\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 63, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/63\hfil Existence of solutions]
{Existence of solutions for a variable exponent system without PS conditions}

\author[L. Yin, Y. Liang, Q. Zhang, C. Zhao \hfil EJDE-2015/63\hfilneg]
{Li Yin, Yuan Liang, Qihu Zhang, Chunshan Zhao}

\address{Li Yin \newline
College of Information and Management Science,
Henan Agricultural University, \newline
Zhengzhou, Henan 450002, China}
\email{mathsr@163.com}

\address{Yuan Liang \newline
Junior College, Zhejiang Wanli University,
Ningbo, Zhejiang 315100, China}
\email{ly0432@163.com}

\address{Qihu Zhang (corresponding author)\newline
College of Mathematics and Information Science,
Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China}
\email{zhangqihu@yahoo.com, zhangqh1999@yahoo.com.cn}

\address{Chunshan Zhao \newline
Department of Mathematical Sciences,
Georgia Southern University, Statesboro, GA 30460, USA}
\email{czhao@GeorgiaSouthern.edu}

\thanks{Submitted September 29, 2014. Published March 13, 2015.}
\subjclass[2000]{35J47}
\keywords{Variable exponent system; integral functional;
PS condition; \hfill\break\indent variable exponent Sobolev space}

\begin{abstract}
 In this article, we study the existence of solution for the following elliptic
 system of variable exponents with perturbation terms
 \begin{gather*}
 -\operatorname{div}| \nabla u| ^{p(x)-2}\nabla u)+|u| ^{p(x)-2}u
 =\lambda a(x)| u| ^{\gamma(x)-2}u+F_{u}(x,u,v)\quad\text{in }
 \mathbb{R}^N, \\
 -\operatorname{div}| \nabla v| ^{q(x)-2}\nabla v)+|v| ^{q(x)-2}v
 =\lambda b(x)| v| ^{\delta(x)-2}v+F_{v}(x,u,v)\quad
 \text{in }\mathbb{R}^N, \\
 u\in W^{1,p(\cdot )}(\mathbb{R}^N),v\in W^{1,q(\cdot )}(\mathbb{R}^N),
 \end{gather*}
 where the corresponding functional does not satisfy PS conditions.
 We obtain a sufficient condition for the existence of solution and
 also present a result on asymptotic behavior of solutions at infinity.
\end{abstract}

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

\section{Introduction}

The study of differential equations and variational problems with variable
exponent has attracted intense research interests in recent years. Such
problems arise from the study of electrorheological fluids, image
processing, and the theory of nonlinear elasticity
\cite{1,9,30,40}. The following variable exponent flow is an important model
in image processing \cite{9}:
\begin{gather*}
u_{t}-\operatorname{div}| \nabla u| ^{p(x)-2}\nabla u)+\lambda
(u-u_0)=0,\quad \text{in }\Omega \times [ 0,T], \\
u(x,t)=g(x),\quad \text{on }\partial \Omega \times [ 0,T], \\
u(x,0)=u_0.
\end{gather*}
The main benefit of this flow is the manner in which it accommodates the
local image information. We refer to \cite{16,22,36} for the
existence of solution of variable exponent problems on bounded domain.

In this article, we consider the existence of solutions for the system
\begin{equation} \label{eP}
\begin{gathered}
-\operatorname{div}| \nabla u| ^{p(x)-2}\nabla u)+|
u| ^{p(x)-2}u=\lambda a(x)| u| ^{\gamma
(x)-2}u+F_{u}(x,u,v)\quad \text{in }\mathbb{R}^N, \\
-\operatorname{div}| \nabla v| ^{q(x)-2}\nabla v)+|
v| ^{q(x)-2}v=\lambda b(x)| v| ^{\delta
(x)-2}v+F_{v}(x,u,v)\quad \text{in }\mathbb{R}^N, \\
u\in W^{1,p(\cdot )}(\mathbb{R}^N),\quad v\in W^{1,q(\cdot )}(\mathbb{R}^N),
\end{gathered}
\end{equation}
where $p,q\in C(\mathbb{R}^N)$ are Lipschitz continuous and
$p(\cdot ),q(\cdot )>>1$, the notation
$h_1(\cdot )>>h_2(\cdot )$ means
$\operatorname{ess\,inf}_{x\in\mathbb{R}^N} (h_1(x)-h_2(x))>0$,
\[
-\Delta _{p(x)}u:=-\operatorname{div}| \nabla u| ^{p(x)-2}\nabla u)
\]
which is called the $p(x)$-Laplacian. When $p(\cdot )\equiv p$ (a constant),
 $p(x)$-Laplacian becomes the usual $p$-Laplacian.
The terms $\lambda a(x)| u| ^{\gamma (x)-2}u$ and
$\lambda b(x)|v| ^{\delta (x)-2}v$ are the perturbation terms. The $p(x)$
-Laplacian possesses more complicated nonlinearities than the $p$-Laplacian
(see \cite{17}). Many methods and results for $p$-Laplacian are invalid
for $p(x)$-Laplacian.

The PS condition is very important in the study of the existence of solution
via variational methods. According to \cite[Theorem 2.8]{j31}, if a
$C^1(X,\mathbb{R})$ functional $f$ satisfies the Mountain Pass Geometry,
then it has a PS sequence $\{x_n\}$ which satisfies $f(x_n)\to c$ which is the
mountain pass level and $f'(x_n)\to 0$. By \cite[Theorem 2.9]{j31}
it follows that if $f$ also satisfies the PS condition, passing
to a subsequence, then $x_n\to x_0$ in $X$, and then $x_0$ is
a critical point of $f$, that is $f'(x_0)=0$. In the study of
this problems in the bounded domain, since we have the compact embedding
from a Sobolev space to a Lebesgue space, so we have the PS condition when
we study the case of subcritical growth condition. For the unbounded domain,
we cannot get the compact embedding in general, so we do not have the PS
condition.

It is well known that a main difficulty in the study of elliptic equations
in $\mathbb{R}^N$ is the lack of compactness.
Many methods have been used to overcome
this difficulty. One type of methods is that under some additional
conditions we can recover the required compact imbedding theorem, for
example, the weighting method \cite{18,38}, and the symmetry method
\cite{35}. If equations are periodic, the corresponding energy functionals
are invariant under period-translation. We refer to \cite{3}--\cite{6} and
references cited therein for the applications of this method to the
$p$-Laplacian equations, the Schr\"{o}dinger equations and the biharmonic
equations etc.

