 
Remark 3.2. By Definition 3.1, unlike the uniform convexity of the space
X, the (S)+-property of the mapping T is independent of the choice of the
equivalent norms in X.
For a k-Musielak-Orlicz function A :
 × Rk ! R, we shall use the
following condition (B):
(B) There exist a positive constant C and a nonnegative function h 2
L1l
oc(
) such that
||  CA(x, ) + h(x) for a.e. x 2
 and all  2 Rk.
The following proposition is obvious.
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三思论文代写网
Proposition 3.1. Let A :
 × Rk ! R be a k-Musielak-Orlicz function
satisfying the conditions (
2,(x)) and (B). Then there exists a continuous
imbedding LA(
,Rk) ,! L1l
oc(
,Rk).
Let A0 :
×R ! R be a 1-Musielak-Orlicz function and A1 :
×RN ! R
an N-Musielak-Orlicz function. Suppose that both A0 and A1 satisfy the
conditions (B) and (
2,(x)). Define
W1,(A0,A1)(
) = u 2 LA0(
,R) : ru 2 LA1(
,RN) , (3.3)
with the norm
kuk(A0,A1) = kukW1,(A0,A1)(
) = kukA0
+ krukA1
, 8u 2 W1,(A0,A1)(
).
(3.4)
Remark 3.3. Let us explain the meaning of ru appearing in (3.3) and
(3.4). Let u 2 LA0(
,R). Then, by Proposition 3.1, u 2 L1l
oc(
). Thus u
can be regarded as a generalized function or distribution on
 (see [2]). @u
@xi
denotes the derivative of the distribution u. ru = ( @u
@x1
, @u
@x2
, · · · , @u
@xN
).
It is clear that un ! u inW1,(A0,A1)(
) if and only if un ! u in LA0(
,R)
and run ! ru in LA1(
,RN).
Theorem 3.2. The space W1,(A0,A1)(
) is a Banach space.
Proof. Let {un} be a Cauchy sequence in W1,(A0,A1)(
). Then {un} is a Cauchy sequence in LA0(
,R) and {run} is a Cauchy sequence in
LA1(
,RN). Since the spaces LA0(
,R) and LA1(
,RN) are complete, there
are u 2 LA0(
,R) and v 2 LA1(
,RN) such that un ! u in LA0(
,R) and
run ! v in LA1(
,RN). By Proposition 3.1, un ! u in L1l
oc(
,R) and
run ! v in L1l
oc(
,RN). This shows that un ! u and run ! v in the
sense of the distributions. Using the standard arguments (see e.g. [2]) we
can prove that v = ru. Hence u 2 W1,(A0,A1)(
) and the proof is complete.

Theorem 3.3. Let A0, A1 and W1,(A0,A1)(
) be as above. Define
(u) = A0(u) + A1(ru), 8u 2 W1,(A0,A1)(
). (3.5)
If A0 and A1 are uniformly convex, then @ : W1,(A0,A1)(
) ! 2(W1,(A0,A1)(
))
is of type (S)+.
Proof. Denote X = W1,(A0,A1)(
), X0 = LA0(
,R), X1 = LA1(
,RN).
Define J0 : X ! X0 and J1 : X ! X1 by J0(u) = u and J1(u) = ru
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三思论文代写网
for every u 2 X respectively and define '0 and '1 : X ! R by '0(u) =
A0(J0(u)) and '1(u) = A1(J1(u)) for every u 2 X respectively. Then
(u) = '0(u) + '1(u). The functionals '0 and '1 : X ! R are continuous
and convex. Thus the functional  : X ! R is continuous and convex, and
@(u) = @'0(u) + @'1(u) for every u 2 X (see [6]). Now let {un}  X,
un * u in X and limn!1(@(un), un − u)  0. Then un * u in X0,
run * ru in X1 and
limn!1(@'0(un) + @'1(un), un − u)  0.
Since @'0 and @'1 are monotone, by Remark 3.1, it follows that
lim
n!1
(@'0(un), un − u) = 0 and lim
n!1
(@'1(un), un − u) = 0,
that is,
lim
n!1
(@A0(un), un − u) = 0 and lim
n!1
(@A1(run),run − ru) = 0.
By Theorem 3.1, @A0 and @A1 are of type (S)+, consequently un ! u in
X0 and run ! ru in X1. Thus un ! u in X and the proof is complete. 
Remark 3.4. We may regard  defined by (3.5) as a convex modular in
L1l
oc(
). kuk defined by
kuk = inf n > 0 : (u

