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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 137, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/137\hfil Existence of solutions]
{Existence of solutions for nonlocal elliptic systems with
nonstandard growth conditions}

\author[G. Dai\hfil EJDE-2011/137\hfilneg]
{Guowei Dai}

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

\thanks{Submitted July 9, 2011. Published October 19, 2011.}
\thanks{Supported by grants 11061030 from the NSFC, and NWNU-LKQN-10-21}
\subjclass[2000]{35D05, 35J60, 35J70}
\keywords{Variational method;
 nonlinear elliptic systems; nonlocal condition}

\begin{abstract}
 This article concerns the existence and multiplicity of solutions
 for a $p(x)$-Kirchhoff-type systems with Dirichlet boundary condition.
 By a direct variational approach and the theory of the variable
 exponent Sobolev spaces, under growth conditions on the reaction terms,
 we establish the existence and multiplicity of solutions.
\end{abstract}

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


\section{Introduction}

  In this article, we study the following nonlocal elliptic
systems of gradient type with nonstandard growth conditions
\begin{equation}
\begin{gathered}
-M_1\Big(\int_\Omega\frac{1}{p(x)}|\nabla
u|^{p(x)}\,dx\Big)\operatorname{div}\big(|\nabla
u|^{p(x)-2}\nabla u\big)
= \frac{\partial F}{\partial u} (x,u,v)\quad \text{in } \Omega,\\
-M_2\Big(\int_\Omega\frac{1}{q(x)}|\nabla v|^{q(x)}\,dx\Big)
\operatorname{div}\big(|\nabla v|^{q(x)-2}\nabla v\big)
=  \frac{\partial F}{\partial v} (x,u,v)\quad \text{in }\Omega,\\
u=0,\quad v=0\quad \text{on }\partial\Omega,
\end{gathered}  \label{e1.1}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with a smooth
boundary $\partial\Omega$,
$p(x), q(x)\in C_+(\overline{\Omega})$ with
\begin{gather*}
1<p^{-}:=\min_{\overline{\Omega}}p(x)\leq p^{+}
:=\max_{\overline{\Omega}}p(x)<+\infty,\\
1<q^{-}:=\min_{\overline{\Omega}}q(x)\leq q^{+}
:=\max_{\overline{\Omega}}q(x)<+\infty,
\end{gather*}
$M_1(t)$, $M_2(t)$ are continuous functions. We confine ourselves
to the case where $M_1 = M_2$ for simplicity.
Notice that the results of this paper remain valid for
$M_1\neq M_2$ by adding some slight changes in the hypothesis
(H4) and (H5). The function
$F:\Omega \times \mathbb{R}\times\mathbb{R}\to \mathbb{R}$
is assumed to be continuous in $x \in\overline{\Omega}$
and of class $C^1$ in $u, v\in \mathbb{R}$.

The operator $-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)$ is
called the $p(x)$-Laplacian, and becomes $p$-Laplacian when
$p(x)\equiv p$ (a constant). The $p(x)$-Laplacian possesses more
complicated nonlinearities 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 electrorheological fluids, which are characterized by
their ability to drastically change the mechanical properties under
the influence of an exterior electromagnetic field
\cite{a1,r1,z1}.
Problems with variable exponent growth conditions also appear in the
mathematical modeling of stationary thermo-rheological viscous flows
of non-Newtonian fluids and in the mathematical description of the
processes filtration of an ideal barotropic gas through a porous
medium \cite{a5,a6}. 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{d2,h2,s1,z2,z3} for an overview of and references on this subject,
and to \cite{a2,f1,f2,f3,f4,f5,f6,f7}
 for the study of the $p(x)$-Laplacian equations
and the corresponding variational problems.

Problem \eqref{e1.1} is related to the stationary version of a
 model introduced by Kirchhoff \cite{k1}. More precisely, Kirchhoff
proposed the model
\begin{equation}
\rho\frac{\partial^2u}{\partial
t^2}-\Big(\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L  |
\frac{\partial u}{\partial
x}|^2\,dx\Big)\frac{\partial^2u}{\partial
x^2}=0,  \label{e1.2}
\end{equation}
where $\rho, \rho_0, h, E, L$ are constants, which extends the
classical D'Alembert's wave equation, by considering the effects of
the changes in the length of the strings during the vibrations.
A distinguishing feature of equation \eqref{e1.2} is that the equation
contains a nonlocal coefficient
$\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L
|\frac{\partial u}{\partial x}|^2\,dx$ which depends
on the average $\frac{1}{2L}\int_0^L  |
\frac{\partial u}{\partial x}|^2\,dx$, and hence the
equation is no longer a pointwise identity.
Some early classical studies of Kirchhoff equations
were Bernstein \cite{b1} and Poho\v{z}aev \cite{p1}.
The equation
\begin{equation}
\begin{gathered}
-\Big(a+b\int_\Omega |\nabla
u|^{2}\,dx\Big)\Delta u=  f(x,u)\quad \text{in } \Omega,\\
u=0\quad \text{on } \partial\Omega,
\end{gathered}  \label{e1.3}
\end{equation}
is related to the stationary analogue of the equation \eqref{e1.2}.
 Equation \eqref{e1.3} received much attention only after
Lions \cite{l1} proposed an abstract framework to the problem.
Some important and interesting results can be found, for example,
in \cite{a3,c1,d6}. More recently Alves et al. \cite{a4} and
Ma and Rivera \cite{m1} obtained positive
solutions of such problems by variational methods.
The study of Kirchhoff type equations has already been extended
to the case involving the $p$-Laplacian (for details, see
\cite{c3,d7,d8})and $p(x)$-Laplacian  (see \cite{d1,d4,d6,f8}).
In \cite{d1}, by a direct
variational approach, we establish conditions ensuring the existence
and multiplicity of solutions for the problem
\begin{gather*}
-M\Big(\int_\Omega\frac{1}{p(x)}|\nabla
u|^{p(x)}\,dx\Big)\operatorname{div}(|\nabla
u|^{p(x)-2}\nabla u)=  f(x,u)\quad \text{in }\Omega,\\
u=0\quad \text{on }\partial\Omega.
\end{gather*}
In \cite{h1}, the author established that existence and multiplicity
results for a class of elliptic systems with
nonstandard growth conditions.

