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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 139, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/139\hfil Existence of multiple solutions]
{Existence of multiple solutions for a $p(x)$-biharmonic equation}

\author[L. Li, L. Ding, W.-W. Pan \hfil EJDE-2013/139\hfilneg]
{Lin Li, Ling Ding, Wen-Wu Pan}  % in alphabetical order

\address{Lin Li \newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China}
\email{lilin420@gmail.com}

\address{Ling Ding\newline
School of Mathematics and Computer Science,
Hubei University of Arts and Science, Hubei 441053, China}
\email{591517149@qq.com}

\address{Wen-Wu Pan\newline
Department of Science, 
Sichuan University of Science and Engineering,
Zigong 643000, China}
\email{23973445@qq.com}

\thanks{Submitted December 30, 2012. Published June 21, 2013.}
\subjclass[2000]{35J65, 35J60, 47J30, 58E05}
\keywords{$p(x)$-biharmonic equation; Navier boundary condition;
\hfill\break\indent  Multiple solutions; three critical points theorem; 
variational methods}

\begin{abstract}
 In this article, we show  the existence of at least three solutions
 to a Navier boundary problem involving the $p(x)$-biharmonic operator.
 The technical approach is mainly base on a three critical points theorem
 by Ricceri.
\end{abstract}

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

\section{Introduction and statement of the main result}

In this article, we consider the fourth-order quasilinear elliptic equation
\begin{equation}\label{p1}
\begin{gathered}
\Delta_{p(x)}^2u+|u|^{p(x)-2}u=\lambda f(x,u)+\mu g(x,u), \quad 
  \text{in }\Omega, \\
u=0,\quad \Delta u=0, \quad  \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Delta_{p(x)}^2u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the
 $p(x)$-biharmonic operator of fourth order, $\lambda$, 
$\mu\in [0,\infty)$, $\Omega\subset\mathbb{R}^N(N > 1)$ is a
nonempty bounded open set with a sufficient smooth boundary $\partial\Omega$. 
$f$, $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ 
are Carath\'{e}odory functions. Next, let
$F(x,u)=\int_0^uf(x,s)ds$ and $G(x,u)=\int_0^ug(x,s)ds$.
For  $p\in C(\overline{\Omega})$, denote
$1<p^- = \min_{x\in \overline{\Omega }} p(x)\leq p^+ 
=\max_{x\in\overline{\Omega}}p(x)<+\infty$.
Moreover,
\[
p_2^*(x)= \begin{cases}
\frac{Np(x)}{N-2p(x)} & p(x)<\frac{N}{2},\\
\infty & p(x)\geq \frac{N}{2},
\end{cases}
\]
is the critical exponent just as in many papers. 
Obviously, $p(x)<p^*(x)$ for all
$x\in\overline{\Omega}$. In the sequel, $X$ will denote the 
Sobolev space $W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)$.

The energy functional corresponding to problem \eqref{p1} is defined on $X$ 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 }\frac{1}{p(x)}( | \Delta u| ^{p(x)}+ | u| ^{p(x)}) dx,\\
\label{psi}
\Psi(u)=-\int_{\Omega}F(x,u)dx, \\
\label{:J}
J(u)=-\int_{\Omega}G(x,u)dx.
\end{gather}
Let us recall that a weak solution of  \eqref{p1} is any $u\in X$ such that
\[
\begin{aligned}
& \int_{\Omega }( | \Delta u| ^{p(x)-2}\Delta u\Delta v+ | u| ^{p(x)-2}uv)
dx\\
& =\lambda\int_{\Omega}f(x,u)vdx+\mu\int_{\Omega}g(x,u)vdx \quad \text{for all }
 v\in X.
\end{aligned}
\]

In recent years, the study of differential equations and variational problems 
with $p(x)$-growth conditions has been an interesting topic, 
which arises from nonlinear electrorheological fluids and
elastic mechanics. In that context we refer the reader to 
Ruzicka \cite{Rocircuvzivcka2000}, Zhikov \cite{Zhikov1986}, 
and the references therein. Moreover, we
point out that elliptic equations involving the $p(x)$-biharmonic 
equations are not trivial generalizations of similar problems studied 
in the constant case since the $p(x)$-biharmonic operator
is not homogeneous and, thus, some techniques which can be applied 
in the case of the $p$-biharmonic operators will fail in that new situation, 
such as the Lagrange Multiplier Theorem.