Sometimes we can compare the original equation with its limiting equation at
infinity. Especially, we can compare the corresponding critical values of
the functionals for these two equations when the existence of the ground
state solution for the limiting equation is known. Usually the limiting
equations are homogeneous, but in \cite{3}-\cite{6} the limiting equations
are periodic. We also refer to \cite{15} for the existence of solution for
$p(x)$-Laplacian equations with periodic conditions.

In this article we consider the existence and the asymptotic behavior of
solutions near infinity for a variable exponent system with perturbations
that does not satisfy periodic conditions, which implies the corresponding
functional does not satisfy PS conditions on unbounded domain. We will also
give a sufficient condition for the existence of solutions for the system
\eqref{eP}. Our method is to compare the original equation with its limiting
equation at infinity without perturbation. These results also partially
generalize the results in \cite{15} and \cite{32}.

In this article, we make the following assumptions.
\begin{itemize}
\item[(A0)] $p(\cdot ),q(\cdot )$ are Lipschitz continuous,
$1<<p(\cdot ),q(\cdot )<<N$, $1<<\gamma (\cdot )<<p(\cdot )$, $a(\cdot )\in
L^{\frac{p(\cdot )}{p(\cdot )-\gamma (\cdot )}}(\mathbb{R}^N)$,
$1<<\delta (\cdot )<<q(\cdot )$,
$b(\cdot )\in L^{\frac{q(\cdot )}{q(\cdot )-\delta (\cdot )}}(\mathbb{R}
^N)$, $F$ $\in C^1(\mathbb{R}^N\times\mathbb{R}^2,
\mathbb{R})$ satisfies
\begin{gather*}
|F_{u}(x,u,v)| \leq C(|u|^{p(x)-1}+|u|^{\alpha (x)-1}+|v|^{q(x)/\alpha
^0(x)}+|v|^{\beta (x)/\alpha ^0(x)}), \\
|F_{v}(x,u,v)| \leq C(|v|^{q(x)-1}+|v|^{\beta (x)-1}+|u|^{p(x)/\beta
^0(x)}+|u|^{\alpha (x)/\beta ^0(x)}),
\end{gather*}
where $F_{u}=\frac{\partial }{\partial u}F$,
$F_{v}=\frac{\partial }{\partial v}F$,
 $\alpha ,\beta \in C(\mathbb{R}^N)$ satisfy
\begin{equation*}
p(\cdot )<<\alpha (\cdot )<<p^{\ast }(\cdot ), q(\cdot )<<\beta
(\cdot )<<q^{\ast }(\cdot ),
\end{equation*}
$h^0(\cdot )$ denotes the conjugate function of $h(\cdot )$, that is
$\frac{1}{h(x)}+\frac{1}{h^0(x)}\equiv 1$, and
\begin{equation*}
p^{\ast }(x)=\begin{cases}
Np(x)/(N-p(x)), & p(x)<N, \\
\infty, &p(x)\geq N.
\end{cases}
\end{equation*}

\item[(A1)] There exist constants $\theta _1>p^{+}$ and $
\theta _2>q^{+}$, such that $F$ satisfies the following conditions
\begin{gather*}
0\leq sF_{s}(x,s,t), 0\leq tF_{t}(x,s,t),\quad \forall (x,s,t)\in
\mathbb{R}^N\times\mathbb{R}\times\mathbb{R}, \\
0<F(x,s,t)\leq \frac{1}{\theta _1}sF_{s}(x,s,t)+\frac{1}{\theta _2}
tF_{t}(x,s,t),\quad \forall x\in\mathbb{R}^N,\forall (s,t)\in
\mathbb{R}\times\mathbb{R}\backslash \{(0,0)\}.
\end{gather*}

\item[(A2)] For  $(s,t)\in\mathbb{R}^2$, the function
 $sF_{s}(x,\tau ^{1/\theta_1}s,\tau ^{1/\theta_2}t)
 /\tau ^{\frac{\theta _1-1}{\theta _1}}$ and the function
 $tF_{t}(x,\tau ^{1/\theta_1} s,
\tau ^{1/\theta_2} t)/\tau ^{\frac{\theta _2-1}{\theta _2}}$
are increasing with respect to $\tau >0$.

\item[(A3)] There is a measurable function
$\widetilde{F} (s,t)$ such that
\begin{equation*}
\lim_{| x| \to +\infty } F(x,s,t)= \widetilde{F}(s,t)
\end{equation*}
for bounded $| s| +| t| $ uniformly,
and
\begin{equation*}
| \widetilde{F}(s,t)| +| \widetilde{F}
_{s}(s,t)s| +| \widetilde{F}_{t}(s,t)t| \leq
C(| s| ^{p^{+}}+| s| ^{\alpha^{-}}+| t| ^{q^{+}}+| t| ^{\beta^{-}}),\quad
\forall (s,t)\in\mathbb{R}^2,
\end{equation*}
and when $| x| \geq R$ the following inequalities hold
\begin{gather*}
| F(x,s,t)-\widetilde{F}(s,t)| \leq \varepsilon (R)(|
s| ^{p(x)}+| s| ^{p^{\ast }(x)}+|
t| ^{q(x)}+| t| ^{q^{\ast }(x)}), \\
\begin{aligned}
&| F_{s}(x,s,t)-\widetilde{F}_{s}(s,t)| \\
&\leq \varepsilon (R)(|s| ^{p(x)-1}+| s| ^{p^{\ast }(x)-1}
 +|t| ^{q(x)(p^{\ast }(x)-1)/p^{\ast }(x)}
 +| t|^{q^{\ast }(x)(p^{\ast }(x)-1)/p^{\ast }(x)}),
\end{aligned} \\
\begin{aligned}
&| F_{t}(x,s,t)-\widetilde{F}_{t}(s,t)| \\
&\leq \varepsilon (R)(|s| ^{p(x)(q^{\ast }(x)-1)/q^{\ast }(x)}+| s|
^{p^{\ast }(x)(q^{\ast }(x)-1)/q^{\ast }(x)}+| t|
^{q(x)-1}+| t| ^{q^{\ast }(x)-1}),
\end{aligned}
\end{gather*}
where $\varepsilon (R)$ satisfies
$\lim_{R\to +\infty } \varepsilon (R)=0$.
\end{itemize}

This article is organized as follows.
In Section 2, we introduce some basic
properties of the Lebesgue-Sobolev spaces with variable exponents and
$p(x)$-Laplacian.
In Section 3, we give the main results and the proofs.

