\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 44, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/44\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for a differential
inclusion problem involving the $p(x)$-Laplacian}

\author[G. Dai\hfil EJDE-2010/44\hfilneg]
{Guowei Dai}

\address{Guowei Dai \newline
 Department of Mathematics, Northwest Normal University,
 Lanzhou, 730070,  China}
\email{daigw06@lzu.cn}

\thanks{Submitted December 31, 2009. Published March 26, 2010.}
\thanks{Supported by grants NNSFC 10971087 and NWNU-LKQN-09-1.}
\subjclass[2000]{35J20, 35J70, 35R70}
\keywords{$p(x)$-Laplacian; nonsmooth mountain pass theorem;
\hfill\break\indent differential inclusion}

\begin{abstract}
 In this article we consider the differential inclusion
 \begin{gather*}
 -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)\in \partial F(x,u)
 \quad\text{in }\Omega,\\
 u=0 \quad \text{on }\partial \Omega
 \end{gather*}
 which involves the $p(x)$-Laplacian.
 By applying  the nonsmooth Mountain Pass Theorem, we obtain at
 least one nontrivial solution; and by applying the
 symmetric Mountain Pass Theorem, we obtain $k$-pairs of
 nontrivial solutions in $W_{0}^{1,p(x)}(\Omega)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

  Let $\Omega$ be bounded open subset of
 $\mathbb{R}^{N}$ with a $C^1$-boundary $\partial\Omega$.
 We consider the  differential inclusion problem
 \begin{equation}
\begin{gathered}
-\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)\in \partial F(x,u) \quad
\text{in }\Omega , \\
u=0\quad \text{on }\partial \Omega,
\end{gathered}  \label{e1.1}
\end{equation}
where $p\in C( \overline{\Omega} )$ with
$1<p^{-}:=\inf_{\Omega}p(x)\leq p^{+}:=\sup_{\Omega }p(x)<+\infty$,
$F(x,u)$ is measurable
with respect to $x$ (for every $u\in\mathbb{R}$) and locally
Lipschitz with respect to $u$ (for a.e. $x\in\Omega$), and $\partial
F(x,u)$ is the Clarke sub-differential of $F(x,\cdot)$.

The operator $-\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)$ is
said to be $p(x)$-Laplacian, and becomes $p$-Laplacian when
$p(x)\equiv p$ (a constant). The $p(x)$-Laplacian possesses more
complicated properties than the $p$-Laplacian; for example, it
is inhomogeneous. The study of various mathematical problems with
variable exponent growth condition has been received considerable
attention in recent years. These problems are interesting in
applications and raise many difficult mathematical problems. One of
the most studied models leading to problem of this type is the model
of motion of electro-rheological fluids, which are characterized by
their ability to drastically change the mechanical properties under
the influence of an exterior electromagnetic field \cite{r1,z1}.
Problem with variable exponent growth conditions also appear in the
mathematical modelling of stationary therm-rheological viscous flows
of non-Newtonian fluids and in the mathematical description of the
processes filtration of an ideal baro-tropic gas through a porous
medium \cite{a1,a2}. Another field of application of equations with
variable exponent growth conditions is image processing \cite{c2}. The
variable nonlinearity is used to outline the borders of the true
image and to eliminate possible noise. We refer the reader to
\cite{d1,h1,s1,z2,z3}
for an overview of and references on this subject, and to
\cite{d1,d2,d3,d4,d5,f1,f2,f3,f4,f5,f6,g3}
 for the study of the $p(x)$-Laplacian equations and the
corresponding variational problems.

Since many free boundary problems and obstacle problems may be
reduced to partial differential equations with discontinuous
nonlinearities, the existence of multiple solutions for Dirichlet
boundary value problems with discontinuous nonlinearities has been
widely investigated in recent years. In 1981, Chang \cite{c1} extended
the variational methods to a class of non-differentiable
functionals, and directly applied the variational methods for
non-differentiable functionals to prove some existence theorems for
PDE with discontinuous nonlinearities. Later, in 2000,
Kourogenis and  Papageorgiou \cite{k1} obtained some non-smooth
critical point theories and applied these to nonlinear elliptic
equations at resonance, involving the $p$-Laplacian with
discontinuous nonlinearities.

