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

\begin{document}
\title[ Existence and  multiplicity results...]
{Existence and  multiplicity results for elliptic problems with
Nonlinear Boundary Conditions and variable exponents}

\author[A. Zerouali, B. Karim, O. Chakrone and A. Anane ]
{A. Zerouali, B. Karim, O. Chakrone and A. Anane}  % in alphabetical order

\address{A. Zerouali \newline
Centre R\'egional des M\'etiers de l'\'{E}ducation et de la
Formation, F\`es, Maroc} \email{abdellahzerouali@yahoo.fr}

\address{B. Karim\newline
Universit\'e Moulay Isma\"{\i}l, Facult\'e des Sciences et
T\'echniques, Arrachidia, Maroc} \email{karembelf@gmail.com}

\address{O. Chakrone and A. Anane \newline
Universit\'e Mohamed Premier, Facult\'e des Sciences, Oujda,
Maroc} \email{chakrone@yahoo.fr; ananeomar@yahoo.fr}

\thanks{Submitted  XX. Published XX.}
\subjclass[2000]{35J48, 35J65, 35J60, 47J30, 58E05}
\keywords{Variable exponents; Elliptic problem; Nonlinear boundary
conditions;\hfill\break\indent Multiple solutions; Three critical
points theorem; Variational methods}

\begin{abstract}
By applaying the Ricceri's three critical points theorem, we show
the existence of at least three solutions
 to the following elleptic problem:
\begin{equation*}
\begin{gathered}
-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad
  \text{in }\Omega, \\
a(x, \nabla u).\nu=\mu g(x,u), \quad  \text{on } \partial\Omega,
\end{gathered}
\end{equation*}

where $\lambda$, $\mu \in \mathbb{R}^{+},$
$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain of
smooth boundary $\partial\Omega$ and $\nu$ is the outward normal
vector on $\partial\Omega$. $p: \overline{\Omega} \mapsto
\mathbb{R}$, $a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto
\mathbb{R}^{N},$ $f: \Omega\times\mathbb{R} \mapsto \mathbb{R}$
and $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ are
fulfilling appropriate conditions.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and main result}

In this article, we consider the elliptic problem with nonlinear
boundary conditions and variable exponents
\begin{equation}\label{p1}
\begin{gathered}
-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad
  \text{in }\Omega, \\
a(x, \nabla u).\nu=\mu g(x,u), \quad  \text{on }
\partial\Omega,
\end{gathered}
\end{equation}
where  $\lambda$, $\mu\in [0,\infty)$,
$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain with
smooth boundary $\partial\Omega$ and $\nu$ is the outward normal
vector on $\partial\Omega$. $f: \Omega\times\mathbb{R} \mapsto
\mathbb{R}$ and $g:\partial\Omega\times\mathbb{R} \mapsto
\mathbb{R}$ are two Carath\'{e}odory functions.
%Next, let
%$F(x,u)=\int_0^uf(x,s)ds$ and $G(x,u)=\int_0^ug(x,s)ds$.
 $p\in C(\overline{\Omega})$ is the variable exponent.\\ Throughout this paper, we denote $$p^- = \min_{x\in
\overline{\Omega }} p(x);~~~~p^+
=\max_{x\in\overline{\Omega}}p(x);$$ $$ p^{*}(x)=\left\{%
\begin{array}{ll}
    Np(x)/[N-p(x)]  & \hbox{ if } p(x)<N,\\
    +\infty  & \hbox{ if }p(x)\geq N,
\end{array}%
\right.$$
$$p^{\partial}(x)=\left\{%
\begin{array}{ll}
    (N-1)p(x)/[N-p(x)]  & \hbox{ if } p(x)<N,\\
   + \infty  & \hbox{ if }p(x)\geq N,
\end{array}%
\right. $$ and  $$ C_{+}(\overline{\Omega})=\{p\in
C(\overline{\Omega}): 1<p^{-}<p^{+}<\infty\}.$$ Our variable
exponent $p$ fulfills $p\in C_{+}(\overline{\Omega})$ and for this
$p$ we introduce a characterization the Carath\'eodory function
$a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto
\mathbb{R}^{N}.$
\begin{itemize}
\item[($H_{0}$)] $a(x,-s)=-a(x,s)$ for a.e.
$x\in\overline{\Omega}$ and all $s\in\mathbb{R}^{N}.$
\item[$(H_{1})$] There exists a Carath\'eodory function $A: \overline{\Omega}\times \mathbb{R}^{N} \mapsto \mathbb{R}$
continuously differentiable with respect to its second argument,
such that $a(x, s)= \nabla_{s}A(x, s)$ all $s\in\mathbb{R}^{N}$
and a.e. $x\in\overline{\Omega}.$
\item[$(H_{2})$] $A(x, 0)=0$ for a.e. $x\in\overline{\Omega}.$
\item[$(H_{3})$] There exists $c>0$ such that the function $a$ satisfies the
growth condition\\ $|a(x, s)|\leq c(1+|s|^{p(x)-1})$ for a.e.
$x\in\overline{\Omega}$ and all $s\in\mathbb{R}^{N},$ where $|.|$
denotes the Euclidean norm.
\item[$(H_{4})$] The monotonicity condition $0\leq [a(x, s_{1})-a(x,
s_{2})](s_{1}-s_{2})$ holds for a.e. $x\in\overline{\Omega}$ and
all $s_{1}$, $s_{2}\in\mathbb{R}^{N}.$ With equality if and only
if $s_{1}=s_{2}.$
\item[$(H_{5})$] The inequalities $|s|^{p(x)}\leq a(x, s)s\leq
p(x)A(x, s)$ hold for a.e. $x\in\overline{\Omega}$ and all
$s\in\mathbb{R}^{N}.$
\end{itemize}