\section{Notation and preliminary results}

Throughout this paper, the letters $c$, $c_i$, $C_i$, $i=1,2,\dots $,
denote positive constants which may vary from line to line but are
independent of the terms which will take part in any limit process.
To discuss problem \eqref{eP}, we need some preparations on space
$W^{1,p(\cdot )}(\Omega )$ which we call variable exponent Sobolev space,
where $\Omega \subset\mathbb{R}^N$ is an open domain.
Firstly, we state some basic properties of spaces
$W^{1,p(\cdot )}(\Omega )$ which we will use later
(for details, see \cite{11,13,14,16}). Denote
\begin{gather*}
C_{+}(\overline{\Omega }) =\{ h\in C(\overline{\Omega })
, h(x)\geq 1\text{ for }x\in \overline{\Omega } \} , \\
h_{\Omega }^{+} =\operatorname{ess\,sup}_{x\in \Omega } h(x),
h_{\Omega}^{-}=\operatorname{ess\,inf}_{x\in \Omega } h(x),\text{ for any }
h\in L^{\infty}(\Omega ), \\
h^{+} =\operatorname{ess\,sup}_{x\in \mathbb{R}^N}h(x),
h^{-}=\operatorname{ess\,inf}_{x\in \mathbb{R}^N}h(x),
\text{ for any }h\in L^{\infty }(\mathbb{R}^N), \\
S(\Omega ) = \{ u: u\text{ is a real-valued measurable function on }
\Omega \} , \\
L^{p(\cdot )}(\Omega ) = \{ u\in S(\Omega ):
 \int_{\Omega }| u(x)| ^{p(x)}\,dx<\infty \} .
\end{gather*}

In this section, $p(\cdot )$ and $p_i(\cdot )$ are Lipschitz continuous
unless otherwise noted. We introduce the norm on $L^{p(\cdot )}(\Omega )$ by
\begin{equation*}
| u| _{p(\cdot ),\Omega }=\inf \{ \lambda
>0: \int_{\Omega }| \frac{u(x)}{\lambda }| ^{p(x)}\,dx\leq 1 \} ,
\end{equation*}
 and ($L^{p(\cdot )}(\Omega )$, $| \cdot |
_{p(\cdot ),\Omega }$) becomes a Banach space, we call it variable exponent
Lebesgue space.

If $\Omega =\mathbb{R}^N$, we will simply denote by $| \cdot | _{p(\cdot )}$
the norm on $L^{p(\cdot )}(\mathbb{R}^N)$.

\begin{proposition}[\cite{11}] \label{prop2.1}
(i) The space $(L^{p(\cdot )}(\Omega ),| \cdot | _{p(\cdot ),\Omega })$ is a
separable, uniform convex Banach space, and its conjugate space is
$ L^{p^0(\cdot )}(\Omega )$, where
$\frac{1}{p(x)}+\frac{1}{p^0(x)}\equiv 1$.
For any $u\in L^{p(\cdot )}(\Omega )$ and $v\in L^{p^0(\cdot )}(\Omega)$,
we have
\begin{equation*}
| \int_{\Omega }uv\,dx| \leq (\frac{1}{p_{\Omega }^{-}}+
\frac{1}{(p^0)_{\Omega }^{-}})| u| _{p(\cdot ),\Omega
}| v| _{p^0(\cdot ),\Omega }.
\end{equation*}

(ii) If $\Omega $ is bounded, $p_1$, $p_2\in C_{+}(\overline{\Omega })$,
$p_1(\cdot )\leq p_2(\cdot )$ for any $x\in \overline{\Omega }$, then $
L^{p_2(\cdot )}(\Omega )\subset L^{p_1(\cdot )}(\Omega )$, and the
imbedding is continuous.
\end{proposition}

\begin{proposition}[\cite{11}] \label{prop2.2}
If $f:\Omega \times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function
and satisfies
\begin{equation*}
| f(x,s)| \leq h(x)+d| s| ^{p_1(x)/p_2(x)}\quad
 \text{for any }x\in \Omega ,s\in \mathbb{R},
\end{equation*}
where $p_1$, $p_2\in C_{+}(\overline{\Omega })$ ,
$h\in L^{p_2(\cdot)}(\Omega )$, $h(x)\geq 0$, $d\geq 0$,
then the Nemytskii operator from
$L^{p_1(\cdot )}(\Omega )$ to $L^{p_2(\cdot )}(\Omega )$ defined by
$(N_{f}u)(x)=f(x,u(x))$ is continuous and bounded.
\end{proposition}

\begin{proposition}[\cite{11}] \label{prop2.3}
 If we denote
\begin{equation*}
\rho (u)=\int_{\Omega }| u| ^{p(x)}\,dx, \quad \forall u\in L^{p(\cdot )}(\Omega ),
\end{equation*}
then
\begin{itemize}
\item[(i)] $| u| _{p(\cdot ),\Omega}<1\; (=1;>1)\Longleftrightarrow
 \rho (u)<1\;(=1;>1)$;

\item[(ii)] $| u| _{p(\cdot ),\Omega }>1\Longrightarrow
| u| _{p(\cdot ),\Omega }^{p^{-}}\leq \rho (u)\leq
| u| _{p(\cdot ),\Omega }^{p^{+}}$;
 $|u| _{p(\cdot ),\Omega }<1\Longrightarrow | u|
_{p(\cdot ),\Omega }^{p^{-}}\geq \rho (u)\geq | u|
_{p(\cdot ),\Omega }^{p^{+}}$;

\item[(iii)] $| u| _{p(\cdot ),\Omega }\to
0\Longleftrightarrow \rho (u)\to 0;$ $| u|
_{p(\cdot ),\Omega }\to \infty \Longleftrightarrow \rho
(u)\to \infty $.
\end{itemize}
\end{proposition}

\begin{proposition}[\cite{11}] \label{prop2.4}
If $u$, $u_n\in L^{p(\cdot)}(\Omega )$, $n=1,2,\dots $, then the following
statements are equivalent.
\begin{itemize}
\item[(1)] $\lim_{n\to\infty} | u_n-u|_{p(\cdot ),\Omega }=0$;

\item[(2)] $\lim_{n\to\infty} \rho ( u_n-u) =0;$

\item[(3)] $u_n\to  u$ in measure in $\Omega $ and
$\lim_{n\to \infty} \rho ( u_n) =\rho (u)$.
\end{itemize}
\end{proposition}

Denote $Y=\prod_{i=1}^k L^{p_i(\cdot )}(\Omega )$
with the norm
\begin{equation*}
\| y\| _Y=\sum_{i=1}^k y^i| _{p_i(\cdot ),\Omega }, \forall y=(y^1,\dots
,y^{k})\in Y,
\end{equation*}
where $p_i(\cdot )\in C_{+}(\overline{\Omega })$, $i=1,\dots ,m$, then $Y$
is a Banach space.