Motivated by above, we consider the nonlocal
elliptic systems \eqref{e1.1}. We establish the existence
and multiplicity of solutions for system \eqref{e1.1}.
Local elliptic systems with standard growth conditions have
 been the subject of a sizeable literature.
We refer to the excellent survey article by De Figueiredo
\cite{d3}. We also refer to \cite{c4} about nonlocal elliptic systems of
$p$-Kirchhoff-type.

This paper is organized as follows. In Section 2, we present some
necessary preliminary knowledge on variable exponent Sobolev spaces.
In Sections 3, we give some existence results of weak solutions of
problem \eqref{e1.1} and their proofs.

\section{Preliminaries}

To discuss problem \eqref{e1.1}, we need some theory on
$W_{0}^{1,p(x) }( \Omega ) $  which is called
variable exponent Sobolev space. Firstly we state some basic
properties of spaces $W_{0}^{1,p(x) }( \Omega)$ which will
be used later (for details, see \cite{f6}).
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 of ${\mathbf{S}}(\Omega )$ when they
are equal almost everywhere.
Write
\begin{gather*}
C_+(\overline{\Omega})=\{h:h\in C(\overline{\Omega}), h(x)>1
\,\ \text{for}\,\ \text{any} \,\ x\in\overline{\Omega}\},
\\
h^{-}:=\min_{\overline{\Omega}}h(x),\quad
h^{+}:=\max_{\overline{\Omega}}h(x)\quad \text{for every }
 h\in C_+(\overline{\Omega}).
\end{gather*}
Define
\begin{equation*}
L^{p(x)}( \Omega ) =\{u\in {\mathbf{S}}(\Omega
):\int_{\Omega }|u(x)|^{p(x)}\,dx<+\infty \text{ for }
p\in C_+ (\overline{\Omega})\}
\end{equation*}
with the norm
\begin{equation*}
|u|_{L^{p(x)}( \Omega ) }=|u|_{p(x)}
=\inf \{ \lambda >0:\int_{\Omega }
|\frac{ u(x)}{\lambda}|^{p(x)}\,dx\leq 1\},
\end{equation*}%
and
\begin{equation*}
W^{1,p(x) }( \Omega ) =\{ u\in
L^{p(x) }( \Omega ) :|\nabla
u|\in L^{p(x) }( \Omega ) \}
\end{equation*}
with the norm
\begin{equation*}
\| u\| _{W^{1,p(x)}(\Omega )}= |u|_{L^{p(x)}(\Omega )} +|\nabla u|
_{L^{p(x)}(\Omega )}.
\end{equation*}

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}
Set $\rho (u)=\int_{\Omega }|u(x)|^{p(x)}\,dx$.
For any $u\in L^{p(x)}( \Omega ) $, then
\begin{itemize}
\item[(1)] for $u\neq 0$, $|u|_{p(x)}=\lambda$
if and only if $\rho (\frac{u}{\lambda })=1$;

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

\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
\begin{equation*}
|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{equation*}
\end{proposition}

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

\begin{proposition}[\cite{f3,f6}] \label{prop2.4}
If $q\in C_+(\overline{\Omega})$
and $q(x)\leq p^{\ast }(x)$ ($ q(x)< p^{\ast }(x)$) for
$x\in \overline{\Omega}$, then there is a continuous (compact)
embedding $W_0^{1,p(x)}(\Omega )\hookrightarrow
L^{q(x)}(\Omega )$, where
\begin{equation*}
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)}& \text{if } p(x)<N,\\
+\infty &\text{if }p(x)\geq N.
\end{cases}
\end{equation*}
\end{proposition}

\begin{proposition}[\cite{f4,f6}] \label{prop2.5}
 The conjugate space of $L^{p(x)}(\Omega)$ is $L^{q(x)}(\Omega)$,
where $\frac{1}{q(x)}+\frac{1}{p(x)}=1$ holds a.e. in
$\Omega$. For any $u\in L^{p(x)}(\Omega)$
and $v\in L^{q(x)}(\Omega)$, we have the following H\"{o}lder-type
inequality
\begin{equation*}
\big|\int_{\Omega}uv\,dx \big|
\leq(\frac{1}{p^{-}}+\frac{1}{q^{-}}) |u|_{p(x)}|v|_{q(x)}.
\end{equation*}
We write
\begin{equation*}
I(u)=\int_{{\Omega }}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx.
\end{equation*}
\end{proposition}


\begin{proposition}[\cite{f4}] \label{prop2.6}
 The functional $I:X\to \mathbb{R}$ is convex. The mapping
$I':X\to X^{\ast }$  is a strictly monotone, bounded
homeomorphism, and is of $(S_{+})$ type, namely
\begin{equation*}
u_{n}\rightharpoonup u\text{ and }\limsup_{n\to +\infty }
I'(u_{n})(u_{n}-u)\leq 0\text{ implies }u_{n}\to u,
\end{equation*}
where $X=W_{0}^{1,p(x)}(\Omega )$, $X^*$ is the dual space of $X$.
\end{proposition}