)  1o
is a norm in W1,(A0,A1)(
), which is equivalent to the norm kukW1,(A0,A1)(
).
Now let
 be a bounded open subset of RN and A1 :
 × RN ! R
be an N-Musielak-Orlicz function satisfying the conditions (B), (
2,(x))
and (UC). Suppose that A|_| 1 is locally integrable (see Remark 2.4 for
the definition). Then L1(
,RN)  LA1(
,RN). Define kukW
1,A1
0 (
)
=
krukLA1(
,RN) for every u 2 C10 (
). Then kukW
1,A1
0 (
)
is a norm in the
vector space C10 (
). The completion of C10 (
) with respect to the norm
k·kW
1,A1
0 (
)
is denoted by W1,A1
0 (
).
From the local integrability of A|_| 1 and the condition (B) we can see
that there is a continuous imbedding W1,A1
0 (
) ,! W1,1
0 (
). If define J1 :
C10 (
) ! LA1(
,RN) by J1(u) = ru for every u 2 C10 (
), then the
mapping J1 is injective and, W1,A1
0 (
) and (J1(C10 (
))), the closure of
J1(C10 (
)) in LA1(
,RN), are linearly homeomorphic. Since LA1(
,RN) is
a uniformly convex Banach space, so is W1,A1
0 (
). Define ' : W1,A1
0 (
) ! R
by
'(u) = A1(ru) = Z

A1(x,ru)dx, 8u 2 W1,A1
0 (
). (3.6)
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三思论文代写网
Analogously to Theorem 3.3, we can prove that @' : W1,A1
0 (
) ! 2(W
1,A1
0 (
))
is of type (S)+. Let us formulate these in the form of the following theorem.
Theorem 3.4. Let
 be a bounded open subset of RN and A1 :
×RN ! R
an N-Musielak-Orlicz function satisfying the conditions (B), (
2,(x)) and
(UC). Suppose that A|_| is locally integrable. Then
1). W1,A1
0 (
), k·kW
1,A1
0 (
) is a uniformly convex Banach space and
there is a continuous imbedding W1,A1
0 (
) ,! W1,1
0 (
).
2). @' : W1,A1
0 (
) ! 2(W
1,A1
0 (
)) is of type (S)+, where ' : W1,A1
0 (
) !
R is as in (3.6).
Remark 3.5. We may regard ' defined by (3.6) as a convex modular in
W1,1
0 (
). W1,A1
0 (
) is just the modular space generated by ' and k·kW
1,A1
0 (
)
is the Luxemburg norm.
Now let
 be a bounded open subset of RN and A0,A1 be as in Theorem
3.3. In the calculus of variations we need to consider the integral functional
of the form
I(u) = Z

(A1(x,ru) + A0(x, u))dx − Z

F(x, u)dx, 8u 2 W1,(A0,A1)(
),
(3.7)
where F :
 × R ! R. Suppose that the functions A1(x, ), A0(x, t) and
F(x, t) are continuously differentiable in the second variable and denote
a1(x, ) = A0
1(x, ), a0(x, t) = A0
0(x, t) and f(x, t) = F0(x, t), where 0 denotes
the derivative with respect to the second variable. Under the appropriate
hypotheses, I 2 C1(W1,(A0,A1)(
),R) and
I0(u)v = Z

(a1(x,ru)rv+a0(x, u)v)dx−Z

f(x, u)vdx, 8u, v 2 W1,(A0,A1)(
).
In such a case, the critical points of I are just the weak solutions of the
Neumann boundary value problem
 −div(a1(x,ru)) + a0(x, u) = f(x, u) in

a1(x,ru)n = 0 on @
,
where n is the unit outward normal vector on @
. Define for u 2 W1,(A0,A1)(
),
(u) = Z