 Problem \eqref{e1.1} has been studied extensively when
$p(x)\equiv p$ (a constant); see \cite{k1,k2}.
If $f$ is a C\'{a}ratheodory
function and $F(x,u)=\int_0^u f(x,t)\mathrm{d}t$,
then problem \eqref{e1.1} becomes
\begin{equation}
\begin{gathered}
-\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \quad \text{in }
 \Omega , \\
u=0\quad \text{on }\partial \Omega.
\end{gathered}  \label{e1.2}
\end{equation}
which has also been studied extensively; see \cite{f3,f5}.
We emphasize that in our approach, no continuity with respect to
the second argument will be required on the function $f$. So
\eqref{e1.2} need not have a solution. To avoid this situation,
we consider functions $f(x,\cdot)$ which are locally essentially
bounded and fills the discontinuity gaps of $f(x,\cdot)$,
replacing $f$ by an interval $[f_1, f_2]$, where
\begin{gather*}
f_1(x,s)=\lim_{\delta\to 0^{+}}\mathop{\rm ess\,inf}_{|t-s|<\delta}f(x,t), \\
f_2(x,s)=\lim_{\delta\to 0^{+}}\mathop{\rm ess\,sup}_{|t-s|<\delta}f(x,t).
\end{gather*}
It is well known that if
$F(x,u)=\int_{0}^{u}f(x,t)\mathrm{d}t$, then $F$ becomes locally
Lipschitz and $\partial F(x,u)=[f_1(x,u),f_2(x,u)]$ (see \cite{k3}).
This fact motivates the formulation of the \emph{differential
inclusion} problem \eqref{e1.1}.

This paper is organized as follows: In Section 2, we present some
necessary preliminary knowledge on variable exponent Sobolev spaces
and generalized gradient of locally Lipschitz function; In Section
3, we give the main results of this paper. In Section 4; we use the
nonsmooth Mountain Pass Theorem and symmetric Mountain Pass Theorem
to prove our main results.

\section{Preliminaries}


 To discuss problem \eqref{e1.1}, we need some properties of
$W_{0}^{1,p(x) }(\Omega ) $ (see \cite{f6}) and of the
generalized gradient of locally Lipschitz functions,
which will be used later.
 Denote by ${\mathbf{S}}(\Omega )$ the set of all measurable
real functions defined on $\Omega$. Two functions in
${\mathbf{S}}(\Omega )$ are considered as the same
element  when they are equal almost everywhere.
Let
\[
L^{p(x)}(\Omega ) =\{u\in {\mathbf{S}}(\Omega
):\int_{\Omega }|u(x)|
^{p(x)}\mathrm{d}x<+\infty \}
\]
with the norm
\[
|u|_{L^{p(x)}(\Omega ) }=|u|_{p(x)}=\inf \big\{ \lambda >0:\int_{\Omega }|\frac{
u(x)}{\lambda }|^{p(x)}\mathrm{d}x\leq 1\big\} ,
\]
and let
\[
W^{1,p(x) }(\Omega ) =\{ u\in
L^{p(x) }(\Omega ) :|\nabla u|\in L^{p(x) }(\Omega ) \}
\]
with the norm
\[
\| u\| _{W^{1,p(x)}(\Omega )}
= |u|_{L^{p(x)}(\Omega )}+|\nabla u|_{L^{p(x)}(\Omega )}.
\]
Denote by $W_{0}^{1,p(x) }(\Omega ) $ the
closure of $C_{0}^{\infty }(\Omega ) $ in $W^{1,p(x) }(\Omega )$.