For every $(u, v)$ and $(\varphi,\psi)$ in
$W :=W_0^{1,p(x)}(\Omega)\times W_0^{1,q(x)}(\Omega)$, let
\begin{equation*}
\mathcal{F}(u,v):=\int_\Omega F(x,u,v)\,dx.
\end{equation*}
Then
\begin{equation*}
\mathcal{F}'(u,v)(\varphi,\psi)
=D_1\mathcal{F}(u,v)(\varphi)+D_2\mathcal{F}(u,v)(\psi),
\end{equation*}
where
\begin{gather*}
D_1\mathcal{F}(u,v)(\varphi)
=\int_\Omega \frac{\partial F}{\partial u}(x,u,v)\varphi\,dx,\\
D_2\mathcal{F}(u,v)(\psi)
=\int_\Omega \frac{\partial F}{\partial v}(x,u,v)\psi\,dx.
\end{gather*}
The Euler-Lagrange functional associated to \eqref{e1.1} is given by
\begin{equation*}
J(u,v):=\widehat{M}\Big(\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}\,dx
\Big)+\widehat{M}(\int_\Omega \frac{1}{q(x)}|\nabla v|^{q(x)}\,dx)
 -\mathcal{F}(u,v),
\end{equation*}
where $\widehat{M}(t):=\int_0^t M(\tau)\,d\tau$.
It is easy to verify that $J \in C^1(W,\mathbb{R})$ is weakly lower
semi-continuous and $(u,v)\in W$
is a weak solution of \eqref{e1.1} if and only if $(u,v)$ is a critical
point of $J$. Moreover, we have
\begin{equation}
J'(u,v)(\varphi,\psi)=D_1 J(u,v)(\varphi)+D_2 J(u,v)(\psi),
\end{equation}
where
\begin{gather*}
D_1 J(u,v)(\varphi)=M\Big(\int_\Omega \frac{1}{p(x)}
|\nabla u|^{p(x)}\,dx\Big)\int_\Omega |\nabla u|^{p(x)-2}
\nabla u\nabla\varphi\,dx-D_1\mathcal{F}(u,v)(\varphi),
\\
D_2 J(u,v)(\psi)=M\Big(\int_\Omega \frac{1}{q(x)}
 |\nabla v|^{q(x)}\,dx\Big)
\int_\Omega |\nabla v|^{q(x)-2}\nabla v\nabla\psi\,dx
-D_2\mathcal{F}(u,v)(\psi).
\end{gather*}
Let us choose on $W$ the norm $\|\cdot\|$ defined by
\begin{equation*}
\| (u,v)\|:=\max\{\| u\|_p,\| v\|_q\}.
\end{equation*}
The dual space of $W$ will be denoted by $W^*$ and $\|\cdot\|_*$
will stand for its norm. Therefore
\begin{equation*}
\| J'(u,v)\|_*=\| D_1 J(u,v)\|_{*,p}+\| D_2 J(u,v)\|_{*,q}
\end{equation*}
where $\|\cdot\|_{*,p}$ (respectively $\|\cdot\|_{*,q}$) is the
norm of $(W_0^{1,p(x)}(\Omega))^*$
(respectively $(W_0^{1,q(x)}(\Omega))^*$).

\section{Existence of solutions}

 In this section we  discuss the existence of weak
solutions of \eqref{e1.1}. For simplicity, we use $c$, $c_i$,
$i=1,2,\dots$ to denote the general positive constant
(the exact value may change from line to line).

Before stating our results, we introduce some natural growth
hypotheses on the right-hand side of \eqref{e1.1} and the nonlocal
coefficient $M(t)$. These hypotheses will ensure the mountain pass
geometry and the Palais-Smale condition for the Euler-Lagrange
functional $J$.
\begin{itemize}
\item[(H1)] For all $(x,s,t)\in\Omega\times\mathbb{R}^2$,  we assume
\begin{equation*}
|F(x,s,t)|\leq c\Big(1+|s|^{p_1(x)}+|t|^{q_1(x)}
+|s|^{\alpha(x)}|t|^{\beta(x)}\Big),
\end{equation*}
where $c$ is a positive constant,
$(p_1(x),q_1(x),\alpha(x),\beta(x))\in (C_+(\overline{\Omega}))^4$
 such that
\begin{gather*}
p_1(x)<p^*(x),\quad q_1(x)<q^*(x),\quad
\frac{2\alpha(x)}{p^*(x)}+\frac{2\beta(x)}{q^*(x)}<1\quad
 \text{in }\overline{\Omega}, \\
p_1^-,\quad 2\alpha^->p^+,\quad  q_1^-,\quad
2\beta^->q^+.
\end{gather*}

\item[(H2)] There exist $M >0$, $\theta_1 >\frac{p^+}{1-\mu}$,
$\theta_2 >\frac{q^+}{1-\mu}$ such that for all  $x\in\Omega$,
and all $(s, t)\in \mathbb{R}^2$ with
$|s|^{\theta_1}+ |t|^{\theta_2} \geq 2M$, one has
\begin{equation*}
0<F(x,s,t)\leq \frac{s}{\theta_1}
\frac{\partial F}{\partial s}(x,s,t)
+\frac{t}{\theta_2}\frac{\partial F}{\partial t}(x,s,t),
\end{equation*}
where $\mu$ comes from (H5) below.

\item[(H3)] $F(x,s,t)=o(|s|^{p^+}+|t|^{q^+})$ as $(s,t)\to (0,0)$
uniformly with respect to to  $x \in\Omega$.

\item[(H4)] There exists $m_{0}>0$, such that
$ M(t)\geq m_{0}$.