With a proof similar to proof in \cite{8}, we have:

\begin{proposition} \label{prop2.5}
 Suppose $f(x,y):\Omega \times\mathbb{R}^{k}\to\mathbb{R}^{m}$ is a Caratheodory
function; that is, $f$ satisfies
\begin{itemize}
\item[(i)] For a.e. $x\in \Omega $, $y\to f(x,y)$ is a continuous function
from $\mathbb{R}^{k}$ to $\mathbb{R}^{m}$,

\item[(ii)] For any $y\in\mathbb{R}^{k}$, $x\to f(x,y)$ is measurable.
\end{itemize}
If there exist $p_1(\cdot ),\dots ,p_{k}(\cdot )\in C_{+}(\overline{
\Omega })$, $1\leq \beta (\cdot )\in C(\overline{\Omega })$,
 $\rho (\cdot )\in L^{\beta (\cdot )}(\Omega )$ and positive constant $c>0$ such that
\begin{equation*}
| f(x,y)| \leq \rho (x)+c\underset{i=1}{\overset{k}{\sum }
}| y_i| ^{p_i(x)/\beta (x)}\quad \text{for any }x\in
\Omega ,y\in \mathbb{R}^{k},
\end{equation*}
then the Nemytskii operator from $Y$ to $(L^{\beta (\cdot )}(\Omega ))^{m}$
defined by $(N_{f}u)(x)=f(x,u(x))$ is continuous and bounded.
\end{proposition}

The space $W^{1,p(\cdot )}(\Omega )$ is defined by
\begin{equation*}
W^{1,p(\cdot )}(\Omega )=\{ u\in L^{p(\cdot )}( \Omega )
: \nabla u\in (L^{p(\cdot )}( \Omega ) )^N\} ,
\end{equation*}
with the norm
\begin{equation*}
\| u\| _{p(\cdot ),\Omega }
=| u|_{p(\cdot ),\Omega }+| \nabla u| _{p(\cdot ),\Omega},\quad
\forall u\in W^{1,p(\cdot )}( \Omega ) .
\end{equation*}
If $\Omega =\mathbb{R}^N$, we will denote the norm on
$W^{1,p(\cdot )}(\mathbb{R}^N)$ as $\| u\| _{p(\cdot )}$.

Denote
\begin{gather*}
\| u\| _{p(\cdot ),\Omega }' =\inf \{
\lambda >0: \int_{\Omega }| \frac{\nabla u}{\lambda }
| ^{p(x)}\,dx+\int_{\Omega }| \frac{u(x)}{\lambda }
| ^{p(x)}\,dx\leq 1 \} , \\
\| v\| _{q(\cdot ),\Omega }'
 =\inf \{\lambda >0: \int_{\Omega }| \frac{\nabla v}{\lambda }
| ^{q(x)}\,dx+\int_{\Omega }| \frac{v(x)}{\lambda }
| ^{q(x)}\,dx\leq 1 \} .
\end{gather*}
It is easy to see that the norm
$\| \cdot \| _{p(\cdot),\Omega }'$ is equivalent to
$\| \cdot \|_{p(\cdot ),\Omega }$ on $W^{1,p(\cdot )}(\Omega )$, and
$\| \cdot \| _{q(\cdot ),\Omega }'$ is equivalent to
$\|\cdot \| _{q(\cdot ),\Omega }$ on $W^{1,q(\cdot )}(\Omega )$. In
the following, we will use $\| \cdot \| _{p(\cdot ),\Omega}'$ instead of
$\| \cdot \| _{p(\cdot ),\Omega }$ on $W^{1,p(\cdot )}(\Omega )$, and use
$\| \cdot \|_{q(\cdot ),\Omega }'$ instead of
$\| \cdot \| _{q(\cdot ),\Omega }$ on $W^{1,q(\cdot )}(\Omega )$.
We denote by $W_0^{1,p(\cdot )}(\Omega )$ the closure of
$C_0^{\infty }( \Omega) $ in $W^{1,p(\cdot )}(\Omega )$.

\begin{proposition}[\cite{z1,11,13}] \label{prop2.6}
 (i) $ W^{1,p(\cdot )}(\Omega )$ and $W_0^{1,p(\cdot )}(\Omega )$ are separable
reflexive Banach spaces;

(ii) If $p(\cdot )$ is Lipschitz continuous, $\alpha (\cdot )$ is
measurable, and satisfies $p(\cdot )\leq \alpha (\cdot )\leq p^{\ast }(\cdot
)$ for any $x\in \Omega $, then the imbedding from
$W^{1,p(\cdot )}(\mathbb{R}^N)$ to $L^{\alpha (\cdot )}(\mathbb{R}^N) $
is continuous;

(iii) If $\Omega $ is bounded, $\alpha \in C_{+}( \overline{\Omega }) $
and $\alpha (\cdot )<p^{\ast }(\cdot )$ for any $x\in \overline{
\Omega }$, then the imbedding from $W^{1,p(\cdot )}(\Omega )$ to
$L^{\alpha (\cdot )}( \Omega ) $ is compact and continuous.
\end{proposition}

\begin{proposition}[{\cite[Lemma 3.1]{14}}]  \label{prop2.7}
 Assume that $p:\mathbb{R}^N\to\mathbb{R}$ is a uniformly continuous function,
if $\{u_n\}$ is bounded in $W^{1,p(\cdot )}(\mathbb{R}^N)$ and
\begin{equation*}
\sup_{y\in\mathbb{R} ^N} \int_{B(y,r)}| u_n| ^{\rho
(x)}\,dx\to 0, n\to +\infty ,
\end{equation*}
for some $r>0$ and some $\rho \in L_{+}^{\infty }(\mathbb{R}^N)$ satisfying
\begin{equation*}
p(\cdot )\leq \rho (\cdot )<<p^{\ast }(\cdot ),\text{ }
\end{equation*}
then $u_n\to 0$ in $L^{\alpha (\cdot )}(\mathbb{R}^N)$ for any $\alpha $
satisfying $p(\cdot )<<\alpha (\cdot )<<p^{\ast}(\cdot )$, where $B(y,r)$
is an open ball with center $y$ and radius $r$.
\end{proposition}