\begin{proposition}[\cite{f6}] \label{prop2.1}
 The spaces $L^{p(x)}(\Omega) $, $W^{1,p(x)}(\Omega )$ and
$W_{0}^{1,p(x) }(\Omega ) $ are separable and
reflexive Banach spaces.
\end{proposition}

\begin{proposition}[\cite{f6}] \label{prop2.2}
 Let $\rho (u)=\int_{\Omega }|u(x)|^{p(x)}\mathrm{d}x$,
for $u\in L^{p(x)}(\Omega )$. Then:
\begin{itemize}
\item[(1)] For $u\neq 0$, $|u|_{p(x)}=\lambda$ implies
$ \rho (\frac{u}{\lambda })=1$

\item[(2)] $|u|_{p(x)}<1$ $(=1;>1) \Leftrightarrow \rho
(u)<1$ $(=1;>1)$

\item[(3)] If $|u|_{p(x)}>1$, then
$|u|_{p(x)}^{p^{-}}\leq \rho (u)
\leq |u|_{p(x)}^{p^{+}}$

\item[(4)] If $|u|_{p(x)}<1$, then
$|u|_{p(x)}^{p^{+}}\leq \rho (u)
\leq |u|_{p(x)}^{p^{-}}$

\item[(5)] $\lim_{k\to  +\infty }|u_{k}|
_{p(x)}=0$ if and only if $\lim_{k\to  +\infty }\rho (u_{k})=0$

\item[(6)] $\lim_{k\to  +\infty }|u_{k}|_{p(x)}=
+\infty$ if and only if $\lim_{k\to  +\infty }\rho
(u_{k})= +\infty$.
\end{itemize}
\end{proposition}

\begin{proposition}[\cite{f6}] \label{prop2.3}
In $W_{0}^{1,p(x) }(\Omega ) $ the Poincar\'e inequality holds; that
is, there exists a positive constant $C_{0}$ such that
\[
|u|_{L^{p(x)}(\Omega )}\leq C_{0}|
\nabla u|_{L^{p(x)}(\Omega )}, \quad \forall u\in
W_{0}^{1,p(x) }(\Omega ).
\]
\end{proposition}

So $|\nabla u|_{L^{p(x)}(\Omega )}$ is an
equivalent norm in $W_{0}^{1,p(x) }(\Omega)$.
We will use the equivalent norm in the following
discussion and write $\| u\|=|\nabla u|_{L^{p(x)}(\Omega )}$
for simplicity.

\begin{proposition}[\cite{f6}] \label{prop2.4}
 (1) Assume that the boundary of $\Omega$ possesses the cone
property and $p\in C(\overline{\Omega })$.
If $q\in C(\overline{\Omega })$ and $1\leq q(x)\leq p^{\ast }(x)$
for $x\in \overline{\Omega }$, then there is a continuous
embedding $W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega )$.
When $1\leq q(x)< p^{\ast}(x)$, the embedding is compact, where
$p^*(x)=\frac{Np(x)}{N-p(x)}$ if $p(x)<N$, $p^*(x)=\infty$ if
$p(x)\geq N$.

(2) If $p_1(x), p_2(x)\in C(\overline{\Omega})$, and
$1<p_1(x)\leq p_2(x)$, then
$L^{p_2(x)}\hookrightarrow L^{p_1(x)}$, and the embedding is continuous.
\end{proposition}

\begin{proposition}[\cite{f6}] \label{prop2.5}
The conjugate space of $L^{p(x)}(\Omega)$ is $L^{q(x)}(\Omega)$, where
$\frac{1}{q(x)}+\frac{1}{p(x)}=1$. For any $u\in L^{p(x)}(\Omega)$
and $v\in L^{q(x)}(\Omega)$, we have
\[
\big|\int_{\Omega}uv\mathrm{d}x \big|
\leq \big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\big) |u|_{p(x)}|v|_{q(x)}.
\]
\end{proposition}