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}.
To our best of knowledge, there no result of multiple solutions 
of $p(x)$-biharmonic equation under sublinear condition. 
The aim of this paper is to prove the following result

\begin{theorem}\label{thm}
Assume that $\sup_{(x,s)\in\Omega\times\mathbb{R}}
\frac{|f(x,s)|}{1+|s|^{t(x)-1}}<+\infty$, where 
$t \in C(\overline{\Omega})$ and $t(x)<p^*(x)$ for all 
$x\in\overline{\Omega}$ and there exist two positive constants 
$\varrho$, $\vartheta$ and a function $\gamma(x)\in C(\overline{\Omega})$
with $1<\gamma^-\leq\gamma^+<p^-$, such that
\begin{itemize}
\item[(I1)] $F(x,s)> 0$ for a.e. $x\in\Omega$ and all $s\in ]0,\varrho]$;
\item[(I2)] 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[(I3)] $|F(x,s)|\leq \vartheta (1+|s|^{\gamma(x)})$ for a.e. $x\in\Omega$ 
and all $s\in\mathbb{R}$.
\end{itemize}
Then, there exist an open interval
$\Lambda\subseteq(0,+\infty)$ and a positive real number $\rho$ with the
 following property: for
each $\lambda\in\Lambda$ and each function
 $g(x,s)\colon\Omega\times\mathbb{R}\to\mathbb{R}$ satisfying
\[
\sup_{(x,s)\in\Omega\times\mathbb{R}}\frac{|g(x,s)|}{1+|s|^{p_2(x)-1}}<+\infty,
\]
where $p_2\in C(\overline{\Omega})$ and $p_2(x)<p^*(x)$ for all 
$x\in\overline{\Omega}$, there
exists $\delta>0$ such that, for each $\mu\in[0,\delta]$, 
problem \eqref{p1} has at least three weak
solutions whose norms in $X$ are less than $\rho$.
\end{theorem}

\begin{remark} \rm
  The conclusion of Theorem \ref{thm} gives a precise information 
about the $p(x)$-biharmonic equation \eqref{p1} with parameter, 
namely, one can see that \eqref{p1} is stable with respect to small
 perturbations.
\end{remark}

This article is divided into four sections. 
In Section 2, we recall some basic facts about the
variable exponent Lebesgue and Sobolev spaces. In the third section, 
we present some important
properties of the $p(x)$-biharmonic operator. 
In section 4, we recall B. Ricceri's three critical
points theorem at first, then prove our main result.

\section{Preliminaries}

To study $p(x)$-biharmonic problems, we need some results on the spaces 
$L^{p(x)}( \Omega ) $ and $W^{k,p(x)}( \Omega )$,  and properties 
of $p(x)$-biharmonic operator, which we will use later.

Define the generalized Lebesgue space by
\[
L^{p(x)}( \Omega ) :=\big\{ u:\Omega \to \mathbb{R} \text{ measurable and }
\int_{\Omega }| u( x) | ^{p(x)}dx<\infty \big\},
\]
where $p(x)\in C_{+}( \overline{\Omega }) $ and
\[
C_{+}( \overline{\Omega }) :=\big\{ p\in C( \overline{ \Omega })
:p(x)>1\big\}, \text{  for any } x\in \overline{\Omega }.
\]
Denote
\[
p^{+}=\max_{x\in \overline{\Omega }} p(x),\quad
p^{-}=\min_{x\in \overline{\Omega }} p(x),
\]
and for any $x\in \overline{\Omega }$, $k\geq 1$,
\begin{gather*}
p^{\ast }( x) :=
\begin{cases}
\frac{Np(x)}{N-p(x)} & \text{if }p(x)<N, \\
+\infty & \text{if }p(x)\geq N,
\end{cases}
\\
p_{k}^{\ast }( x) :=
\begin{cases}
\frac{Np(x)}{N-k p(x)} & \text{if }k p(x)<N, \\
+\infty & \text{if }k p(x)\geq N.
\end{cases}
\end{gather*}
One introduces in $L^{p(x)}( \Omega ) $ the norm
\[
| u| _{p(x)}=\inf \big\{ \alpha >0 : \int_{\Omega }| \frac{u( x) }{\alpha }
| ^{p(x)}dx\leq 1\big\}\,.
\]
The space $( L^{p(x)}( \Omega ) ,|\cdot |_{p(x)}) $ is a Banach space.