A first remark is that hypothesis $(H_{0})$ is only needed to
obtain the multiplicity of solutions.
 As in \cite{Boureanu2013}, we have decided to use this kind
of function $a$ satisfying $(H_{0})$--$(H_{5})$ because we want to
assure a high degree of generality to our work. Here we invoke the
fact that, with appropriate choices of $a$, we can obtain many
types of operators. We give, in the following, two examples of
well known operators which are present in lots of papers.


\textbf{Examples:}
\begin{enumerate}
\item If $a(x, s)=|s|^{p(x)-2}s,$ we have $A(x,
s)=\frac{1}{p(x)}|s|^{p(x)}.$ \\$(H_{0})$--$(H_{5})$ are verified,
and we arrive to the $p(x)$-Laplace operator\\ $div(a(x,\nabla
u))= div(|\nabla u|^{p(x)-2}\nabla u)=\triangle_{p(x)}u.$
\item If $a(x, s)= (1+|s|^{2})^{(p(x)-2)/2}s,$ we have  $A(x,
s)=\frac{1}{p(x)}[(1+|s|^{2})^{p(x)/^{2}}-1].$
\\$(H_{0})$--$(H_{5})$
are verified, and we find a generalized mean curvature operator\\
$div(a(x,\nabla u))= div((1+|\nabla u|^{2})^{(p(x)-2)/2}\nabla
u).$

\end{enumerate}



The energy functional corresponding to problem \eqref{p1} is
defined on $W^{1, p(x)}(\Omega)$ as
\begin{equation}\label{erfuc}
H(u)=\Phi(u)+\lambda\Psi(u)+\mu J(u),
\end{equation}
where
\begin{gather}\label{phi}
\Phi(u)=\int_{\Omega} A(x, \nabla u)dx+ \int_{\Omega}\frac{1}{p(x)}| u| ^{p(x)}dx,\\
\label{psi}
\Psi(u)=-\int_{\Omega}F(x,u)dx, \\
\label{:J} J(u)=-\int_{\partial\Omega}G(x,u)d\sigma,
\end{gather}
where $F(x,u)=\int_{0}^{u}f(x, s)ds,$ $G(x,u)=\int_{0}^{u}g(x,
s)ds,$ and $d\sigma$ is the $N-1$ dimensional Hausdorff measure.
Let us recall that a weak solution of \eqref{p1} is any $u\in
W^{1, p(x)}(\Omega)$ such that
\[
\begin{aligned}
& \int_{\Omega }a(x, \nabla u)\nabla v dx+ \int_{\Omega }| u|
^{p(x)-2}uv dx\\
&
=\lambda\int_{\Omega}f(x,u)vdx+\mu\int_{\partial\Omega}g(x,u)vd\sigma
\quad \text{for all }
 v\in W^{1, p(x)}(\Omega).
\end{aligned}
\]