Let $(Y,\| \cdot\|)$ be a real Banach space and $Y^*$ be its
topological dual. A function $f:Y \to  \mathbb{R}$ is
called locally Lipschitz if each point $u\in Y$ possesses a
neighborhood $\Omega_{u}$ such that $|f(u_1)-f(u_2)|\leq L\| u_1-u_2\|$
for all $u_1,u_2\in \Omega_{u}$, for a constant $L>0$  depending on
$\Omega_{u}$. The generalized directional derivative of $f$ at the
point $u\in Y$ in the direction $v\in X$ is
\[
f^{0}(u,v)=\limsup_{w\to u,t\to 0}\frac{1}{t}(f(w+tv)-f(w)).
\]
The generalized gradient of $f$ at $u\in Y$ is
\[
\partial f(u)=\big\{u^*\in X^*:\langle u^*,\varphi\rangle
\leq f^{0}(u;\varphi) \text{ for all } \varphi\in Y\big\},
\]
which is a non-empty, convex and $w^*$-compact subset of $Y^*$,
where $\langle\cdot,\cdot\rangle$ is the duality pairing between
$Y^*$ and $Y$. We say that $u\in Y$ is a critical point of $f$ if
$0\in \partial f(u)$. For further details, we refer the reader to Chang
\cite{c1} or Clarke \cite{c3}.

\section{Main results}

 In this section we give two existence theorems for problem
\eqref{e1.1}. For simplicity we write $X=W_{0}^{1,p(x)}(\Omega )$,
denote by $c$, $c_i$, $l$ and $M$ the general positive constant
(the exact value may change from line to line).
The precise hypotheses are the followings:
\begin{itemize}
\item[(HF)]  $F:\Omega\times \mathbb{R}\to  \mathbb{R}$
is a Borel measurable locally Lipschitz function with $F(x,0)=0$ for
 a.e. $x\in \Omega$ such that
\begin{itemize}
\item[(i)] there exists a constant $c>0$
such that for a.e. $x\in\Omega$, all $u\in \mathbb{R}$ and
all $\xi(u)\in\partial F(x,u)$
\[
|\xi(u)|\leq c(1+|u|^{\alpha(x)-1}),
\]
where $\alpha\in C(\overline{\Omega})$ and
$p^{+}<\alpha^{-}\leq\alpha(x)<p^*(x)$;

\item[(ii)] There exist $M>0$, $\theta>p^+$ such that
\begin{equation}
0<\theta F(x,u)\leq\langle \xi,u\rangle,\quad \text{a.e. }
 x\in \Omega, \text{ all } u\in X,\; |u|\geq M,\,\;
 \xi\in \partial F(x,u);
\end{equation}

\item[(iii)] $F(x,t)=o(|t|^{p^{+}}), t\to 0$,
uniformly for a.e. $x\in\Omega$.
\end{itemize}
\end{itemize}

Because $X$ be a reflexive and separable Banach space, there exist
$e_i\in X$ and $e_j^*\in X^*$ such that
\begin{gather*}
X=\overline{\mathop{\rm span}\{e_i:i=1,2, \ldots\}},\quad
X^*=\overline{\mathop{\rm span}\{e_j^*:j=1,2,\ldots\}},\\
\langle e_i,e_j^*\rangle =
\begin{cases}1,& i=j,\\
0, &i\neq j.
\end{cases}
\end{gather*}
For convenience, we write
$X_i = \mathop{\rm span}\{e_i\}$,
$Y_k =\oplus_{i=1}^kX_i$, $Z_k = \overline{\oplus_{i=k}^\infty X_i}$.
In the following we need the nonsmooth version of
\emph{Palais-Smale}
condition.

\begin{definition} \label{def3.1} \rm
 We say that $I$ satisfies the nonsmooth
$(\mathrm{PS})_{c}$ condition if any sequence $\{u_{n}\}\subset X$
such that $I(u_{n})\to  c$ and $m(u_{n})\to  0$, as
$n\to +\infty$, has a strongly convergent subsequence, where
$m(u_{n})=\inf\{\| u^*\|_{X^*}:u^*\in \partial I(u_{n})\}$.
\end{definition}