\begin{proposition}[\cite{Fan2001}] \label{prop1.1}
 The space $(L^{p(x)}( \Omega ) ,| \cdot| _{p(x)}) $ is
separable, uniformly convex, reflexive and its conjugate space is
$L^{q(x)}( \Omega ) $  where $q(x)$ is the conjugate function of
$p(x)$; i.e.,
\[
\frac{1}{p(x)}+\frac{1}{q(x)}=1,
\]
for all $x\in \Omega$.
For  $u\in L^{p(x)}( \Omega ) $ and $v\in L^{q(x)}(\Omega ) $ we have
\[
\Big| \int_{\Omega }u( x) v(x)dx\Big| \leq 
\Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)| u| _{p(x)}|v| _{q(x)}.
\]
\end{proposition}

The Sobolev space with variable exponent $W^{k,p(x)}( \Omega ) $  is defined
as
\[
W^{k,p(x)}( \Omega ) =\big\{ u\in L^{p(x)}( \Omega ) :D^{\alpha }u\in
L^{p(x)}( \Omega ) ,| \alpha | \leq k\big\} ,
\]
where $D^{\alpha }u=\frac{\partial ^{| \alpha | }}{ \partial x_{1}^{\alpha
_{1}}\partial x_{2}^{\alpha _{2}}\dots \partial x_{N}^{\alpha _{N}}}u$ with 
$\alpha =(\alpha _{1},\dots ,\alpha_{N})$ is a multi-index and 
$| \alpha| =\sum_{i=1}^{N}\alpha _{i}$. The
space  $W^{k,p(x)}(\Omega ) $, equipped with the norm
\[
\| u\| _{k,p(x)}:=\sum_{| \alpha | \leq k}| D^{\alpha }u| _{p(x)},
\]
also becomes a Banach, separable and reflexive space. 
For more details, we refer the reader to
\cite{Edmunds1999,Edmunds2000,Fan2001a,Fan2001}.

\begin{proposition}[\cite{Fan2001}] \label{prop1.2}
For  $p,r\in C_{+}( \overline{\Omega }) $  such that
$r(x)\leq p_{k}^{\ast }( x)$ for all
$x\in \overline{\Omega }$,
there is a continuous and compact embedding
\[
W^{k,p(x)}( \Omega ) \hookrightarrow L^{r(x)}( \Omega) .
\]
\end{proposition}

We denote by $W_{0}^{k,p(x)}( \Omega ) $ the closure of  
$C_{0}^{\infty }(\Omega )$\ in $W^{k,p(x)}(\Omega ) $.

\section{Properties of the $p(x)$-biharmonic operator}

Note that the weak solutions of  \eqref{p1} are considered in the 
generalized Sobolev space
\[
X:=W^{2,p(x)}( \Omega ) \cap W_{0}^{1,p(x)}( \Omega ),
\]
equipped with the norm
\[
\| u\| =\inf \Big\{ \alpha >0:\int_{\Omega }
\Big(| \frac{\Delta u( x) }{\alpha } | ^{p(x)}+| \frac{u( x) }{\alpha } 
| ^{p(x)}\Big) dx\leq 1\Big\}.
\]

\begin{remark}\label{rq1.1} \rm
(1) According to \cite{Zang2008}, the norm $\| \cdot \| _{2,p(x)}$, 
cited in the preliminaries, is
equivalent to the norm $| \Delta \cdot | _{p(x)}$ in the space $X$. 
Consequently, the norms $\|\cdot \|_{2,p(x)},\| \cdot \| $ and 
$| \Delta \cdot | _{p(x)}$ are equivalent.

(2) By the above remark and Proposition \ref{prop1.2}, there is
a continuous and compact embedding of $X$ into $L^{q(x)}( \Omega ) $,
where $q(x)<p_{2}^{\ast }(x)$ for all $x\in \overline{\Omega }$.
\end{remark}

We consider the functional
\[
\Phi( u) =\int_{\Omega }\frac{1}{p(x)}
\big( | \Delta u| ^{p(x)}+ | u| ^{p(x)}\big) dx,
\]
It is well known that $\Phi(u)$ is well defined and continuous differentiable
 in $X$. Now we give the following fundamental proposition.