Denote $X_1=W^{1,p(\cdot )}(\mathbb{R}^N)$, $X_2=W^{1,q(\cdot )}(\mathbb{R}^N)$,
$X=X_1\times X_2$. Let us endow the norm $\| \cdot\| $ on $X$ as
\begin{equation*}
\| (u,v)\| =\max \{\| u\| _{p(\cdot )},\| v\| _{q(\cdot )}\}.
\end{equation*}
The dual space of $X$ will be denoted by $X^{\ast }$, then for any
$\Theta \in X^{\ast }$, there exist $f\in (W^{1,p(\cdot )}(\mathbb{R}^N))^{\ast }$
and $g\in (W^{1,q(\cdot )}(\mathbb{R}^N))^{\ast }$ such that
$\Theta (u,v)=f(u)+g(v)$. We denote $\|\cdot \| _{\ast }$,
$\| \cdot \| _{\ast ,p(\cdot)} $ and $\| \cdot \| _{\ast ,q(\cdot )}$
 the norms of $ X^{\ast },(W^{1,p(\cdot )}(\mathbb{R}^N))^{\ast }\ $
and $(W^{1,q(\cdot )}(\mathbb{R} ^N))^{\ast }$, respectively.
 Obviously
$X^{\ast }=(W^{1,p(\cdot )}(\mathbb{R}^N))^{\ast }\times (W^{1,q(\cdot )}(
\mathbb{R}^N))^{\ast }$ and
\begin{equation*}
\| \Theta \| _{\ast }=\| f\| _{\ast
,p(\cdot )}+\| g\| _{\ast ,q(\cdot )},\quad \forall \Theta \in X.
\end{equation*}
For every $(u,v)$ and $(\varphi ,\psi )$ in $X$, set
\begin{gather*}
\Phi _1(u) =\int_{\mathbb{R}^N}\frac{1}{p(x)}| \nabla u| ^{p(x)}\,dx
+\int_{\mathbb{R}^N}\frac{1}{p(x)}| u| ^{p(x)}\,dx, \\
\Phi _2(v) =\int_{\mathbb{R}^N}\frac{1}{q(x)}| \nabla v| ^{q(x)}\,dx
+\int_{\mathbb{R}^N}\frac{1}{q(x)}| v| ^{q(x)}\,dx, \\
\Phi (u,v) =\Phi _1(u)+\Phi _2(v), \\
\Psi (u,v) =\int_{\mathbb{R}^N}\{\lambda [ \frac{a(x)}{\gamma (x)}| u|
^{\gamma (x)}+\frac{b(x)}{\delta (x)}| v| ^{\delta
(x)}]+F(x,u,v)\}\,dx, \\
\Psi _1(u,v) =\int_{\mathbb{R}^N}\lambda \frac{a(x)}{\gamma (x)}| u|
^{\gamma(x)}\,dx
+\int_{\mathbb{R}^N}\lambda \frac{b(x)}{\delta (x)}| v| ^{\delta (x)}\,dx.
\end{gather*}

It follows from Proposition \ref{prop2.5} that $\Phi \in C^1(X,\mathbb{R})$, then
\begin{gather*}
\Phi '(u,v)(\varphi ,\psi ) = D_1\Phi (u,v)(\varphi )+D_2\Phi
(u,v)(\psi ),\forall (\varphi ,\psi )\in X, \\
\Psi '(u,v)(\varphi ,\psi ) = D_1\Psi (u,v)(\varphi )+D_2\Psi
(u,v)(\psi ),\forall (\varphi ,\psi )\in X,
\end{gather*}
where
\begin{gather*}
D_1\Phi (u,v)(\varphi )
= \int_{\mathbb{R}^N}| \nabla u| ^{p(x)-2}\nabla u\nabla \varphi \,dx
+\int_{\mathbb{R}^N}| u| ^{p(x)-2}u\varphi \,dx=\Phi _1'(u)(\varphi ),\\
\forall \varphi \in X_1, \\
D_2\Phi (u,v)(\psi ) =\int_{\mathbb{R}^N}| \nabla v| ^{p(x)-2}\nabla v\nabla \psi \,dx
+\int_{\mathbb{R}^N}| v| ^{p(x)-2}v\psi \,dx=\Phi _2'(v)(\psi),\\
\forall \psi \in X_2, \\
D_1\Psi (u,v)(\varphi ) =\int_{\mathbb{R}^N}[\lambda a(x)| u| ^{\gamma (x)-2}u
+\frac{\partial }{\partial u}F(x,u,v)]\varphi \,dx,\quad \forall \varphi \in X_1, \\
D_2\Psi (u,v)(\psi )=\int_{\mathbb{R}^N}[\lambda b(x)| v| ^{\delta (x)-2}v
+\frac{\partial }{\partial v}F(x,u,v)]\psi \,dx,\quad \forall \psi \in X_2.
\end{gather*}

The integral functional associated with the problem \eqref{eP} is
\begin{equation*}
J(u,v)=\Phi (u,v)-\Psi (u,v).
\end{equation*}

Without loss of generality, we may assume that $F(x,0,0)=0$, then we have
\begin{equation*}
F(x,u,v)=\int_0^1[u\partial _2F(x,tu,tv)+v\partial _{3}F(x,tu,tv)]dt,
\end{equation*}
where $\partial _{j}$ denotes the partial derivative of $F$ with respect to its
$j$-th variable. The condition (A0) holds
\begin{equation}
|F(x,u,v)|\leq c(|u|^{p(x)}+|u|^{\alpha (x)}+|v|^{q(x)}+|v|^{\beta (x)}).
\label{a1}
\end{equation}

From Proposition \ref{prop2.5} and condition (A0), it is easy
to see that $J\in C^1(X,\mathbb{R})$ and satisfies
\begin{equation*}
J'(u,v)(\varphi ,\psi )=D_1J(u,v)(\varphi )+D_2J(u,v)(\psi
),\quad \forall (\varphi ,\psi )\in X,
\end{equation*}
where
\begin{gather*}
D_1J(u,v)(\varphi ) = D_1\Phi (u,v)(\varphi )-D_1\Psi (u,v)(\varphi
),\quad \forall \varphi \in X_1, \\
D_2J(u,v)(\psi )  = D_2\Phi (u,v)(\psi )-D_2\Psi (u,v)(\psi ),\quad \forall
\psi \in X_2.
\end{gather*}
Obviously,
\begin{equation*}
\| J'(u,v)\| _{\ast }=\|D_1J(u,v)\| _{\ast ,p(\cdot )}
+\| D_2J(u,v)\|_{\ast ,q(\cdot )}.
\end{equation*}
We say $(u,v)\in X$ is a critical point of $J$ if
\begin{equation*}
J'(u,v)(\varphi ,\psi )=0,\quad \forall (\varphi ,\psi )\in X.
\end{equation*}

\begin{proposition}[\cite{33}] \label{prop2.8}
 (i) $\Phi $ is a convex functional;

(ii) $\Phi '$ is strictly monotone, that is, for any $(u_1,v_1)$
, $(u_2,v_2)\in X$ with $(u_1,v_1)\neq (u_2,v_2)$, we have
\begin{equation*}
(\Phi '(u_1,v_1)-\Phi '(u_2,v_2))(u_1-u_2,v_1-v_2)>0,
\end{equation*}

(iii) $\Phi '$ is a mapping of type (S$_{+}$), that is if
$(u_n,v_n)\rightharpoonup (u,v)$ in $X$ and
\begin{equation*}
\limsup_{n\to \infty } [\Phi '(u_n,v_n)-\Phi '(u,v)](u_n-u,v_n-v)\leq 0,
\end{equation*}
then $(u_n,v_n)\to (u,v)$ in $X$.