(A1(x,ru) + A0(x, u))dx,  (u) = Z

F(x, u)dx.
Then I(u) = (u) −  (u) and I0(u) = 0(u) −  0(u). By Theorem 3.3,
0 : W1,(A0,A1)(
) ! (W1,(A0,A1)(
)) is of type (S)+. A common case is
that the mapping  0 is weakly-strongly continuous, that is, un * u implies
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三思论文代写网
 0(un) !  0(u). In this case the mapping I0 is of type (S)+ because that the
sum of a mapping of type (S+) and a weakly-strongly continuous mapping
is also of type (S)+. It is well known that in the critical point theory a key
step is to verify that the functional I satisfies (PS) condition. When I0 is of
type (S)+, the verification of (PS) condition becomes easier thanks to the
following principle.
Theorem 3.5. Let X be a reflexive Banach space and I 2 C1(X,R). Suppose
that I0 : X ! X is of type (S+). Then any bounded (PS) sequence,
that is a bounded sequence {un}  X such that {I(un)} is bounded and
I0(un) ! 0, has a convergent subsequence.
Proof. Suppose that {un}  X, {kunk} and {I(un)} are bounded, and
I0(un) ! 0. Since X is reflexive, there is a subsequence {unj} of {un} such
that unj * u in X. It follows from I0(un) ! 0 that limn!1(I0(unj ), unj −
u) = 0. Because I0 is of (S)+, we have unj ! u in X. 
By Theorem 3.5, when I0 : X ! X is of type (S+), in order to verify
that I satisfies (PS) condition, (that is, any (PS) sequence has a convergent
subsequence), it is sufficient to show that any (PS) sequence is bounded.
The above discussion for the integral functional I with form (3.7) is also
suitable for the following integral functional with form
I0(u) = Z

A1(x,ru)dx − Z

F(x, u)dx, 8u 2 W1,A1
0 (
),
where A1 is as in Theorem 3.4 and F :
 × R ! R. Under the appropriate
conditions, I0 2 C1(W1,A1
0 (
),R) and the critical points of I0 are just the
weak solutions of the Dirichlet boundary value problem
 −div(a1(x,ru)) = f(x, u) in

u = 0 on @
.
Theorem 3.4 ensures the (S)+-property of the operator −div(a1(x,ru).
4 Operators of p(x)-Laplacian type
The theory established in Section 3 possesses the generality. In the present
section we shall study a special subclass of the k-Musielak-Orlicz functions,
which is called of type with variable exponent p(x), and introduce the notion
of the p(x)-Laplacian type operators which include the p(x)-Laplacian operator
−div|ru|p(x)−2 ruas a special case, where p 2 S(
,R) and satisfies
the condition
1 < p− = essinf
x2

p(x)  p+ = esssup
x2

p(x) < 1. (4.1)
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三思论文代写网
Let
 be an open subset of RN and p 2 S(
,R) satisfy (4.1). For
a positive integer k, the variable exponent Lebesgue space Lp(x)(
,Rk) is
defined by
Lp(x) 
,Rk= {u 2 S 
,Rk: Z
 |u|p(x) dx < 1}
with the norm
|u|Lp(x)(
,Rk) = |u|p(x) = inf  > 0 : Z

u

p(x)
dx  1.
The variable exponent Sobolev space W1,p(x)(
) is defined by
W1,p(x) (
) = nu 2 Lp(x) (
,R) : ru 2 Lp(x) 􀀀
,RN
o
with the norm
kukW1,p(x)(
) = |u|Lp(x)(
) + |ru|Lp(x)(
,RN) .
W1,p(x)
0 (
) denotes the closure of C10 (
) in W1,p(x) (
). For the basic properties
of the spaces Lp(x) (
) and W1,p(x)(
) we refer to [11,13,17,24,32].
It is easy to see that Lp(x) 􀀀
,Rk
= LA 􀀀
,Rk
with A :
 × Rk ! R
such that A(x, ) = ||p(x) , and W1,p(x) (
) = W1,(A0,A1) (
) with A0 :