\item[(H5)] There exists $0<\mu<1$
 such that
$\widehat{M}(t)\geq (1-\mu)M(t)t$.
\end{itemize}

As an example, we let  $M(t)=a+bt:\mathbb{R}^+\to \mathbb{R}$
 with $a, b$ are
two positive constants. It is clear that
$M(t)\geq a>0$.
Taking $\mu=1/2$, we have
\begin{equation*}
\widehat{M}(t)=\int_0^tM(s)\,ds=at+\frac{1}{2}bt^2\geq
 \frac{1}{2}(a+bt)t=(1-\mu)M(t)t.
\end{equation*}
So  conditions (H4), (H5) are satisfied.

\begin{theorem} \label{thm3.1}
If $M$ satisfies {\rm (H4)} and
\begin{equation*}
|F(x,s,t)|\leq c_1(1+|s|^{\alpha_1}+|t|^{\beta_1}),
\end{equation*}
where $\alpha_1$, $\beta_1$ are two constants with
$1\leq\alpha_1<\min\{p^-,q^-\}$, $1\leq\beta_1<\min\{p^-,q^-\}$
then \eqref{e1.1} has a weak solution.
\end{theorem}

\begin{proof} From (H4) we have $\widehat{M}(t)\geq m_{0}t$.
For $(u_n,v_n)\in W$ such that $\|(u_n,v_n)\|\to+\infty$, we have
\begin{align*}
&J(u_n,v_n)\\
&= \widehat{M}(\int_{\Omega}\frac{1}{p(x)}|\nabla
u_n|^{p(x)}\,dx)+\widehat{M}(\int_{\Omega}\frac{1}{q(x)}|\nabla
v_n|^{q(x)}\,dx)-\int_{\Omega}F(x,u_n,v_n)\,dx \\
&\geq m_{0}\int_{\Omega}\frac{1}{p(x)}|\nabla
u_n|^{p(x)}\,dx+m_{0}\int_{\Omega}\frac{1}{q(x)}|\nabla
v_n|^{q(x)}\,dx\\
&\quad -c_1\int_{\Omega}|u_n|^{\alpha_1}\,dx
-c_1\int_{\Omega}|v_n|^{\beta_1}\,dx-c_1|\Omega|\\
&\geq \frac{m_{0}}{p^+}\| u_n\|_p^{p^-}+\frac{m_{0}}{q^+}\| v_n\|_q^{q^-}-c_3\| u_n\|_p^{\alpha_1}-c_2\|
v_n\|_q^{\beta_1}-c_1|\Omega|,
\end{align*}
where $|\Omega|$ denotes the measure of $\Omega$.
Without loss of generality, we may assume
$\| u_n\|_p\geq \| v_n\|_q$. Hence,
\begin{equation} \label{e3.1}
J(u_n,v_n)
\geq\frac{m_{0}}{p^+}\| u_n\|_p^{p^-}-c_3\| u_n\|_p^{\alpha_1}-c_2\|
u_n\|_p^{\beta_1}-c_1|\Omega|,
\end{equation}
By the definition of norm on $W$, we have
$\|(u_n,v_n)\|=\| u_n\|_p\to+\infty$.
In view of \eqref{e3.1} and the assumptions on $\alpha_1$ and
$\beta_1$, we can easily see that $J(u_n,v_n)\to +\infty$ as
$n\to+\infty$; i.e., $J$ is a coercive functional.
Since $J$ also is weakly lower semi-continuous, $J$ has a minimum
point $(u,v)$ in $W$, and $(u,v)$ is a weak solution pair which
may be trivial of \eqref{e1.1}. The proof is
completed.
\end{proof}

\begin{lemma} \label{lem3.1}
Let $(u_n, v_n)$ be a Palais-Smale sequence for the Euler-Lagrange
functional $J$.
If {\rm (H2), (H4), (H5)} are satisfied then $(u_n, v_n)$ is bounded.
\end{lemma}

\begin{proof}
 Let $(u_n, v_n)$ be a Palais-Smale sequence for the functional $J$.
This means that $J(u_n, v_n)$ is bounded and $\| J'(u_n, v_n)\|_*\to 0$
as $n\to +\infty$. Then, there is a positive constant $c_0$ such that
\begin{align*}
c_0
&\geq  J(u_n,v_n)\\
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla {u_n}|^{p(x)}
 \,dx\Big)
 +\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla {v_n}|^{q(x)}\,dx\Big)
 -\int_\Omega F(x,u_n,v_n)\,dx\\
&\geq  (1-\mu)M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla {u_n}|^{p(x)}
 \,dx\Big)
 \int_{\Omega}\frac{1}{p(x)}|\nabla {u_n}|^{p(x)}\,dx\\
&\quad  -\int_\Omega\frac{u_n}{\theta_1}
 \frac{\partial F}{\partial u}(x,u_n,v_n)\,dx
 +(1-\mu)M\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla {v_n}|^{q(x)}
 \,dx\Big)\\
&\quad\times \int_{\Omega}\frac{1}{q(x)}|\nabla {v_n}|^{q(x)}\,dx
 -\int_\Omega\frac{v_n}{\theta_2}\frac{\partial F}{\partial v}
 (x,u_n,v_n)\,dx-c_4,
\end{align*}
where $c_4$ is some positive constant. Then
\begin{align*}
c_0&\geq  J(u_n,v_n)\\
&\geq \big(\frac{1-\mu}{p^+}-\frac{1}{\theta_1}\big)
 M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla
 u_n|^{p(x)}\,dx\Big)
 \int_\Omega|\nabla u_n|^{p(x)}\,dx
 +\frac{1}{\theta_1}D_1 J(u_n,v_n)(u_n)\\
& \quad +\big(\frac{1-\mu}{q^+}-\frac{1}{\theta_2}\big)
 M\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla v_n|^{q(x)}\,dx\Big)
 \int_\Omega|\nabla v_n|^{q(x)}\,dx\\
 &\quad +\frac{1}{\theta_2}D_2 J(u_n,v_n)(v_n)-c_4
 \\
&\geq \big(\frac{1-\mu}{p^+}-\frac{1}{\theta_1}\big)m_{0}
 \int_\Omega|\nabla u_n|^{p(x)}\,dx
 +\big(\frac{1-\mu}{q^+}-\frac{1}{\theta_2}\big)m_{0}
 \int_\Omega|\nabla v_n|^{q(x)}\,dx\\
&\quad -\frac{1}{\theta_1}\| D_1J(u_n,v_n)\|_{*,p}\| u_n\|_p
 -\frac{1}{\theta_2}\| D_2J(u_n,v_n)\|_{*,q}\| v_n\|_q-c_4.
\end{align*}
Now, suppose that the sequence $(u_n, v_n)$ is not bounded.
Without loss of generality, we may assume $\| u_n\|_p\geq \| v_n\|_q$.