In what follows we write the $(\mathrm{PS})_{c}$-condition as simply
the $\mathrm{PS}$-condition if it holds for every level $c\in
\mathbb{R}$ for the \emph{Palais-Smale} condition at level $c$.
Let
\[
J(u)=\int_{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\mathrm{d}x,\quad
\Psi (u)=\int_{\Omega}F(x,u)\mathrm{d}x.
\]

By a solution of \eqref{e1.1}, we mean a function $u\in X$ to which there
corresponds a mapping $\Omega\ni x\to  g(x)$ with $g(x)\in
\partial F(x,u)$ for a.e. $x\in \Omega$ having the property
that for every $\varphi\in X$, the function $x\to
g(x)\varphi(x)\in L^1(\Omega)$ and
\[
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla \varphi
\mathrm{d}x=\int_{\Omega}g(x)\varphi(x)\mathrm{d}x.
\]

By standard argument, we show that $u\in X$ is a solution of
\eqref{e1.1} if and only if $0\in I(u)$, where $I(u)=J(u)-\Psi(u)$.
Below we give a proposition that will be used later.

\begin{proposition}[\cite{f5}] \label{prop3.1}
 The functional $J:X\to  \mathbb{R}$ is convex. The mapping
$J':X\to  X^{\ast }$  is a strictly monotone, bounded
homeomorphism, and is of $(S_{+})$ type; namely
$u_{n}\rightharpoonup u$ and
$\overline{\lim}_{n\to  \infty }(J'(u_{n},u_{n}-u)\leq 0$
implies $u_{n}\to  u$.
\end{proposition}

\begin{theorem} \label{thm3.1} If {\rm (HF)} holds, then
\eqref{e1.1} has at least one nontrivial solution.
\end{theorem}

\begin{theorem} \label{thm3.2} If  {\rm (HF)} holds and
$F(x,-u)=F(x,u)$ for a.e. $x\in \Omega$ and all $u\in \mathbb{R}$,
then \eqref{e1.1} has at least $k$-pairs of nontrivial solutions.
\end{theorem}

To prove Theorems \ref{thm3.1} and  \ref{thm3.2} we need the following
generalizations of the classical Mountain pass Theorem
(see \cite{c1,g1,k1,k2}) and of the
symmetric Mountain pass Theorem \cite{g1,g2}.

\begin{lemma} \label{lem3.1}
If $X$ is a reflexive Banach space,
$I:X\to  \mathbb{R}$ is a locally Lipschitz function which
satisfies the nonsmooth {\rm (PS)c}-condition, and for some
$r>0$ and $e_1\in X $ with $\| e_1\|>r$,
$\max\{I(0),I(e_1)\}\leq\inf\{ I(u):\| u\|=r\}$. Then $I$
has a nontrivial critical $u\in X$ such that the critical value
$c=I(u)$ is characterized by the following minimax principle
\[
  c=\inf_{\gamma\in\Gamma}\max_{t\in [0,1]}I(\gamma(t),
\]
where $\Gamma=\{\gamma\in C([0,1],X): \gamma(0)=0,\gamma(1)=e_1\}$.
\end{lemma}

\begin{lemma} \label{lem3.2}
If $X$ is a reflexive Banach space and
$I:X\to \mathbb{R}$ is even locally Lipschitz functional
satisfying the nonsmooth {\rm (PS)c}-condition and
\begin{itemize}
\item[(i)] $I(0) = 0$;

\item[(ii)] there exists a subspace $Y\subseteq X$ of finite codimension
and number $\beta, \gamma > 0$, such that $\inf \{I(u) : u \in Y
\cap \partial B_\gamma (0)\}\geq\beta$, where $B_\gamma
 = \{u \in X : \| u\| < \gamma\}$
and $\partial B_\gamma=\{u \in X : \| u\| = \gamma\}$;