(iv) $\Phi '$ $:X\to X^{\ast }$ is a bounded homeomorphism.
\end{proposition}

\begin{theorem} \label{thm2.9}
 $\Psi _1\in C^1(X, \mathbb{R})$ and $\Psi _1$,$\Psi _1'$
are weakly-strongly continuous, that is,
$(u_n,v_n)\rightharpoonup (u,v)$ implies
$\Psi _1(u_n,v_n)\to \Psi _1(u,v)$ and $\Psi _1'(u_n,v_n)\to \Psi _1'(u,v)$.
\end{theorem}

The proof is similar to the proof of  \cite[Theorem 3.2]{38},
we omit it here.

\section{Main results and their proofs}

In this section, we state the main results at first, and using the critical
point theory, we prove the existence of solutions for problem \eqref{eP}, and the
asymptotic behavior of solutions near infinity.

We say that $(u,v)\in X$ is a weak solution for \eqref{eP}, if
\begin{align*}
&\int_{\mathbb{R}^N}| \nabla u| ^{p(x)-2}\nabla u\cdot \nabla \varphi\,dx
+\int_{\mathbb{R}^N}| u| ^{p(x)-2}u\cdot \varphi \,dx\\
&=\int_{\mathbb{R}^N}\{\lambda a(x)| u| ^{\gamma(x)-2}u+F_{u}(x,u,v)\}\varphi \,dx,
\quad \forall \varphi \in X_1,
\\
&\int_{\mathbb{R}^N}| \nabla v| ^{q(x)-2}\nabla v\cdot \nabla \psi\,dx
+\int_{\mathbb{R}^N}| v| ^{q(x)-2}v\cdot \psi \,dx\\
&=\int_{\mathbb{R}^N}\{\lambda b(x)| v| ^{\delta (x)-2}v+F_{v}(x,u,v)\}\psi \,dx,\quad
\forall \psi \in X_2.
\end{align*}
It is easy to see that the critical point of $J$ is a solution for \eqref{eP}.

Similar to the proof of \cite[Theorem 5]{22}, from (A1) we have
\begin{gather}
F(x,\tau ^{1/\theta_1}s,\tau ^{1/\theta_2} t)
\geq \tau F(x,s,t),\quad
\forall (x,s,t)\in\mathbb{R}^N\times\mathbb{R}^2,\;\tau \geq 1,  \label{b1} \\
F(x,\tau ^{1/\theta_1}s,\tau ^{1/\theta_2} t)
\leq \tau F(x,s,t),\quad \forall (x,s,t)\in\mathbb{R}^N\times\mathbb{R}^2,\;
0\leq \tau \leq 1.  \label{b2}
\end{gather}
In fact, from (A0) and (A1) we have
\begin{itemize}
\item[(A0')]  $0\leq F(x,s,t)\leq \sigma |
s| ^{p(x)}+C(\sigma )| s| ^{\alpha (x)}+\sigma
| t| ^{q(x)}+C(\sigma )| t| ^{\beta
(x)}$, where $\sigma $ is a small enough positive constant.
\end{itemize}
Denote
\begin{gather*}
\Gamma =\{\gamma \in C([0,1],X): \gamma (0)=(0,0), \gamma
(1)=(u^{\ast },v^{\ast })\}, \\
c=\inf_{\gamma \in \Gamma } \max_{(u,v)\in \gamma } J(u,v),
\end{gather*}
where $(u^{\ast },v^{\ast })\in X$ satisfies $J(u^{\ast },v^{\ast })<0$.

Denote
\begin{gather*}
\begin{aligned}
\widehat{J}(u,v)
&=\int_{\mathbb{R}^N}\frac{1}{p(x)}(| \nabla u| ^{p(x)}+|u| ^{p(x)})\,dx\\
&\quad +\int_{\mathbb{R}^N}\frac{1}{q(x)}(| \nabla v| ^{q(x)}+|v| ^{q(x)})\,dx
-\int_{\mathbb{R}^N}\widetilde{F}(u,v)\,dx,
\end{aligned} \\
\mathcal{N}=\{ (u,v)\in X: \widehat{J}'(u,v)(\frac{1}{
\theta _1}u,\frac{1}{\theta _2}v)=0, (u,v)\neq 0\} ,
\\
J^{\infty }=\inf_{(u,v)\in \mathcal{N}} \widehat{J}(u,v).
\end{gather*}
Now our results can be stated as follows.

\begin{theorem} \label{thm3.1}
If $F$ satisfies {\rm (A0)--(A3)},
 the positive parameter $\lambda $ is small enough and $
c<J^{\infty }$, then \eqref{eP} possesses a nontrivial solution.
\end{theorem}

Next, we give an application of Theorem \ref{thm3.1}, that is, a sufficient condition
for $c<J^{\infty }$.

We say $h(x)$ is periodic and its period is
$A=\{a_1,a_2,\dots ,a_{N}\}$  where $a_i\geq 0$,
$1\leq i\leq N$,
if
\begin{equation*}
h(x)=h(x+n_ia_ie_i),\quad  \forall x\in \mathbb{R}^N,
\end{equation*}
where $n_i$ are integers and
$\langle e_1,\dots ,e_{N}\rangle $ is the standard basis of $\mathbb{R}^N$.

Denote
\begin{equation*}
Q(x_{o},A)=\{x\in \mathbb{R}^N: (x-x_{o})e_i\in [ 0,a_i]\}.
\end{equation*}

\begin{theorem} \label{thm3.2}
If $p(\cdot ),q(\cdot )$ are periodic and their periods
is $A$, $F(x,s,t)$ satisfies {\rm (A0)--(A3)},
and there exist $\tau ,\delta >0$ and $p(\cdot )<<\alpha
_{o}(\cdot )<<p^{\ast }(\cdot )$, $q(\cdot )<<\beta _{o}(\cdot )<<q^{\ast
}(\cdot )$ such that
\begin{equation}
\begin{gathered}
F(x,s,t)\geq \widetilde{F}(s,t),\quad \forall (x,s,t)\in\mathbb{R}^N
\times\mathbb{R}^{+}\times\mathbb{R}^{+},
\\
 F(x,s,t)\geq \widetilde{F}(s,t)+\tau s^{\alpha _{o}(x)-1}+\tau t^{\beta
_{o}(x)-1},\\
\forall (x,s,t)\in B(Q(x_{o},A),\delta )\times\mathbb{R}^{+}\times\mathbb{R}^{+},
\end{gathered}
\label{3.51}
\end{equation}
then \eqref{eP} possesses at least one nontrivial solution when $\lambda $
is small enough.
\end{theorem}