The study of differential and partial differential equation with
variable exponent has been received considerable attention in
recent years. This importance reflects directly into a various
range of applications. There are applications concerning elastic
materials \cite{Zhikov1986}, image restoration \cite{Chen2006},
thermorheological and electrorheological fluids \cite{
Antontsev2006, Ruzicka2002} and mathematical biology
\cite{Fragnelli2010}.

Ricceri's  three critical points theorem is a powerful tool to
study boundary
 problem of differential equation (see, for example,
\cite{Afrouzi2007,Bonanno2003a,Bonanno2003,Bonanno2011}).
Particularly, Mihailescu \cite{Mihuailescu2007} use three critical
points theorem of
 Ricceri \cite{Ricceri2000} study a particular $p(x)$-Laplacian equation.
He proved existence of three solutions for the problem. Liu
\cite{Liu2008} study the solutions of the general $p(x)$-Laplacian
equations with Neumann or Dirichlet boundary condition on a
bounded domain, and obtain three solutions under appropriate
hypotheses. Shi \cite{Shi2009} generalizes the corresponding
result of \cite{Mihuailescu2007}. The multiple solutions of
$p(x)$-biharmonic equation under sublinear condition has been
studied in \cite{Li2013} by L. Li, L. Ding and W.W. Pan. To our
knowledge, there is no result of multiple solutions of elliptic
problems with nonlinear boundary conditions and variable
exponents.

We enumerate the hypotheses concerning the functions $f,$ $F$ and
$g$.

\begin{itemize}
\item[(I1)] For $t \in
C(\overline{\Omega})$ and $t(x)<p^*(x)$ for all
$x\in\overline{\Omega},$ we have
$$\sup_{(x,s)\in\Omega\times\mathbb{R}}
\frac{|f(x,s)|}{1+|s|^{t(x)-1}}<+\infty;$$
\item[(I2)]There exist positive constant $c_{1}$ such that $F(x,s)> 0$ for a.e. $x\in\Omega$ and all $s\in ]0,c_{1}]$;
\item[(I3)] there exist $p_1(x)\in C(\overline{\Omega})$ and
 $p^+<p_1^-\leq p_1(x)<p^*(x)$,
such that
\[
\limsup_{s\to
0}\sup_{x\in\Omega}\frac{F(x,s)}{|s|^{p_1(x)}}<+\infty;
\]
\item[(I4)] There exist positive constant $c_{2}$ and a function $\gamma(x)\in
C(\overline{\Omega})$ with $1<\gamma^-\leq\gamma^+<p^-$, such that
$|F(x,s)|\leq c_{2} (1+|s|^{\gamma(x)})$ for a.e. $x\in\Omega$ and
all $s\in\mathbb{R}$;
\item[(I5)] For  $p_2\in C(\overline{\Omega})$ and $p_2(x)<p^\partial(x)$ for all
$x\in\overline{\Omega}$, we have \[
\sup_{(x,s)\in\partial\Omega\times\mathbb{R}}\frac{|g(x,s)|}{1+|s|^{p_2(x)-1}}<+\infty,
\].
\end{itemize}



This article is divided into three sections. In Section 2, we
recall some basic facts about the variable exponent Lebesgue and
Sobolev spaces at first and we recall B. Ricceri's three critical
points theorem (Theorem \ref{thm:ricceri}). In the third section,
we prove the following theorem which is the main result of this
paper.
\begin{theorem}\label{thm}
Assume $(H_{0})$--$(H_{5})$ and (I1)--(I4). Then there exist an
open interval $\Lambda\subseteq(0,+\infty)$ and a positive real
number $\omega$ such that, for each $\lambda\in\Lambda$ and each
function
 $g\colon\partial\Omega\times\mathbb{R}\to\mathbb{R}$
 satisfying (I5), there exists $\delta>0$ which satisfies, for
each $\mu\in[0,\delta]$, the problem \eqref{p1} has at least three
weak solutions whose norms in $W^{1, p(x)}(\Omega)$ are less than
$\omega$.
\end{theorem}