\item[(iii)] there is a finite dimensional subspace $V$ of $X$ with
$\mathrm{dim}V>\mathrm{codim}Y$, such that $I(v) \to -\infty$ as
$\| v\|\to +\infty$ for any $v\in V$.
\end{itemize}
Then $I$ has at least $\mathrm{dim} V-\mathrm{codim} Y$ pairs of
nontrivial critical points.
\end{lemma}

\section{Proof main results}

 Let $\widehat{\Psi}$ denote its extension
to $L^{\alpha(x)}(\Omega)$. We  know that $\widehat{\Psi}$ is
locally Lipschitz on $L^{\alpha(x)}(\Omega)$. In fact, by
Proposition \ref{prop2.5}, for $u$, $v\in L^{\alpha(x)}(\Omega)$, we have
\begin{equation}
|\widehat{\Psi}(u)-\widehat{\Psi}(v)|
\leq\Big(C_1|1|_{\alpha'(x)}+C_2\max_{w\in
U}|w^{\alpha(x)-1}|_{\alpha'(x)}\Big)|u-v|_{\alpha(x)},
\end{equation}
where $U$ is an open neighborhood involving $u$ and $v$, $w$ in the
open segment joining $u$ and $v$. However, since
$\rho(1)=|\Omega|$, by Proposition \ref{prop2.2}, we have
\begin{equation}
|1|_{\alpha'(x)}<\infty.
\end{equation}
 Meanwhile, since
\begin{align*}
\rho(w^{\alpha(x)-1})
&= \int_\Omega |w^{\alpha(x)-1}|^{\alpha'(x)}\mathrm{d}x\\
&\leq \int_\Omega |w|^{\alpha(x)}\mathrm{d}x \\
&\leq 2^{\alpha^+}(\int_\Omega |
u|^{\alpha(x)}\mathrm{d}x+\int_\Omega |
u|^{\alpha(x)}\mathrm{d}x)
< \infty,
\end{align*}
by Proposition \ref{prop2.2}, we also have $|
w^{\alpha(x)-1}|_{\alpha'(x)}<\infty$.
 Then, using Proposition \ref{prop2.4} and
\cite[Theorem 2.2]{c1},
we have that ${\Psi}=\widehat{\Psi}|_{X}$ is also
locally Lipschitz, and
$\partial{\Psi}(u)\subseteq\int_{\Omega}\partial F(x,u)\mathrm{d}x$
(see \cite{k3}), where $\widehat{\Psi}|_{X}$ stands for the restriction
of $\widehat{\Psi}$ to $X$. The interpretation of
$\partial{\Psi}(u)\subseteq\int_{\Omega}\partial F(x,u)\mathrm{d}x$
is as follows: For every $\xi\in\partial{\Psi}(u)$ there corresponds
a mapping $\xi(x)\in\partial F(x,u)$ for a.e. $x\in \Omega$ having
the property that for every $\varphi\in X$ the function
$\xi(x)\varphi(x)\in L^1(\Omega)$ and $\langle g,
\varphi\rangle=\int_{\Omega}\xi(x)\varphi(x)\mathrm{d}x$ (see \cite{k3}).
Therefore, $I$ is a locally Lipschitz
functional and we can use the nonsmooth critical point theory.

\begin{lemma} \label{lem4.1}
If hypotheses {\rm (i)} and
{\rm (ii)} hold, then $I$ satisfies the nonsmooth
{\rm (PS)}-condition.
\end{lemma}

\begin{proof}
 Let $\{u_{n}\}_{n\geq1}\subseteq X$ be a sequence
such that $|I(u_{n})|\leq c$ for all $n\geq1$ and
$m(u_{n})\to 0$ as $n\to \infty$. Then, from
(ii), we have
\begin{align*}
c&\geq I(u_n)=\int_\Omega\frac{|\nabla
u_n|^{p(x)}}{p(x)}\mathrm{d}x-\int_\Omega
F(x,u)\mathrm{d}x\\
&\geq \frac{\| u_n\|^{p^-}}{p^+}-\int_\Omega
\frac{1}{\theta}\langle
\xi(u_n),u_n\rangle\mathrm{d}x-c_1\\
&\geq \big(\frac{1}{p^+}-\frac{1}{\theta}\big)\|
u_n\|^{p^-}+\int_\Omega
\frac{1}{\theta}(\| u_n\|^{p^-}-\langle \xi(u_n),u_n\rangle)\mathrm{d}x-c_1\\
&\geq \big(\frac{1}{p^+}-\frac{1}{\theta}\big)\|
u_n\|^{p^-}- \frac{1}{\theta}\| \xi\|_{X^*}\| u_n\|-c_1.
\end{align*}
Hence $\{u_{n}\}_{n\geq1}\subseteq X$ is bounded.