 × R ! R and A1 :
 × RN ! R such that A0(x, t) = |t|p(x) and
A1(x, ) = ||p(x).
Definition 4.1. Let A :
 × Rk ! R be a k-Musielak-Orlicz function. A
is said to be of type with variable exponent p(x) if the following conditions
are satisfied.
(1) There exists a Carath´eodory function a :
 × Rk ! Rk such that
a(x, ) = A0(x, ) for a.e. x 2
 and all  2 Rk, where A0(x, ) denotes the
derivative of A(x, ) with respect to .
(2) There exist a positive constant c1 and a nonnegative function g 2
L
p(x)
p(x)−1 (
) such that |a(x, )|  c1 ||p(x)−1 +g(x) for a.e. x 2
 and all  2
Rk.
(3) There exists a positive constant
 such that A(x, ) 
 ||p(x) for
a.e. x 2
 and all  2 Rk.
(4) A is uniformly convex, that is A satisfies the condition (UC).
When an N-Musielak-Orlicz function A is of type with variable exponent
p(x), the operator −div(a(x,ru)) is called of p(x)-Laplacian type.
Theorem 4.1. Let A :
×Rk ! R be a k-Musielak-Orlicz function of type
with variable exponent p(x). Then the following statements hold.
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三思论文代写网
1) There exist a positive constant c2 and a nonnegative function h 2
L1(
) such that A(x, )  c2 ||p(x) + h(x) for a.e. x 2
 and all  2 Rk.
2) A|_| is locally integrable and A satisfies the conditions (B) and (
2,(x)).
3) LA(
,Rk) = Lp(x)(
,Rk), the norms k·kLA(
,Rk) and |u|Lp(x)(
,Rk)
are equivalent.
4) (LA(
,Rk), k·kLA(
,Rk)) is a uniformly convex and separable Banach
space.
5) The functional A : LA(
,Rk) ! R defined by
A(u) = Z

A(x, u(x))dx, 8u 2 LA(
,Rk),
is of C1 class,
0
A(u)v = Z

a(x, u(x))v(x)dx, 8u, v 2 LA(
,Rk), (4.2)
and the mapping 0
A : Lp(x)(
,Rk) ! 􀀀Lp(x)(
,Rk)
 is of type (S+).
Proof. 1). It follows from the condition (2) in Definition 4.1 that
A(x, )  c1 ||p(x) + g(x) ||  c2 ||p(x) + c3(g(x))
p(x)
p(x)−1 ,
which shows that statement 1) holds with h(x) = c3(g(x))
p(x)
p(x)−1 .
2). Statement 1) implies that
A|_|(x, t)  c2 |t|p(x) + h(x), 8x 2
, 8t  0,
and hence A|_| is locally integrable. It follows from the condition (3) in
Definition 4.1 that ||  1

A(x, ) + 1, which shows that A satisfies the
condition (B). Let us verify the condition (
2,(x)). Setting (x) = (h(x) 1
p(x) ,
then  2 Lp(x)(
) and
A|_|(x, 2(x))  c2 |2(x)|p(x) + h(x)  2p+c2 |(x)|p(x) + |(x)|p(x) ,
consequently R
 A|_|(x, 2(x))dx < 1. For x 2
 and  2 Rk with || 
(x), we have that
A(x, 2)  c2 |2|p(x) + h(x)  2p+c2 ||p(x) + |(x)|p(x)
 (2p+c2 + 1) ||p(x) 
1

A(x,run)dx − kunk − c2
 1 −
p+
μ 
 kunkp− − kunk − c3,
which shows {kunk} is bounded. By Theorem 3.5, I satisfies the (PS) condition.
Using the standard arguments we can verify that I satisfies the
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三思论文代写网
geometry conditions of the mountain pass lemma (see e.g. [14,15,16]). By
the mountain pass lemma (see e.g. [35,37]), I has a nontrivial critical point
v1 such that I(v1) > 0. When (5.7) holds, using the symmetric mountain
pass lemma (see e.g. [35]) we can obtain the required result. 
Remark 5.2. It is well known that, in the p(x)-Laplacian case, that is
when A(x, ) = 1
p(x) ||p(x) , there holds the equality
a(x, ) = p(x)A(x, ), 8x 2
, 8 2 RN.
As noted in the end of the previous section, the condition (4.13) given in
[10] holds for A(x, ) = 1
p h(1 + ||2) p
2 − 1i if and only if p  2. When A is
as in Example 4.1, it is easy to see that there holds the inequality
a(x, )  p(x)A(x, ) + p(x)b(x)(!(x))p(x), (5.12)
which shows that in this case the condition (5.10) holds. This shows the
condition (5.10) is reasonable.
Remark 5.3. As mentioned in the beginning of the previous section, the
(S)+-property of the operators of p(x)-Laplacian type is only an example of
applying the theory established in Section 3. The theory can also be applied
to other cases. For example, let for i = 1, 2, · · · ,N, Ai :
×R ! R be such
that
Ai(x, t) = bi(x) !2
i (x) + |t|2pi(x)
2 − bi(x)(!i(x))pi(x), 8x 2
, 8t 2 R,
where pi, bi and !i satisfy the same conditions as for p, b and ! in Example
4.1. Define A :
 × RN ! R by
A(x, ) =
N Xi=1
Ai(x, i), 8x 2
, 8 = (1, 2, · · · , N) 2 RN.
Then it is easy to see that the space LA(
,RN) with the Luxemburg norm
is uniformly convex. When
 is bounded, we can define the space W1,A
0 (
)
and can prove that the operator −div(a(x,ru)), as a mapping fromW1,A
0 (
)
into W1,A
0 (
), is of type (S)+. In such a case, the operator −div(a(x,ru))
is anisotropic. For the anisotropic Orlicz-Sobolev spaces and the anisotropic
elliptic equations see [23] and [5].
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(2003) 743-755.
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in RN involving the p(x)-Laplacian, Progress in Nonlinear Differential
Equations and Their Applications 66 (2005) 17–32.
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31