\section{Preliminaries}

\hspace{0.4 cm} We first recall some basic facts about the
variable exponent Lebesgue-Sobolev.\\ For $p\in
C_{+}(\bar{\Omega}),$ we introduce the variable exponent Lebesgue
space
$$L^{p(x)}(\Omega):=\left\{u; u:\Omega\rightarrow \mathbb{R}
\mbox{ is a measurable and } \int_{\Omega}|u|^{p(x)}dx <
+\infty\right\},$$ endowed with the Luxemburg norm
$$ |u|_{p(x)}:= \inf\left\{\alpha>0; \int_{\Omega}\left|\frac{u(x)}{\alpha}\right|^{p(x)}dx\leq 1
\right\},$$ which is separable and reflexive Banach space (see
\cite{KR}).\\
 Let us define the space $$ W^{1, p(x)}(\Omega):= \{u\in
L^{p(x)}(\Omega); |\nabla u|\in L^{p(x)}(\Omega)\},$$ equipped
with the norm $$ \|u\|= \inf\left\{\alpha>0; \int_{\Omega}
\left(\left|\frac{\nabla u(x)}{\alpha}\right|^{p(x)}+
\left|\frac{u(x)}{\alpha}\right|^{p(x)}\right)dx\leq 1
\right\};~~~~\forall u\in W^{1, p(x)}(\Omega).$$ Let $W^{1,
p(x)}_{0}(\Omega)$ be the closure of $C_{0}^{\infty}(\Omega)$ in
$W^{1, p(x)}(\Omega).$


\begin{proposition}\cite{F, FZ, HKV, KR}\label{P1}
\begin{itemize}
\item[(1)]$W^{1, p(x)}_{0}(\Omega)$ is separable reflexive Banach
space;
\item[(2)] If $q \in C_{+}(\bar{\Omega})$ and $q(x)<p^{*}(x)$ for
any $x\in \bar{\Omega},$ then the embedding from $W^{1,
p(x)}(\Omega)$ to $L^{q(x)}(\Omega)$ is compact and continuous;
\item[(3)] If $q \in C_{+}(\bar{\Omega})$ and $q(x)<p^{\partial}(x)$ for
any $x\in \bar{\Omega},$ then the embedding from $W^{1,
p(x)}(\Omega)$ to $L^{q(x)}(\partial\Omega)$ is compact and
continuous;
\item[(4)] (Poincar\'e) There is a constant $C>0$, such that
$$|u|_{p(x)}\leq C|\nabla u|_{p(x)}~~~~\forall u \in W^{1,
p(x)}_{0}(\Omega).$$
\end{itemize}
\end{proposition}

An important role in manipulating the generalized Lebesgue-Sobolev
spaces is played by the mapping $\rho$ defined by
$$\rho(u):=\int_{\Omega}\left[|\nabla u|^{p(x)}+|u|^{p(x)}\right]dx,~~~~\forall u\in
W^{1, p(x)}(\Omega).$$
\begin{proposition} \cite{FH} \label{P2}
For $u, u_{k}\in W^{1, p(x)}(\Omega); k=1,2,...,$ we have
\begin{itemize}
\item[(1)] $\|u\|\geq1$ if and only if $\|u\|^{p^{-}}\leq
\rho(u)\leq\|u\|^{p^{+}};$
\item[(2)] $\|u\|\leq1$ if and only if  $\|u\|^{p^{-}}\geq
\rho(u)\geq\|u\|^{p^{+}};$
\item[(3)] $\|u_{k}\| \rightarrow 0$ as $k\rightarrow +\infty$ if and only if
$\rho(u_{k})\rightarrow 0$ as $k\rightarrow +\infty$;
\item[(4)] $\|u_{k}\| \rightarrow +\infty$ as $k\rightarrow +\infty$ if and only if
$\rho(u_{k})\rightarrow +\infty$ as $k\rightarrow +\infty$.
\end{itemize}
\end{proposition}
In the sequel, we recall the revised form of Ricceri's three
critical points theorem \cite[Theorem 1]{Ricceri2009} and
\cite[Proposition 3.1]{Ricceri2000a}.