Next we give the behavior of solutions near infinity.

\begin{theorem} \label{thm3.3}
Suppose {\rm (A0)--(A3)} hold, $a,b\in L^{\infty }(\mathbb{R}^N)$.
If $u$ is a weak solution for problem \eqref{eP}, then
$u,v \in C^{1,\alpha }(\mathbb{R}^N)$, $u(x)\to 0,|\nabla u(x)|\to 0,v(x)\to 0$ and
$|\nabla v(x)|\to 0$ as $|x|\to \infty $.
\end{theorem}

\subsection{Proof of Theorem \ref{thm3.1}}

For the proof,  we need to do some preparations.

\begin{lemma} \label{lem3.4}
 If $F$ satisfies {\rm (A0)--(A2)},
and the  parameter $\lambda $ is small enough, then $J$ satisfies
the Mountain Pass Geometry, that is,
\begin{itemize}
\item[(i)] There exist positive numbers $\rho $ and $\alpha $ such that $J(u,v)\geq
\alpha $ for any $(u,v)\in X$ with $\| (u,v)\| =\rho $;

\item[(ii)] $J(0,0)=0$, and there exists $(u,v)\in X$ with $\| (u,v)\| >\rho $
such that $J(u,v)<0$.
\end{itemize}
\end{lemma}

\begin{proof} (i) Recall that (A0)--(A1) imply (A0'). Then from (A0') we have
\begin{equation*}
|F(x,u,v)|\leq \varepsilon (|u|^{p^{+}}+|v|^{q^{+}})+C(\varepsilon
)(|u|^{\alpha (x)}+|v|^{\beta (x)}).
\end{equation*}
Suppose $\varepsilon $ and $\lambda $ are small enough. We have
\begin{align*}
J(u,v) &=\int_{\mathbb{R}^N}\frac{1}{p(x)}(| \nabla u| ^{p(x)}
+|u| ^{p(x)})\,dx+\int_{\mathbb{R}^N}\frac{1}{q(x)}(| \nabla v| ^{q(x)}
+|v| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}^N}\lambda [ \frac{a(x)}{\gamma (x)}| u|
^{\gamma(x)}+\frac{b(x)}{\delta (x)}| v| ^{\delta (x)}]\,dx
-\int_{\mathbb{R}^N}F(x,u,v)\,dx \\
&\geq \Phi (u,v)-\lambda \int_{\mathbb{R}^N}(| u| ^{p(x)}+| v|
^{q(x)})\,dx-\lambda C_1 \\
&\quad-\varepsilon \int_{\mathbb{R}^N}[|u|^{p^{+}}+|v|^{q^{+}}]\,dx
-C(\varepsilon )\int_{\mathbb{R}^N}(|u|^{\alpha (x)}+|v|^{\beta (x)})\,dx \\
&\geq \frac{1}{2}\Phi (u,v)-C(\varepsilon )\int_{\mathbb{R}^N}(|u|^{\alpha (x)}
+|v|^{\beta (x)})\,dx.
\end{align*}
Since
\begin{equation*}
p(\cdot )<<\alpha (\cdot )<<p^{\ast }(\cdot ), q(\cdot )<<\beta
(\cdot )<<q^{\ast }(\cdot ),
\end{equation*}
there exists a positive constant $\varepsilon _0$ such that
\begin{equation*}
\alpha (\cdot )-p(\cdot )\geq 2\varepsilon _0\quad\text{and}\quad
\beta (\cdot)-q(\cdot )\geq 2\varepsilon _0.
\end{equation*}

Since $p$ is Liptchitz cntinuous, we can divide
$\mathbb{R}^N$ into countable disjoint cube $\Omega _n,n=1,2,\dots $, each one has
the same side length, such that
 $\cup_{n=1}^{\infty }\Omega _n=\mathbb{R}^N$ and for any $n=1,2,\dots $,
the following inequalities hold
\begin{equation*}
\inf_{x\in\Omega_n} \alpha (x)-\underset{x\in \Omega _n}{
\sup }p(x)\geq \varepsilon _0\quad \text{and}\quad
\inf_{x\in\Omega_n}
\beta (x)-\sup_{x\in\Omega_n} q(x)\geq \varepsilon _0.
\end{equation*}
Denote $\alpha _{\Omega _n}^{-}=\inf_{x\in\Omega_n} \alpha
(x),p_{\Omega _n}^{+}=\sup_{x\in\Omega_n} p(x)$. Suppose the
positive number $\rho <1$ is small enough and $\| (u,v)\| =\rho $,
from Propositions \ref{prop2.3} and \ref{prop2.6}, it follows that
\begin{align*}
C(\varepsilon )\int_{\Omega _n}|u|^{\alpha (x)}
&\leq C(\varepsilon )|u|_{\alpha (\cdot ),\Omega _n}^{\alpha _{\Omega _n}^{-}}
\leq \frac{1}{ 8p^{+}}\| u\| _{p(\cdot ),\Omega _n}^{p_{\Omega _n}^{+}}\\
&\leq \frac{ 1}{8}\int_{\Omega _n}\frac{1}{p(x)}(| \nabla u|^{p(x)}+| u| ^{p(x)})\,dx.
\end{align*}
Similarly, we have
\begin{equation*}
C(\varepsilon )\int_{\Omega _n}|v|^{\beta (x)}\,dx\leq \frac{1}{8}
\int_{\Omega _n}\frac{1}{q(x)}(| \nabla v|
^{q(x)}+| v| ^{q(x)})\,dx.
\end{equation*}
Thus, we have
\begin{align*}
J(u,v)
&\geq \frac{1}{2}\Phi (u,v)-C(\varepsilon )\int_{\mathbb{R}^N}
 (|u|^{\alpha (x)}+|v|^{\beta (x)})\,dx \\
&=\frac{1}{2}\underset{n=1}{\overset{\infty }{\sum }}\int_{\Omega _n}
\frac{1}{p(x)}(| \nabla u| ^{p(x)}\,dx+|
u| ^{p(x)})\,dx \\
&\quad +\frac{1}{2}\underset{n=1}{\overset{\infty }{\sum }}\int_{\Omega _n}
\frac{1}{q(x)}(| \nabla v| ^{q(x)}+|
v| ^{q(x)})\,dx \\
&\quad -\underset{n=1}{\overset{\infty }{\sum }}C(\varepsilon )\int_{\Omega
_n}(|u|^{\alpha (x)}+|v|^{\beta (x)})\,dx \\
&\geq \frac{1}{8}\Phi (u,v).
\end{align*}
It means that assertion (i) holds.