u = 0 on @
.
(5.1)
In this section we always assume that f 2 C0(
 × R,R) satisfies the subcritical
growth condition
|f(x, t)|  c1 |t|q(x)−1 + c2, 8x 2
, 8t 2 R, (5.2)
where c1 and c2 are positive constants, q 2 C0(
) and
1 < q(x) < p(x), 8x 2
, (5.3)
p(x) denotes the Sobolev critical exponent for p(x), that is
p(x) = ( Np(x)
N−p(x) if p(x) < N
+1 if p(x)  N.
24
三思论文代写网
Definition 5.1. u 2 W1,p(x)
0 (
) is called a weak solution of (5.1) if
Z

a(x,ru)rvdx = Z

f(x, u)vdx, 8v 2 W1,p(x)
0 (
). (5.4)
Define I : W1,p(x)
0 (
) ! R by
I(u) = Z

A(x,ru)dx − Z

F(x, u)dx, 8u 2 W1,p(x)
0 (
),
where F(x, t) = Rt
0 f(x, s)ds. Then it is easy to see that I 2 C1(W1,p(x)
0 (
),R)
and
I0(u)v = Z

a(x,ru)rvdx − Z

f(x, u)vdx, 8u, v 2 W1,p(x)
0 (
).
Thus, u 2 W1,p(x)
0 (
) is a weak solution of (5.1) if and only if u is a critical
point of I.
Below denote X = W1,p(x)
0 (
), kuk = kukW1,p(x)
0 (
)
, '(u) = R
 A(x,ru)dx
and  (u) = R
 F(x, u)dx. ci denotes a generic positive constant. By Theorem
4.3, the mapping '0 : X ! X is of type (S)+. Since the imbedding
W1,p(x)
0 (
) ,! Lq(x)(
) is compact for the subcritical q(x), the mappings
  : X ! R and  0 : X ! X are weakly-strongly continuous. So the mapping
I0 : X ! X is also of type (S)+.
Theorem 5.1. Let f satisfy the condition
|f(x, t)|  c1 |t| (x)−1 + c2, 8x 2
, 8t 2 R, (5.5)
where c1 and c2 are positive constants, 2 C0(
) and 1 < −  + < p−.
Then:
1) The functional I achieves its infimum at some u 2 X and consequently
u is a solution of (5.1).
2) If there exist a point x0 2
, a neighborhood
0 of x0 in
, and
positive constants "0, k1, k2, r0, s0 with "0 < 1 and r0 > s0 such that
A(x, )  k1 ||r0 and F(x, t)  k2 |t|s0 for x 2
0, ||  "0 and |t|  "0,
(5.6)
then u 6= 0, where u is as in statement 1), consequently (5.1) has a nontrivial
solution.
3) Under the hypotheses of statement 2), if in addition,
f(x,−t) = −f(x, t), 8x 2
, t 2 R, (5.7)
25
三思论文代写网
then (5.1) has a sequence of solutions {±uk : k = 1, 2, · · · } such that
I(±uk) < 0 and I(±uk) ! 0 as k ! 1.
Proof. 1). The space X is reflexive. The functional ' is weakly lower semicontinuous
because it is continuous and convex, consequently I is weakly
lower semi-continuous. The condition (5.5) implies that
|F(x, t)|  c3 |t| (x) + c4, 8x 2
, 8t 2 R.
From this and the condition (3) in Definition 4.1 it follows that, when kuk  1,
I(u) = '(u) −  (u) 
 Z
 |ru|p(x) dx − c3 Z
 |u| (x) dx − c4 |
|