\begin{theorem}[{\cite[Theorem 1]{Ricceri2009}}] \label{thm:ricceri}
Let $X$ be a reflexive real Banach space. $ \Phi\colon X \to
\mathbb{R} $ is a continuously G\^{a}teaux differentiable and
sequentially weakly lower semicontinuous functional whose
G\^{a}teaux derivative admits a continuous inverse on $ X',$ where
$X'$ is the dual of $X,$ and $\Phi$ is bounded on each bounded
subset of $X$; $\Psi\colon X \to \mathbb{R} $ is a continuously
G\^{a}teaux differentiable functional whose G\^{a}teaux derivative
is compact; $ I \subseteq \mathbb{R} $ an interval. Assume that
\begin{equation}\label{qiangzhi}
\lim_{\|x\| \to +\infty } (\Phi(x)+\lambda \Psi (x))=+\infty
\end{equation}
for all $ \lambda \in I $, and that there exists $h\in \mathbb{R}$
such that
\begin{equation}\label{t2}
\sup_{\lambda \in I} \inf_{x \in X} (\Phi (x)+ \lambda (\Psi
(x)+h)) < \inf_{x \in X} \sup_{\lambda \in I} (\Phi (x)+ \lambda
(\Psi (x)+ h)).
\end{equation}
Then, there exists an open interval $ \Lambda \subseteq I $ and a
positive real number $ \rho $ with the following property: for
every $ \lambda \in \Lambda $ and every $ C^1 $ functional $
J\colon X \mapsto \mathbb{R} $ with compact derivative, there
exists $ \delta > 0 $ such that, for each $ \mu \in [0,\delta] $
the equation
\[
\Phi '(x)+\lambda \Psi '(x)+\mu J'(x)=0
\]
has at least three solutions in $X$ whose norms are less than $
\omega$.
\end{theorem}

\begin{proposition}[{\cite[Proposition 3.1]{Ricceri2000a}}] \label{propo}
Let $X$ be a non-empty set and $\Phi, \Psi$ two real functions on
$X$. Assume that there are $r > 0$ and $ x_0, x_1 \in X$ such that
\[
\Phi(x_0)=-\Psi(x_0)=0, \quad \Phi(x_1)>r, \quad
 \sup_{ x \in \Phi^{-1} ( ]-\infty ,r ] ) }
-\Psi(x) < r \frac{-\Psi(x_1)}{\Phi(x_1)}.
\]
Then, for each $ h $ satisfying
\[
\sup_{ x \in \Phi^{-1} ( ]-\infty ,r ] ) } -\Psi(x) < h
 < r \frac{-\Psi(x_1)}{\Phi(x_1)},
\]
one has
\[
\sup_{\lambda \geq 0} \inf_{x \in X}(\Phi(x)+\lambda(h +\Psi(x)))
< \inf_{x\in X} \sup_{\lambda \geq 0} (\Phi(x)+\lambda(h
+\Psi(x))).
\]
\end{proposition}

In our work, we designate by $X$ the Sobolev space with variable
exponent $W^{1, p(x)}(\Omega).$

\section{Proof of main result}
Its clear that from $(H_{1})$, the Fr\'{e}chet derivative of
$\Phi$ is the operator $ \Phi'\colon X \rightarrow X'$ defined as
$$\langle \Phi'(u), v\rangle = \int_{\Omega }a(x, \nabla u)\nabla v dx+ \int_{\Omega }| u|
^{p(x)-2}uv dx \mbox{ for any } u, v\in X.$$

We start by proving some properties of the operator $\Phi'$.

\begin{theorem} \label{thm1.4}
Suppose that the mapping $a$ satisfies $(H_{0})$--$(H_{5})$. Then
the following statements holds.

\begin{itemize}
\item[(1)] $\Phi'$  is continuous, bounded and strictly monotone.