Thus by passing to a subsequence if necessary, we may assume that
$u_{n}\rightharpoonup u$ in $X$ as $n\to \infty$. We have
\[
\langle J'(u_{n}),u_{n}-u\rangle
-\int_{\Omega}\xi_{n}(x)(u_{n}-u)\mathrm{d}x
\leq\varepsilon_{n}\| u_{n}-u\|
\]
with $\varepsilon_{n}\downarrow0$, where $\xi_n\in
\partial\Psi(u_n)$. From Chang \cite{c1} we know that
$\xi_n\in L^{\alpha'(x)}(\Omega)$
($\alpha'(x)=\frac{\alpha(x)}{\alpha(x)-1}$).  Since $X$ is
embedded compactly in $L^{\alpha(x)}(\Omega)$, we have that
$u_{n}\to  u$ as $n\to \infty$ in
$L^{\alpha(x)}(\Omega)$. So using Proposition \ref{prop2.5}, we have
\begin{equation}
\int_{\Omega}\xi_{n}(x)(u_{n}-u)\,dx\to  0 \quad \text{as }
n\to \infty.
\end{equation}
Therefore we obtain
$\limsup_{n\to \infty}\langle J'(u_{n}),u_{n}-u\rangle\leq0$.
But we know that $J'$ is a mapping
of type ($S_{+}$). Thus we have
\[
u_{n}\to  u \quad \text{in } X.
\]
\end{proof}

\begin{lemma} \label{lem4.2}
If hypotheses {\rm (i), (iii)} hold, then there exist $r>0$ and $\delta>0$
such that $I(u)\geq\delta>0$ for every $u\in X$ and $\| u\|=r$.
\end{lemma}

\begin{proof}
Let $\varepsilon>0$ be small enough such that
$\varepsilon c_{0}^{p^{+}}\leq\frac{1}{2p^{+}}$, where $c_{0}$ is
the embedding constant of $X\hookrightarrow L^{p^+}(\Omega)$. From
hypothesis (i) and (iii), we have
\begin{equation} \label{e4.4}
F(x,t) \leq\varepsilon|t|^{p^{+}}+c(\varepsilon)|t|^{\alpha(x)}.
\end{equation}
Therefore, for every $u\in X$, we have
\begin{align*}
I(u)&\geq \frac{1}{p^{+}}\| u\|^{p^{+}}-\varepsilon
c_{0}^{p^{+}}\| u\|^{p^{+}}-c(\varepsilon)\|
u\|^{\alpha^{-}} \\
&\geq \frac{1}{2p^{+}}\| u\|^{p^{+}}-c(\varepsilon)\|
u\|^{\alpha^{-}},
\end{align*}
when $\| u\|\leq 1$.
So we can find $r>0$ small enough and $\delta>0$ such that
$I(u)\geq\delta>0$ for every $u\in X$ and $\| u\|=r$.
\end{proof}

\begin{lemma} \label{lem4.3}
If hypotheses {\rm (ii)} holds, then there exists
$u_1\in X$  such that $I(u_1)\leq0$.
\end{lemma}