\begin{proposition}\label{prop1.3}
For  $u\in X$ we have
\begin{itemize}
\item[(1)] $\| u\| <( =;>) 1 \Leftrightarrow \Phi( u) <( =;>)1 $,

\item[(2)] $\| u\| \leq 1\Rightarrow \| u\|
^{p^{+}}\leq \Phi( u) \leq \| u\| ^{p^{-}}$,

\item[(3)] $\| u\| \geq 1\Rightarrow \|
u\| ^{p^{-}}\leq \Phi( u) \leq \| u\| ^{p^{+}}$, 
for all $u_n\in X$ we have

\item[(4)] $\| u_n\| \to 0\Leftrightarrow \Phi(u_n) \to 0$,

\item[(5)] $\| u_n\| \to \infty \Leftrightarrow \Phi( u_n) \to \infty $.
\end{itemize}
\end{proposition}

The proof of this proposition is similar to the proof in 
\cite[Theorem 1.3]{Fan2001}.
Moreover, the operator $T:=\Phi ':X\to X'$ defined as
\[
\langle T(u),v\rangle =\int_{\Omega }( | \Delta u| ^{p(x)-2}\Delta u\Delta v
+ | u| ^{p(x)-2}uv) dx \quad \text{for any } u,v\in X,
\]
satisfies the assertions of the following theorem.

\begin{theorem} \label{thm1.4}
The following statements hold:
\begin{itemize}
\item[(1)] $T$  is continuous, bounded and strictly monotone.

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

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

\begin{proof}
(1) Since $T$ is the Fr\'{e}chet derivative of $\Phi$, it follows that $T$ 
is continuous and bounded. Let us define the sets
\[
U_p=\{ x\in \Omega :p(x)\geq 2\} ,\quad 
V_p=\{ x\in \Omega :1<p(x)<2\} .
\]
Using 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 $u\neq v$,
\[
\langle T(u)-T(v),u-v\rangle >0,
\]
which means that $T$ is strictly monotone.