Therefore, for $n$ large enough, we have
\begin{equation*}
c_5\geq\big(\frac{1-\mu}{p^+}-\frac{1}{\theta_1}\big)
 m_{0}\| u_n\|_p^{p^-}-\Big(\frac{1}{\theta_1}\| D_1J(u_n,v_n)\|_{*,p}
 +\frac{1}{\theta_2}\| D_2J(u_n,v_n)\|_{*,q}\Big)\| u_n\|_p.
\end{equation*}
But, this cannot hold true since $p^->1$.
Hence, $\{\| (u_n,v_n)\|\}$ is bounded.
\end{proof}

In the following lemma, we show every bounded Palais-Smale sequence
for the functional $J$ contains a convergence subsequence.

\begin{lemma} \label{lem3.2}
Let $(u_n, v_n)$ be a bounded Palais-Smale sequence for the
Euler-La\-grange functional $J$. If {\rm (H1), (H4)} are satisfied,
then $(u_n, v_n)$ contains a convergent subsequence.
\end{lemma}

\begin{proof}
Let $(u_n, v_n)$ be a bounded Palais-Smale sequence for the functional
$J$. Then there is a subsequence still denoted by $(u_n, v_n)$
which converges weakly in $W$. Without loss of generality,
we assume that $(u_n,v_n) \rightharpoonup (u,v)$, then
$J'(u_n,v_n)(u_n-u,v_n-v)\to 0$. Thus, we have
\begin{align*}
&J'(u_n,v_n)(u_n-u,v_n-v)\\
&= M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}\,dx\Big)
\int_\Omega|\nabla u_n|^{p(x)-2}\nabla u_n(\nabla u_n-\nabla u)\,dx\\
&\quad +M\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla v_n|^{q(x)}\,dx\Big)
 \int_\Omega|\nabla v_n|^{q(x)-2}\nabla v_n(\nabla v_n-\nabla v)\,dx\\
&\quad -\int_\Omega \frac{\partial F}{\partial u}(x,u_n,v_n)(u_n-u)\,dx
 -\int_\Omega \frac{\partial F}{\partial v}(x,u_n,v_n)(v_n-v)\,dx
\to 0.
\end{align*}

On the other hand, let $\widetilde{\alpha}$, $\widetilde{\beta}$
be two continuous and positive functions on $\overline{\Omega}$
such that
\begin{equation*}
\frac{2\alpha(x)+\widetilde{\alpha}(x)}{p^*(x)}
+\frac{2\beta(x)+\widetilde{\beta}(x)}{q^*(x)}=1,\quad \forall
x\in\overline{\Omega}.
\end{equation*}
Using the Young inequality, we obtain
\begin{equation*}
|s|^{\alpha(x)}|t|^{\beta(x)}
\leq |s|^{\frac{\alpha(x)p^*(x)}{2\alpha(x)+\widetilde{\alpha}(x)}}+
|t|^{\frac{\beta(x)q^*(x)}{2\beta(x)+\widetilde{\beta}(x)}}
= |s|^{p_2(x)}+|t|^{q_2(x)},
\end{equation*}
where $p_2(x):=\frac{\alpha(x)p^*(x)}{2\alpha(x)+\widetilde{\alpha}(x)}
<p^*(x)$, $q_2(x):=\frac{\beta(x)q^*(x)}{2\beta(x)+\widetilde{\beta}(x)}
<q^*(x)$.
From (H1), we can obtain that there exist
 $p_3(x), q_3(x)\in C_+(\overline{\Omega})$ with $p_3(x)<p^*(x)$,
$q_3(x)<q^*(x)$ in $\overline{\Omega}$ such that
\[
|F(x,s,t)|\leq c_6\big(1+|s|^{p_3(x)}+|t|^{q_3(x)}\big).
\]
From this inequality, Propositions \ref{prop2.4} and \ref{prop2.5},
we can easily obtain
\[
\int_\Omega \frac{\partial F}{\partial u}(x,u_n,v_n)(u_n-u)\,dx \to0
\]
and
\begin{equation}
\int_\Omega \frac{\partial F}{\partial v}(x,u_n,v_n)(v_n-v)\,dx\to0.
\end{equation}
Therefore, we have
\begin{gather*}
M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}\,dx\Big)
\int_\Omega|\nabla u_n|^{p(x)-2}\nabla
u_n(\nabla u_n-\nabla u)\,dx\to 0,
\\
M\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla v_n|^{q(x)}\,dx\Big)
\int_\Omega|\nabla v_n|^{q(x)-2}\nabla
v_n(\nabla v_n-\nabla v)\,dx\to 0.
\end{gather*}
In view of (H4), we have
\begin{gather*}
\int_\Omega|\nabla u_n|^{p(x)-2}\nabla u_n(\nabla u_n-\nabla
u)\,dx\to 0,
\\
\int_\Omega|\nabla v_n|^{q(x)-2}\nabla
v_n(\nabla v_n-\nabla v)\,dx\to0.
\end{gather*}
Using Proposition \ref{prop2.6}, we have $u_n\to u$ in $W_0^{1,p(x)}(\Omega)$
 and $v_n\to v$ in $W_0^{1,q(x)}(\Omega)$, which implies that
$(u_n,v_n)\to (u,v)$ in $W$. This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.2}
If hypotheses {\rm (H1)--(H5)} hold, then \eqref{e1.1} has at least one
weak solution.
\end{theorem}

\begin{proof}
Let us show that $J$ satisfies the conditions of
Mountain Pass Theorem (see \cite[Theorem 2.10]{w1}).
By Lemmas \ref{lem3.1} and  \ref{lem3.2}, $J$
satisfies Palais-Smale condition in $W$.