 kukp− − c5 kuk + − c6.
This shows I is coercive because + < p−. Hence the statement 1) holds.
2). It suffices to show that under the hypothesis (5.6), infu2X I(u) < 0.
Take u0 2 C10 (
0)\{0}, then u0 2 X. From (5.6) and noting that r0 > s0,
we can see that for sufficiently small t > 0,
I(tu0)  tr0k1 Z
 |ru0|r0 dx − ts0k2 Z
 |u0|s0 dx < 0. (5.8)
This shows infu2X I(u) < 0 and u 6= 0 since I(0) = 0. In this case u is a
nontrivial solution of (5.1).
3). Since I is coercive, by Theorem 3.5, I satisfies (PS) condition. (5.7)
implies that the functional I is even. We shall prove statement 3) by using
the genus theory (see e.g. [35]). Set
P= {E  X\{0} : E is compact and −E = E}, Pk = {E 2 P:
(E)  k}, k = 1, 2, · · · , where
(E) denotes the genus
of E,
ck = inf
E2Pk
sup
u2E
I(u), k = 1, 2, · · · .
Then −1 < infX I(u) = c1  c2  · · ·  ck  ck+1  · · · . We claim that
ck < 0 for every k = 1, 2, · · · . (5.9)
To see this, noting that C10 (
0) is an infinite dimension subspace of X, for
each k, we take a k-dimension subspace Yk  C10 (
0) and denote by Sk the
unit sphere in Yk, i.e. Sk = {u 2 Yk : kuk = 1}. From the proof of statement
2) we know that for every u 2 Sk, there exists tu > 0 such that I(tu) < 0
for t 2 (0, tu]. It follows from the continuity of I and the compactness of
Sk that there exists t > 0 such that I(tu) < 0 for all u 2 Sk. Setting
Ek = tSk, then
(Ek) = k and supu2Ek I(u) < 0. This shows ck < 0 and
consequently ck is a critical value of I. Using the standard arguments (see
26
三思论文代写网
e.g. [15]) we can prove that ck ! 0. The proof is complete. 
Remark 5.1. In the p(x)-Laplacian case, that is when A(x, ) = 1
p(x) ||p(x) ,
the growth condition of A(x, ) with respect to  is clear naturally. For a
general function A of type with variable exponent p(x), by the conditions
(2) and (3) in Definition 4.1, the order of infinity for A(x, ) as || ! 1 is clear, but the order of the infinitesimal for A(x, ) as || ! 0 can be
complicated. If A is as in Example 4.1, then it is easy to see that, at x0 2

with !(x0) 6= 0, there holds
A(x0, )  k1 ||2 when ||  1.
Theorem 5.2. Suppose that the following conditions are satisfied.
1 o) There are M1 > 0 and h 2 L1(
) such that
a(x , )  p+A(x , ) + h(x ), 8x 2
, 8 2 RN with || M1. (5.10)
2 o) There are μ > p+ and M2 > 0 such that
0 < μF(x , t)  f (x , t)t, 8x 2
, 8t 2 R with |t| M2. (5.11)
3 o) limt!0
F(x,t)
|t|p+ = 0 uniformly in x 2
.
Then (5.1) has a nontrivial solution which corresponds to the positive
critical value. If in addition the condition (5.7) is also satisfied, then (5.1)
has infinitely many pairs of solutions {±vk : k = 1, 2, · · · } such that I(±vk) ! +1 as k ! 1.
Proof. At first let us verify the (PS) condition. Let {un}  X, I(un) ! c
and I0(un) ! 0. Then for sufficiently large n, we have that
1 + c  I(un) = Z

A(x,run)dx − Z

F(x, un)dx
 Z

A(x,run)dx −
1
μ Z

f(x, un)undx − c1
= Z

A(x,run)dx −
1
μ Z

a(x,run)rundx +
1
μ
I0(un)un − c1
 Z

A(x,run)dx −
p+
μ Z