(2) Let $( u_n) _n$ be a sequence of $X$ such that
\[
u_n\rightharpoonup u \text{  weakly in }X\quad \text{and}\quad 
\limsup_{ n\to +\infty }\langle T(u_n),u_n-u\rangle \leq 0.
\]
From Proposition \ref{prop1.3}, it suffices to shows that
\begin{equation}
\int_{\Omega }( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0.  \label{e15}
\end{equation}
In view of the monotonicity of $T$, we have
\[
\langle T(u_n)-T(u),u_n-u\rangle \geq 0,
\]
and since $u_n\rightharpoonup u $ weakly in $X$, it follows that
\begin{equation}
\limsup_{n\to +\infty } \langle T(u_n)-T(u),u_n-u\rangle =0.  \label{e16}
\end{equation}
Put
\begin{gather*}
\varphi _n( x) =( | \Delta u_n| ^{p(x)-2}\Delta u_n-| \Delta u|
^{p(x)-2}\Delta u) ( \Delta u_n-\Delta u) , \\
\psi _n( x) =( | u_n| ^{p(x)-2}u_n-| u| ^{p(x)-2}u) ( u_n-u) .
\end{gather*}
By the compact embedding of $X$ into $L^{p(x)}( \Omega ) $, it follows that
\begin{gather*}
u_n\to u\quad \text{in }L^{p(x)}( \Omega ), \\
| u_n| ^{p(x)-2}u_n\to | u| ^{p(x)-2}u\quad \text{in }L^{q(x)}(\Omega ),
\end{gather*}
where $1/q(x)+1/p(x)=1$ for all $x\in \Omega $. It results
that
\begin{equation}
\int_{\Omega }\psi _n( x) dx\to 0.  \label{e21}
\end{equation}
It follows by \eqref{e16}\ and \eqref{e21} that
\begin{equation}
\limsup_{n\to +\infty } \int_{\Omega }\varphi _n( x) dx=0.  \label{e18}
\end{equation}
Thanks to the above inequalities,
\begin{gather*}
\int_{U_p}| \Delta u_n-\Delta u_{k}| ^{p(x)}dx\leq
2^{p^{+}}\int_{U_p}\varphi _n( x) dx, \\
\int_{U_p}| u_n-u_{k}| ^{p(x)}dx\leq 2^{p^{+}}\int_{U_p}\psi _n( x)
dx.
\end{gather*}
Then
\begin{equation}
\int_{U_p}\left( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}\right) 
dx\to 0\quad \text{as }n\to +\infty .
\label{e19}
\end{equation}
On the other hand, in $V_p$, setting  $\delta _n=| \Delta u_n| +|
\Delta u| $, we have
\[
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx\leq \frac{1 }{p^{-}-1}%
\int_{V_p}( \varphi _n) ^{\frac{p(x)}{2}}( \delta _n)
^{\frac{p(x)}{2}( 2-p(x)) }dx\,.
\]
For $d > 0$, by Young's inequality,
\begin{equation}
\begin{aligned} d\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx
 &\leq \int_{V_p}[ d( \varphi _n) ^{\frac{p(x)}{2}}] ( \delta _n)
^{\frac{p(x)}{2}( 2-p(x)) }dx, \\ 
&\leq \int_{V_p}\varphi _n( d)
^{\frac{2}{p(x)} }dx+\int_{V_p}( \delta _n) ^{p(x)}dx. \end{aligned}
\label{e17}
\end{equation}
From \eqref{e18} and since $\varphi _n\geq 0$, one can consider that
\[
0\leq \int_{V_p}\varphi _ndx<1.
\]
If $\int_{V_p}\varphi _ndx=0$ then $\int_{V_p}| \Delta u_n-\Delta u|
^{p(x)}dx=0.$\ If $0<\int_{V_p}\varphi _ndx<1$, we choose
\[
d=\Big( \int_{V_p}\varphi _n( x) dx\Big) ^{-1/2}>1,
\]
and the fact that $2/p(x)<2$, inequality \eqref{e17} becomes
\begin{align*}
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx 
&\leq \frac{1}{d}
\Big( \int_{V_p}\varphi _nd^2dx+\int_{\Omega }\delta _n^{p(x)}dx\Big) , \\
&\leq \Big( \int_{V_p}\varphi _ndx\Big) ^{1/2} \Big(1+\int_{\Omega
}\delta _n^{p(x)}dx\Big) .
\end{align*}
Note that, $\int_{\Omega }\delta _n^{p(x)}dx$ is bounded, which implies
\[  
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx\to 0\quad \text{as }n\to
+\infty .
\]
A similar method gives
\[
\int_{V_p}| u_n-u| ^{p(x)}dx\to 0\quad \text{as }n\to +\infty .
\]
Hence, it result that
\begin{equation}
\int_{V_p}( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0\quad \text{as \ }n\to +\infty
. \label{e20}
\end{equation}
Finally, \eqref{e15} is given by combining \eqref{e19} and \eqref{e20}.

(3) Note that the strict monotonicity of $T$ implies its injectivity. 
Moreover, $T$ is a coercive operator. Indeed, since $p^{-}-1>0$, 
for each $u\in X$ such that $\| u\| \geq 1$ we have
\[
\frac{\langle T(u),u\rangle }{\| u\| }=\frac{ \Phi( u) }{\| u\| }\geq \| u\|
^{p^{-}-1}\to \infty \quad \text{as } \| u\|\to \infty .
\]
Consequently, thanks to Minty-Browder theorem \cite{Zeidler1990}, 
the operator $T$ is an surjection and admits an inverse mapping. 
It suffices then to show the continuity of $T^{-1}$. Let $(f_n)_n$ be
a sequence of $X'$ such that $f_n\to f$ in $X'$. Let $u_n$ and $u$ in $X$ 
such that
\[
T^{-1}( f_n) =u_n\quad \text{and}\quad T^{-1}( f) =u.
\]
By the coercivity of $T$, one deducts that the sequence $( u_n)$  is
bounded in the reflexive space $X$. For a subsequence, we have
$u_n\rightharpoonup \widehat{u}$ in $X$, which implies
\[
\lim_{n\to +\infty } \langle T(u_n)-T(u),u_n- \widehat{u}\rangle
=\lim_{n\to +\infty }\langle f_n-f,u_n-\widehat{u}\rangle =0.
\]
It follows by the second assertion and the continuity of $T$ that
\[
u_n\to \widehat{u}\quad \text{in } X\quad \text{and}\quad T(u_n)\to
 T(\widehat{u})=T(u)\quad \text{in } X'.
\]
Moreover, since $T$ is an injection, we conclude that $u=\widehat{u}$.
\end{proof}

\section{Proof of main theorem}

For the reader's convenience, 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'$ 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 $ \rho $.
\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}