\begin{proof}
 From  (ii),  there exist $M>0$, $c_2>0$ such that (see \cite[p. 298]{g1})
\[
F(x,u)\geq c_2|u|^{\theta}
\]
for all $|u|> M$ and a.e. $x\in \Omega$. Thus for
$1<t\in \mathbb{R}$, we have
\begin{align*}
\int_{\Omega}F(x,tu)\mathrm{d}x
&= \int_{\{t|u|>M\}}F(x,tu)\mathrm{d}x+
\int_{\{t|u|\leq M\}}F(x,tu)\mathrm{d}x  \\
&\geq c_2t^{\theta}\int_{\{t|u|> M\}}|
u|^{\theta}\mathrm{d}x-c_3.
\end{align*}
Therefore, for $t>1$, we have
\begin{equation} \label{e4.5}
\begin{aligned}
I(tu)&\leq \frac{1}{p^{-}}t^{p^{+}}\int_{\Omega}|\nabla
u|^{p(x)}\mathrm{d}x-c_2t^{\theta}\int_{\{t|
u|> M\}}|u|^{\theta}\mathrm{d}x+c_3\\
&= \frac{1}{p^{-}}t^{p^{+}}\int_{\Omega}|\nabla
u|^{p(x)}\mathrm{d}x-c_2t^{\theta}\int_{\Omega}|
u|^{\theta}\mathrm{d}x+c_2t^{\theta}\int_{\{t|u|\leq
M\}}|u|^{\theta}\mathrm{d}x+c_3.
\end{aligned}
\end{equation}
Noting that $c_2t^{\theta}\int_{\{t|u|\leq M_{3}\}}|
u|^{\theta}$ is bounded, it follows that
\[
I(tu)\to -\infty\quad  \text{as } t\to +\infty.
\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
  Using Lemma \ref{lem3.1} and Lemmas \ref{lem4.1}-\ref{lem4.3},
we can find an $u\in X$ such that $I(u)>0$
(hence $u\neq0$) and $0\in\partial I(u)$.
Hence $u\in X$ is a nontrivial solution of \eqref{e1.1}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm3.2}]
 Firstly, we can easily see that $I$
is even functional on $X$. We claim that $I(u)\to -\infty$ as
$\| u\|\to +\infty$, for any $u\in Y_k$. We assume
$\| u\|\geq1$. From \eqref{e4.4}, we have
\[
I(u)\leq\frac{1}{p^{-}}\| u\|^{p^{+}}-c_4|
u|_{\theta}^{\theta}+c_4\int_{\{|u|\leq M\}}|
u|^{\theta}\mathrm{d}x+c_5.
\]
Since $Y_k$ is finite dimensional,  all norms of $Y_k$ are
equivalent. For $p^+<\theta$, we get $I(u)\to -\infty$ as
$\| u\|\to +\infty$. We can apply Lemma \ref{lem3.2} with
$V=Y_k$ and $Y=X$. From Lemma \ref{lem4.1} and
Lemma \ref{lem4.2}, we get $k$-pairs of
nontrivial critical points, which  are solutions of \eqref{e1.1}.
\end{proof}

We remark that using the same method as in hte proof of Theorems
\ref{thm3.1} and \ref{thm3.2}, we can obtain the same results
for the corresponding differential inclusion problems with
Neumann boundary data.

As an example of a nonsmooth potential function
$F(x,u)$ satisfying (HF), we have
\[
F(x,u)=\frac{1}{p^+}|u|^{p^+}+\frac{1}{\alpha(x)}| u|^{\alpha(x)}.
\]
Then we can check that it satisfies all hypotheses of
Theorem \ref{thm3.1}. Note that in this case.
$\partial F(x,u)=|u|^{p^+-1}\mathop{\rm sgn}(u)
+|u|^{\alpha(x)-1}\mathop{\rm sgn}(u)$,
where
\[
\mathop{\rm sgn}(u)= \begin{cases}
1, & \text{if }u>0, \\
[-1,1] &\text{if } u=0,\\
-1& \text{if } u<0.
\end{cases}
\]
Moreover, it is obvious that $F(x,-u)=F(x,u)$. So $F$ satisfies
all the hypotheses in Theorem \ref{thm3.2}.

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\end{document}