For $\| (u, v)\|<1$, using the Young's inequality,
the fact $\frac{2\alpha(x)}{p^*(x)}+\frac{2\beta(x)}{q^*(x)}<1$
in $\overline{\Omega}$,
Propositions \ref{prop2.2} and \ref{prop2.4}, we obtain
\[ % \label{e3.3}
\int_\Omega |u|^{\alpha(x)}|v|^{\beta(x)}\,dx
\leq\frac{1}{2}\int_\Omega|u|^{2\alpha(x)}\,dx
+\frac{1}{2}\int_\Omega|v|^{2\beta(x)}\,dx
\leq c_7(\| u\|_p^{2\alpha^-}+\| v\|_q^{2\beta^-}).
\]
On the other hand, assuming (H1),
$W_0^{1,p(x)}(\Omega) \hookrightarrow L^{p^+}(\Omega)$, and
$W_0^{1,q(x)}(\Omega) \hookrightarrow L^{q^+}(\Omega)$.
Then there exists $c_8,c_9>0$ such that
\begin{gather*}
|u|_{p^+}\leq c_8\| u\|_p \quad\text{for } u\in W_0^{1,p(x)}(\Omega) \\
|v|_{q^+}\leq c_9\| v\|_q\quad \text{for } v\in W_0^{1,q(x)}(\Omega),
\end{gather*}
where $|\cdot|_r$ denote the norm on $L^{r(x)}(\Omega)$ with
$r\in C_+(\overline{\Omega})$.
Let $\varepsilon>0$ be small enough such that
$\varepsilon c_8^{p^+} \leq\frac{m_0}{2p^+}$ and
$\varepsilon c_9^{q^+} \leq\frac{m_0}{2q^+}$.
By the assumptions (H1) and (H3), we have
\[ % \label{e3.4}
|F(x,s,t)|\leq\varepsilon\big(|s|^{p^+}+|t|^{q^+}\big)
+c(\varepsilon)(|s|^{p_1(x)}+|t|^{q_1(x)}+|
s|^{\alpha(x)}|t|^{\beta(x)})
\]
for all $(x,s,t)\in\Omega\times\mathbb{R}^2$.
In view of (H4) and and the above inequality, for $\| (u, v)\|$ sufficiently small,
noting Proposition \ref{prop2.2}, we have
\begin{align*}
J(u,v)&\geq \frac{m_0}{p^+}\int_\Omega|\nabla
u|^{p(x)}\,dx+\frac{m_0}{q^+}\int_\Omega|\nabla
v|^{q(x)}\,dx-\varepsilon\int_\Omega|u|^{p^+}\,dx-\varepsilon\int_\Omega|
v|^{q^+}\,dx\\
&\quad -c(\varepsilon)\int_\Omega\Big(|u|^{p_1(x)}+|v|^{q_1(x)}+|
u|^{\alpha(x)}|v|^{\beta(x)}\Big)\,dx\\
&\geq \frac{m_0}{p^+}\| u\|_p^{p^+}-\varepsilon c_8^{p^+}\|
u\|_p^{p^+}+\frac{m_0}{q^+}\| v\|_q^{q^+}-\varepsilon c_9^{q^+}\|
v\|_q^{q^+}\\
&\quad -c(\varepsilon)\Big(\| u\|_p^{p_1^-}
 +\| v\|_q^{q_1^-}+c_7\| u\|_p^{2\alpha^-}
 +c_7\| v\|_q^{2\beta^-}\Big)\\
&\geq \frac{m_0}{2p^+}\| u\|_p^{p^+}+\frac{m_0}{2q^+}\| v\|_q^{q^+}
 -c(\varepsilon)\Big(\| u\|_p^{p_1^-}+\| v\|_q^{q_1^-}
 +c_7\| u\|_p^{2\alpha^-}+c_7\| v\|_q^{2\beta^-}\Big).
\end{align*}
Since $p_1^-, 2\alpha^->p^+$ and $q_1^-, 2\beta^->q^+$,
there exist $r>0$, $\delta>0$ such that
$J(u)\geq\delta>0$ for every $\| (u,v)\| = r$.