Now we can give the proof of our main result.

\begin{proof}[Proof 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}(|\Delta u|^{p(x)-2}\Delta u\Delta 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_{\Omega}g(x,u)v\,dx.
\end{gather*}
From Theorem \ref{thm1.4}, of course, $\Phi$ is a continuous 
G\^{a}teaux differentiable and
sequentially weakly lower semicontinuous functional whose G\^{a}teaux 
derivative admits a continuous
inverse on $X'$, moreover, $\Psi$ and $J$ are continuously 
G\^{a}teaux differentiable functionals
whose G\^{a}teaux derivative is compact. Obviously, $\Phi$ is bounded 
on each bounded subset of $X$ under our assumptions.

From Proposition \ref{prop1.3}, we have: 
if $\|u\|\geq 1$, then
\begin{equation}\label{eq:3.1}
\frac{1}{p^+}\|u\|^{p^-}\leq\Phi(u)\leq\frac{1}{p^-}\|u\|^{p^+}.
\end{equation}
Meanwhile, for each $\lambda\in\Lambda$,
\begin{align*}
\lambda\Psi(u) & =-\lambda\int_{\Omega}F(x,u)dx\\
& \geq -\lambda\int_{\Omega}\vartheta(1+|u|^{\gamma(x)})dx\\
& \geq -\lambda \vartheta(|\Omega|+|u|_{\gamma(x)}^{\gamma^+})\\
& \geq -C_2(1+|u|_{\gamma(x)}^{\gamma^+})\\
& \geq -C_3(1+\|u\|^{\gamma^+})
\end{align*}
 for any $u\in X$, where $C_2$ and $C_3$ are positive constants. 
Here, we use condition (I3) and
(ii) of Proposition \ref{prop1.1}. 
Combining the two inequalities above, we obtain
\[
\Phi(u)+\lambda\Psi(u)\geq \frac{1}{p^+}\|u\|^{p^-}-C_3(1+\|u\|^{\gamma^+}),
\]
because of $\gamma^+<p^-$, 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 $u_0=0$, we can easily have
\[
\Phi(u_0)=-\Psi(u_0)=0.
\]
Now we claim that \eqref{t2} is satisfied.

From (I2), exist $\eta\in[0,1]$, $C_4>0$, such that
\[
F(x,s)<C_4|s|^{p_1(x)}<C_4|s|^{p_1^-}\quad\forall s\in[-\eta,\eta],
\text{ a.e. }x\in\Omega.
\]
Then, from (I3), 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 ($X\hookrightarrow L^{p_1^-}(\Omega)$ 
is continuous), we have (for suitable positive constant $C_5$, $C_6$)
\[
-\Psi(u)=\int_{\Omega}F(x,u)dx<M\int_{\Omega}|u|^{p_1^-}dx
\leq C_5\|u\|^{p_1^-}\leq C_6 r^{p_1^-/p^+},
\]
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 $u_1 \in C^2(\Omega)$ be a function positive in $\Omega$, 
with $u_1|_{\partial \Omega} = 0$ and $\max_{\overline{\Omega}} u_1 \leq d$. 
Then, of course, $u_1 \in X$ and $\Phi(u_1) > 0$. In view of $(i_1)$ 
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, 
$\int_{\Omega} (|\Delta u|^{p(x)} + |u|^{p(x)} ) dx \leq rp^+ <1$ which,
by Proposition \ref{prop1.3}, implies $\|u\|<1$. Consequently,
\begin{equation*}
    \frac{1}{p^+}\|u\|^{p^+} \leq \int_{\Omega} 
\frac{1}{p(x)} (|\Delta u|^{p(x)} + |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}

\subsection*{Acknowledgments}
The authors are very grateful to the anonymous referees for their 
knowledgeable reports, which helped us to improve our manuscript.

The first and the third author were supported by grant XDJK2013D007
from the Fundamental  Research Funds for the Central Universities, 
grant 2011KY03 from the Scientific Research Fund of SUSE,
and grant 12ZB081 from the Scientific  Research Fund of SiChuan Provincial 
Education Department.

The second author was supported by grant 11101347
from the National Natural Science Foundation of China,
grant 2012M510363 from the  Postdoctor Foundation of China, 
and grants D20112605, D20122501 from the 
Key Project in Science and Technology Research Plan of the
 Education Department of Hubei Province.

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