\item[(2)] $\Phi'$ is of $(S_{+})$ type.

\item[(3)] $\Phi'$  is a homeomorphism.
\end{itemize}
\end{theorem}

\begin{proof}
(1) Since $\Phi'$ is the Fr\'{e}chet derivative of $\Phi$, it
follows that $\Phi'$ is continuous and bounded. Using $(H_{4})$
and the elementary inequalities \cite{Simon1978}
\begin{gather*}
| x-y| ^{\gamma }\leq 2^{\gamma }( | x| ^{\gamma -2}x-| y|
^{\gamma -2}y)
( x-y)  \quad  \text{if  }\gamma \geq 2, \\
| x-y| ^2\leq \frac{1}{( \gamma -1) }( | x| +| y| ) ^{2-\gamma }(
| x| ^{\gamma -2}x-| y| ^{\gamma -2}y) ( x-y)  \quad \text{if
}1<\gamma <2,
\end{gather*}
for all $( x,y) \in  \mathbb{R}^{N}\times\mathbb{R}^{N}$,
 we obtain for all $u,v\in X$ such that $u\neq v$,
\[
\langle \Phi'(u)-\Phi'(v),u-v\rangle >0,
\]
which means that $\Phi'$ is strictly monotone.

(2) Let $( u_n) _n$ be a sequence of $X$ such that
\[
u_n\rightharpoonup u \text{  weakly in }X  \text{ as }n
\rightarrow +\infty\quad \text{and}\quad \limsup_{ n\to +\infty
}\langle \Phi'(u_n),u_n-u\rangle \leq 0.
\]

Using  the compact embedding $W^{1,p(x)}(\Omega)\hookrightarrow
L^{p(x)}(\Omega),$  we have
$$\lim_{n\to +\infty}
\int_{\Omega}(|u_{n}|^{p(x)-2}u_{n}-|u|^{p(x)-2}u)(u_{n}-u)dx
=0.$$ Thus
$$\limsup_{n\to +\infty}
\int_{\Omega}a(x, \nabla u_{n})(\nabla u_{n}-\nabla u)dx \leq 0.$$
The following theorem assure that $u_{n} \rightarrow u$ strongly
in $W^{1, p(x)}(\Omega)$ as $n \rightarrow +\infty.$

\begin{theorem} [\cite{Le2009}, Theorem 4.1]
The Carath\'eodory function $a\colon
\overline{\Omega}\times\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$
described by $(H_{0})$--$(H_{5})$ is an operator of type $(S_{+})$
that is, if $u_{n}\rightharpoonup u$ weakly in
$W^{1,p(x)}(\Omega)$ as $n \rightarrow +\infty$ and
$$\limsup_{n\to +\infty} \int_{\Omega}a(x, \nabla u_{n})(\nabla
u_{n}- \nabla u)dx \leq 0,$$ then  $u_{n} \rightarrow u$ strongly
in $W^{1, p(x)}(\Omega)$ as $n \rightarrow +\infty.$
\end{theorem}