On the other hand, we have known that the assumption (H2)
implies the following assertion:
for every $x\in\overline{\Omega}$, $s, t\in \mathbb{R}$, the inequality
\begin{equation}
F(x,s,t)\geq c_{10}(|s|^{\theta_1}+|t|^{\theta_2}-1)
\end{equation}
holds; see \cite{h1}.
When $t>t_0$, from (H5) we can easily obtain that
\[
\widehat{M}(t)\leq \frac{\widehat{M}(t_0)}{t_0^{1/(1-\mu)}}
t^{1/(1-\mu)}:=c_{11}t^{1/(1-\mu)},
\]
where $t_0$ is an arbitrarily positive constant.
For $(\widetilde{u},\widetilde{v})\in W \setminus \{(0,0)\}$ and $t>1$, we have
\begin{align*}
J(t\widetilde{u},t\widetilde{v})
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|t\nabla
\widetilde{u}|^{p(x)}\,dx\Big)
+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|t\nabla
\widetilde{v}|^{q(x)}\,dx\Big)\\
&\quad -\int_\Omega F(x,t\widetilde{u},t\widetilde{v})\,dx\\
&\leq c_{12}(\int_\Omega|t\nabla
\widetilde{u}|^{p(x)}\,dx)^{1/(1-\mu)}-c_{10}t^{\theta_1}\int_\Omega
|\widetilde{u}|^{\theta_1}\,dx\\
& \quad +c_{13}\Big(\int_\Omega|t\nabla
\widetilde{v}|^{q(x)}\,dx\Big)^{1/(1-\mu)}
 -c_{10}t^{\theta_2}\int_\Omega
|\widetilde{v}|^{\theta_2}\,dx-c_{14}\\
&\leq c_{12}t^{\frac{p^+}{1-\mu}}\Big(\int_\Omega|\nabla
\widetilde{u}|^{p(x)}\,dx\Big)^{1/(1-\mu)}
 -c_{10}t^{\theta_1}\int_\Omega
|\widetilde{u}|^{\theta_1}\,dx\\
&\quad +c_{13}t^{\frac{q^+}{1-\mu}}\Big(\int_\Omega|\nabla
\widetilde{v}|^{q(x)}\,dx\Big)^{1/(1-\mu)}
 -c_{10}t^{\theta_2}\int_\Omega
|\widetilde{v}|^{\theta_2}\,dx-c_{14} \\
&\to -\infty,  \quad \text{as } t\to +\infty,
\end{align*}
due to $\theta_1 > \frac{p^+}{1-\mu}$ and
$\theta_2 > \frac{q^+}{1-\mu}$. Since $J(0,0)=0$, considering
Lemmas \ref{lem3.1} and \ref{lem3.2}, we see that $J$ satisfies
the conditions of Mountain Pass Theorem. So $J$ admits at least one
nontrivial critical point.
\end{proof}

Next we will prove under some symmetry condition on the function
$F$ that  \eqref{e1.1} possesses infinitely many nontrivial
weak solutions.

\begin{theorem} \label{thm3.3}
 Assume {\rm (H1), (H2), (H4), (H5)},
and that $F(x,u,v)$ is even in $u$, $v$.
Then \eqref{e1.1} has a sequence of solutions
$\{(\pm u_k,\pm v_k)\}_{k=1}^{\infty}$ such that
$J(\pm u_k,\pm v_k)\to +\infty$ as
$k\to +\infty$.
\end{theorem}

 Because $W_0^{1,p(x)}$ and $W_0^{1,q(x)}$ are a reflexive and
separable Banach space, then $W$ and $W^*$ are too. There exist
$\{e_j\}\subset W$ and $\{e_j^*\}\subset W^*$ such that
\[
W=\overline{\mathrm{span}\{e_j:j=1,2, \dots\}},\quad
W^*=\overline{\mathrm{span}\{e_j^*:j=1,2,\dots\}},
\]
and
\[
\langle e_i,e_j^*\rangle=\begin{cases}
1, & i=j,\\
0, & i\neq j,
\end{cases}
\]
where $\langle\cdot,\cdot\rangle$ denotes the duality product
between $W$ and $W^*$.
For convenience, we write $X_j = \operatorname{span}\{e_j\}$,
$Y_k = \oplus_{j=1}^kX_j , Z_k = \overline{\oplus_{j=k}^\infty X_j}$.
We will use the following ``Fountain
theorem'' to prove Theorem \ref{thm3.3}.

\begin{lemma}[\cite{w1}] \label{lem3.3}
Assume
\begin{itemize}
\item[(A1)] $X$ is a Banach space, $I\in C^1(X,\mathbb{R})$
is an even functional.
\item[(A2)]
For each $k = 1, 2, \dots$, there exist $\rho_k >r_k >0$ such
that
\item[(A2)] $\inf_{u\in Z_k, \| u\| =r_k} I(u)\to+\infty$
as $k\to+\infty$.

\item[(A3)] $\max_{u\in Y_k, \| u\| =\rho_k} I(u)\leq0$.

\item[(A4)] $I$ satisfies Palais-Smale condition for every $c>0$.
\end{itemize}
Then $I$ has a sequence of critical values tending to $+\infty$.
\end{lemma}

 For every $a >1$, $u,v\in L^a(\Omega)$, we define
\[
|(u,v)|_a:=\max\{|u|_a,|v|_a\}.
\]
Set
\begin{gather*}
a:=\max_{x\in\overline{\Omega}}\{2\alpha(x),2\beta(x),p_1(x),q_1(x)\}
>\min\{p^-,q^-\},
\\
b:=\min_{x\in\overline{\Omega}}\{2\alpha(x),2\beta(x),p_1(x),q_1(x)\}>0.
\end{gather*}
Then we have the following result.

\begin{lemma}[\cite{h1}] \label{lem3.2b}
 Denote
\[
\beta_k=\sup\{|(u,v)|_{a}:\| (u,v)\|=1, (u,v)\in Z_k\}.
\]
Then $\lim_{k\to+\infty} \beta_k=0$.
\end{lemma}

 \begin{proof}[Proof of Theorem \ref{thm3.3}]
 According to the assumptions on $F$, Lemmas \ref{lem3.1} and
\ref{lem3.2}, $J$
is an even functional and satisfies Palais-Smale condition.
We will prove that if $k$ is large enough, then there exist
$\rho_k>r_k>0$ such that ($A_2$) and ($A_3$) holding.
Thus, the  conclusion can be obtained from
Fountain theorem.