 (3) Note that the strict monotonicity of $\Phi'$ implies its
injectivity. Moreover, $\Phi'$ is a coercive operator. Indeed,
using $(H_{5})$, Proposition\ref{P2} and since $p^{-}-1>0$, for
each $u\in X$ such that $\| u\| \geq 1$ we have
\[
\frac{\langle \Phi'(u),u\rangle }{\| u\|
}=\frac{\int_{\Omega}[a(x,\nabla u)\nabla u + |u|^{p(x)} ]dx}{\|
u\| }\geq \| u\| ^{p^{-}-1}\to \infty \quad \text{as } \| u\|\to
\infty .
\]
Consequently, the operator $\Phi'$ is a surjection and admits an
inverse mapping. It suffices then to show the continuity of
$\Phi'^{-1}$. Let $(f_n)_n$ be a sequence of $X'$ such that
$f_n\to f$ in $X'$ as $n \rightarrow +\infty$. Let $u_n$ and $u$
in $X$ such that
\[
\Phi'^{-1}( f_n) =u_n\quad \text{and}\quad \Phi'^{-1}( f) =u.
\]
By the coercivity of $\Phi'$, one deducts that the sequence $(
u_n)$ is bounded in the reflexive space $X$. For a subsequence, we
have $u_n\rightharpoonup \widehat{u}$ weakly in $X$ as $n
\rightarrow +\infty$, which implies
\[
\lim_{n\to +\infty } \langle \Phi'(u_n)-\Phi'(u),u_n-
\widehat{u}\rangle =\lim_{n\to +\infty }\langle
f_n-f,u_n-\widehat{u}\rangle =0.
\]
It follows by the property $(S_{+})$ and the continuity of $\Phi'$
that
\[
u_n\to \widehat{u}\quad \text{ strongly in } X\quad
\text{and}\quad \Phi'(u_n)\to
 \Phi'(\widehat{u})=\Phi'(u)\quad \text{in } X' \text{ as } n \rightarrow +\infty.
\]
Moreover, since $\Phi'$ is an injection, we conclude that
$u=\widehat{u}$.
\end{proof}



Now we can give the proof of our main result.

\begin{proof}[Proof of Theorem \ref{thm}]
Set $\Phi(u)$, $\Psi(u)$ and $J(u)$ as \eqref{phi}, \eqref{psi}
and \eqref{:J}. So, for each $u$, $v\in X$, one has
\begin{gather*}
\langle\Phi '(u),v\rangle =\int_{\Omega}[a(x, \nabla u)\nabla v
+ | u|^{p(x)-2}uv]dx, \\
\langle\Psi '(u),v\rangle =-\int_{\Omega}f(x,u)v\,dx, \\
\langle J'(u),v\rangle =-\int_{\partial\Omega}g(x,u)v\,d\sigma.
\end{gather*}
From Theorem \ref{thm1.4}, the functional $\Phi$ is a continuous
G\^{a}teaux differentiable and sequentially weakly lower
semicontinuous functional whose G\^{a}teaux derivative admits a
continuous inverse on $X'$. By (I1) and (I5), $\Psi$ and $J$ are
continuously G\^{a}teaux differentiable functionals. Moreover,
using the compacity of embedding
$W^{1,p(x)}(\Omega)\hookrightarrow L^{p(x)}(\Omega)$ and the trace
embedding $W^{1,p(x)}(\Omega)\hookrightarrow
L^{p(x)}(\partial\Omega)$ (Proposition \ref{P1}), we deduce that
$\Psi'$ and $J'$ are compact. Obviously, $\Phi$ is bounded on
each bounded subset of $X$ under our assumptions.\\
From $(H_{5})$, if $\|u\|\geq 1$ then
\begin{align*}\label{eq:3.1}
\Phi(u)&=\int_{\Omega} A(x, \nabla u)dx+
\int_{\Omega}\frac{1}{p(x)}| u| ^{p(x)}dx\\& \geq
\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+ |u| ^{p(x)})
dx\\&\geq
 \frac{1}{p^{+}}\rho(u)\\& \geq\frac{1}{p^{+}}\|u\|^{p^{-}},
\end{align*}
Meanwhile, for each $\lambda\in\Lambda$,
\begin{align*}
\lambda\Psi(u) & =-\lambda\int_{\Omega}F(x,u)dx\\
& \geq -\lambda\int_{\Omega}c_{2}(1+|u|^{\gamma(x)})dx\\
& \geq -\lambda c_{2}(|\Omega|+\|u\|_{\gamma(x)}^{\gamma^+})\\
& \geq -c'_2(1+\|u\|_{\gamma(x)}^{\gamma^+})\\
& \geq -c''_2(1+\|u\|^{\gamma^+})
\end{align*}
 for any $u\in X$, where $c'_2$ and $c''_2$ are positive constants and $\|.\|_{\gamma(x)}$ is the usual norm of $W^{1,\gamma(x)}(\Omega)$.
 Combining the two inequalities above, we obtain
\[
\Phi(u)+\lambda\Psi(u)\geq
\frac{1}{p^+}\|u\|^{p^-}-c''_2(1+\|u\|^{\gamma^+}),
\]
since $\gamma^+<p^-$, it follows that
\[
\lim_{\|u\| \to +\infty } (\Phi(u)+\lambda \Psi
(u))=+\infty\quad\forall u\in X,\quad\lambda\in[0,+\infty).
\]
Then assumption \eqref{qiangzhi} of Theorem \ref{thm:ricceri} is
satisfied.

Next, we will prove that assumption \eqref{t2} is also satisfied.
It suffices to verify the conditions of Proposition \ref{propo}.
Let $u_0=0$. By $(H_{2})$ and the definition of $F$, we can easily
have
\[
\Phi(u_0)=-\Psi(u_0)=0.
\]
Now we claim that \eqref{t2} is satisfied.

From (I3), there exist $\eta\in[0,1]$, $c_3>0$, such that
\[
F(x,s)<c_3|s|^{p_1(x)}<c_3|s|^{p_1^-}\quad\forall
s\in[-\eta,\eta], \text{ a.e. }x\in\Omega.
\]
Then, from (I4), we can find a constant $M$ such that
\[
F(x,s)<M|s|^{p_1^-}
\]
for all $s\in\mathbb{R}$ and a.e. $x\in\Omega$. Consequently, by
the Sobolev embedding theorem, $W^{1, p(x)}(\Omega)\hookrightarrow
L^{p_1^-}(\Omega)$ is continuous. And for suitable positive
constant $c_4$, $c_5$),\\ we have
\[
-\Psi(u)=\int_{\Omega}F(x,u)dx<M\int_{\Omega}|u|^{p_1^-}dx \leq
c_4\|u\|^{p_1^-}\leq c_5 r^{p_1^-/p^+},
\]
when $\|u\|^{p^+}/p^+\leq r$. Hence, being $p_1^->p^+$, it follows
that
\begin{equation}\label{eq:lim}
\lim_{r\to 0^+}\frac{\sup_{\|u\|^{p^+}/p^+\leq r}-\Psi(u)}{r}=0.
\end{equation}

Let $u_1 \in C^1(\Omega)$ be a function positive in $\Omega$, with
 $\max_{\overline{\Omega}} u_1 \leq c_{1}$. Then, $u_1 \in
X$ and $\Phi(u_1) > 0$. In view of $(I2)$ we also have $-\Psi(u_1)
= \int_{\Omega} F(x,u_1(x)) dx
> 0$. Therefore, from \eqref{eq:lim}, we can find $r \in \big( 0,
\min\{ \Phi(u_1), \frac{1}{p^+} \} \big)$ such that
\begin{equation*}
    \sup_{\|u\|^{p^+}/p^+ \leq r} (-\Psi(u)) < r\frac{-\Psi(u_1)}{\Phi(u_1)}.
\end{equation*}
Now, let $u \in \Phi^{-1} ((-\infty, r])$. Then, $\int_{\Omega}
(p(x)A(x, \nabla u)+ |u|^{p(x)} ) dx \leq rp^+ <1$ which, by
Proposition \ref{P2}, implies $\|u\|<1$. Consequently,
\begin{equation*}
    \frac{1}{p^+}\|u\|^{p^+} \leq \frac{1}{p^{+}}\rho(u)\leq \int_{\Omega}
(p(x)A(x, \nabla u)+ |u|^{p(x)} ) dx  < r.
\end{equation*}
Therefore, we infer that $\Phi^{-1} ((-\infty, r]) \subset \left\{
u \in X : \frac{1}{p^+}\|u\|^{p^+} < r \right\}$, and so
\begin{equation*}
    \sup_{ u \in \Phi^{-1} ( ]-\infty ,r ] ) }(-\Psi(u))
< r \frac{-\Psi(u_1)}{\Phi(u_1)}.
\end{equation*}
At this point, conclusion follows from Proposition \ref{propo} and
Theorem \ref{thm:ricceri}.
\end{proof}


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\end{document}