(A2): For any $(u_k,v_k)\in Z_k$, $\| u_k\|_p\geq1$,
$\| v_k\|_q\geq1$ and $\| (u_k,v_k)\| = r_k$ ($r_k$ will be
specified below),
we have
\begin{align*}
&J(u_k,v_k)\\
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla
 u_k|^{p(x)}\,dx\Big)+\widehat{M}
 \Big(\int_{\Omega}\frac{1}{q(x)}|\nabla v_k|^{q(x)}\,dx\Big)
 -\int_{\Omega}F(x,u_k,v_k)\,dx\\
&\geq  m_0\int_{\Omega}\frac{1}{p(x)}|\nabla u_k|^{p(x)}\,dx
 +m_0\int_{\Omega}\frac{1}{q(x)}|\nabla
 v_k|^{q(x)}\,dx -\int_{\Omega}F(x,u_k,v_k)\,dx\\
&\geq \frac{m_{0}}{p^+}\int_{\Omega}|\nabla
 u_k|^{p(x)}\,dx+\frac{m_{0}}{q^+}\int_{\Omega}|\nabla
 v_k|^{q(x)}\,dx\\
&\quad -c\int_\Omega\Big(1+|u_k|^{p_1(x)}+|v_k|^{q_1(x)}
 +|u_k|^{\alpha(x)}|v_k|^{\beta(x)}\Big)\,dx\\
&\geq \frac{m_{0}}{p^+}\| u_k\|_p^{p^-}
 +\frac{m_{0}}{q^+}\| v_k\|_q^{q^-}
 -c|u_k|_{p_1(x)}^{p_1(\xi_1^k)}-c|v_k|_{q_1(x)}^{q_1(\xi_2^k)}\\
&\quad -c_{15}|u_k|_{2\alpha(x)}^{2\alpha(\eta_1^k)}
-c_{15}|v_k|_{2\beta(x)}^{2\beta(\eta_2^k)}-c|\Omega|,
\end{align*}
where $\xi_1^k,\xi_2^k,\eta_1^k,\eta_2^k\in\Omega$. Therefore,
\begin{align*}
&J(u_k,v_k)\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
 -c|u_k|_{a}^{p_1(\xi_1^k)}-c|v_k|_{a}^{q_1(\xi_2^k)}\\
&\quad -c|u_k|_{a}^{2\alpha(\eta_1^k)}
 -c|v_k|_{a}^{2\beta(\eta_2^k)}-c|\Omega|\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
 -c(\beta_k\| (u_k,v_k)\|)^{p_1(\xi_1^k)}
 -c(\beta_k\| (u_k,v_k)\|)^{q_1(\xi_2^k)}\\
&\quad -c(\beta_k\| (u_k,v_k)\|)^{2\alpha(\eta_1^k)}
 -c(\beta_k\| (u_k,v_k)\|)^{2\beta(\eta_2^k)}-c|\Omega|\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
 -c_{16}\beta_k^b\| (u_k,v_k)\|^a-c|\Omega|,
\end{align*}
where $a$, $b$ are defined above. At this stage, we fix $r_k$
as follows:
\[
r_k:=\Big(\frac{m_0}{2c_{16}\max\{p^+,q^+\}\beta_k^b}\Big)
^{1/(a-\min\{p^-,q^-\})}\to +\infty\quad \text{as }k\to+\infty.
\]
Consequently, if $\|(u_k, v_k)\|=r_k$ then
\[
J(u_k,v_k)\geq\frac{m_{0}}{2\max\{p^+,q^+\}}\|
(u_k,v_k)\|^{\min\{p^-,q^-\}}-c|\Omega|\to+\infty\quad
\text{as }k\to+\infty.
\]

(A3): From (H2), we have
$F(x, u,v)\geq c_{10}(|u|^{\theta_1}+|v|^{\theta_2}-1)$
for every $x\in\Omega$ and $u,v\in \mathbb{R}$.
Therefore, for any $(u,v)\in Y_k$ with $\| (u,v)\|=1$ and
$1<\rho_k=t_k$ with $t_k\to +\infty$, we have
\begin{align*}
&J(t_ku,t_kv)\\
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|t_k\nabla
 u|^{p(x)}\,dx\Big)
 +\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|t_k\nabla v|^{q(x)}\,dx\Big)
 -\int_\Omega F(x,t_ku,t_kv)\,dx. \\
&\leq c_{17}\Big(\int_\Omega|t_k\nabla u|^{p(x)}\,dx\Big)^{1/(1-\mu)}
 +c_{18}\Big(\int_\Omega|t_k\nabla v|^{q(x)}\,dx\Big)^{1/(1-\mu)}\\
&\quad -c_{10}t_k^{\theta_1}\int_\Omega |u|^{\theta_1}\,dx
 -c_{10}t_k^{\theta_2}\int_\Omega |v|^{\theta_2}\,dx+c_{19},\\
&\leq c_{17}t_k^{\frac{p^+}{1-\mu}}
 \Big(\int_\Omega|\nabla  u|^{p(x)}\,dx\Big)^{1/(1-\mu)}
 -c_{10}t_k^{\theta_1}\int_\Omega |u|^{\theta_1}\,dx\\
&\quad +c_{18}t_k^{\frac{q^+}{1-\mu}}
 \Big(\int_\Omega|\nabla v|^{q(x)}\,dx\Big)^{1/(1-\mu)}
 -c_{10}t_k^{\theta_2}\int_\Omega |v|^{\theta_2}\,dx+c_{19}.
\end{align*}
By $\theta_1>\frac{p^+}{1-\mu}$, $\theta_2>\frac{q^+}{1-\mu}$ and $\text{dim} Y_k =k$, it is easy to
see that $J(t_ku,t_kv)\to-\infty$ as $\|
(t_ku,t_kv)\|\to+\infty$ for $(u,v) \in Y_k$.

The proof of Theorem \ref{thm3.3} is completed by the Fountain theorem.
\end{proof}

\subsection*{Acknowledgments}
The author wishes to express his gratitude to the anonymous referee for
reading the original manuscript carefully and making several
corrections and remarks.

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