(ii) Obviously, $J(0,0)=0$.
When $t\geq 1$, by \eqref{b1}, we have
\begin{align*}
&J(t^{1/\theta_1}u,t^{1/\theta_2} v) \\
&=\int_{\mathbb{R}^N}\frac{1}{p(x)}t^{\frac{p(x)}{\theta _1}}(| \nabla
u| ^{p(x)}+| u| ^{p(x)})\,dx
+\int_{\mathbb{R}^N}\frac{1}{q(x)}t^{\frac{q(x)}{\theta _2}}(| \nabla
v| ^{q(x)}+| v| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}
^N}\lambda [ \frac{a(x)}{\gamma (x)}t^{\frac{\gamma (x)}{\theta _1}
}| u| ^{\gamma (x)}+\frac{b(x)}{\delta (x)}t^{\frac{
\delta (x)}{\theta _2}}| v| ^{\delta (x)}]\,dx
-\int_{\mathbb{R}^N}F(x,t^{1/\theta_1}u,t^{1/\theta_2} v)\,dx \\
&\leq \int_{\mathbb{R}^N}\frac{1}{p(x)}t^{\frac{p(x)}{\theta _1}}(| \nabla
u| ^{p(x)}+| u| ^{p(x)})\,dx
+\int_{\mathbb{R}^N}\frac{1}{q(x)}t^{\frac{q(x)}{\theta _2}}(| \nabla
v| ^{q(x)}+| v| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}^N}\lambda [ \frac{a(x)}{\gamma (x)}
 t^{\frac{\gamma (x)}{\theta _1}}| u| ^{\gamma (x)}
+\frac{b(x)}{\delta (x)}t^{\frac{ \delta (x)}{\theta _2}}| v| ^{\delta (x)}]\,dx
-\int_{\mathbb{R}^N}tF(x,u,v)\,dx.
\end{align*}
Note that $\gamma (x)<<p(x)$, $\delta (x)<<q(x)$, $\theta _1>p^{+}$ and
$\theta _2>q^{+}$, then for any nontrivial $(u,v)\in X$, it is not hard to
check
\begin{equation*}
J(t^{1/\theta_1}u,t^{1/\theta_2} v)\to -\infty\quad
\text{as }t\to +\infty .
\end{equation*}
\end{proof}

We remark that it is easy to see that $J^{\infty }>0$.

\begin{lemma} \label{lem3.5}
If $F$ satisfies {\rm (A0)--(A2)}, $\{(u_n,v_n)\}$ is a
PS sequence of $J$, that is $J(u_n,v_n)\to c$ which is the
mountain pass level, and $J'(u_n,v_n)\to 0$, then $
\{(u_n,v_n)\}$ is bounded.
\end{lemma}

\begin{proof}
 Since $1<<\gamma (\cdot )<<p(\cdot )$,
$a(\cdot )\in L^{\frac{p(\cdot )}{p(\cdot )-\gamma (\cdot )}}(\mathbb{R}^N)$,
$1<<\delta (\cdot )<<q(\cdot )$,
$b(\cdot )\in L^{\frac{q(\cdot )}{q(\cdot )-\delta (\cdot )}}(\mathbb{R}^N)$,
we have
\begin{align*}
&\big| \int_{\mathbb{R}^N}\lambda [ \frac{a(x)}{\gamma (x)}| u| ^{\gamma
(x)}+\frac{b(x)}{\delta (x)}| v| ^{\delta
(x)}]\,dx\big| \\
&\leq \int_{\mathbb{R}^N}[| \lambda a(x)| | u| ^{\gamma(x)}
+| \lambda b(x)| | v| ^{\delta (x)}]\,dx \\
&\leq \int_{\mathbb{R}^N}[\frac{\gamma (x)}{p(x)}(\varepsilon _1)
^{\frac{p(x)}{\gamma (x)}}| u| ^{p(x)}+\frac{p(x)-\gamma (x)}{p(x)}|
\frac{1}{\varepsilon _1}\lambda a(x)| ^{\frac{p(x)}{p(x)-\gamma
(x)}}]\,dx \\
&\quad +\int_{\mathbb{R}^N}[\frac{\delta (x)}{q(x)}
(\varepsilon _1)^{\frac{q(x)}{\delta (x)}}| v| ^{q(x)}
+\frac{q(x)-\delta (x)}{q(x)}|
\frac{1}{\varepsilon _1}\lambda b(x)| ^{\frac{q(x)}{q(x)-\delta(x)}}]\,dx \\
&\leq \varepsilon _1\int_{\mathbb{R}^N}[| u| ^{p(x)}+| v|^{q(x)}]\,dx
 +C(\varepsilon _1),
\end{align*}
where $\varepsilon _1$ is a positive small enough constant.

By (A1), we have for large values of $n$
\begin{align*}
&c+1 \\
&\geq J(u_n,v_n) \\
&=\int_{\mathbb{R}^N}\frac{1}{p(x)}(| \nabla u_n| ^{p(x)}
+|u_n| ^{p(x)})\,dx
+\int_{\mathbb{R}^N}\frac{1}{q(x)}(| \nabla v_n| ^{q(x)}
+|v_n| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}^N}\lambda [ \frac{a(x)}{\gamma (x)}| u_n|
^{\gamma (x)}+\frac{b(x)}{\delta (x)}| v_n| ^{\delta(x)}]\,dx
-\int_{\mathbb{R}^N}F(x,u_n,v_n)\,dx \\
&\geq \int_{\mathbb{R}^N}\frac{1}{p(x)}\{| \nabla u_n| ^{p(x)}+|
u_n| ^{p(x)}-\frac{u_n}{\theta _1}F_{u}(x,u_n,v_n)\}\,dx \\
&\quad +\int_{\mathbb{R}^N}\frac{1}{q(x)}\{| \nabla v_n| ^{q(x)}+|
v_n| ^{q(x)}-\frac{v_n}{\theta _2}F_{v}(x,u_n,v_n)\}\,dx \\
&\quad -\int_{\mathbb{R}^N}\lambda [ \frac{a(x)}{\gamma (x)}| u_n|
^{\gamma (x)}+\frac{b(x)}{\delta (x)}| v_n| ^{\delta(x)}]\,dx, \\
&=\int_{\mathbb{R}^N}(\frac{1}{p(x)}-\frac{1}{\theta _1})(| \nabla
u_n| ^{p(x)}+| u_n| ^{p(x)})\,dx
+J'(u_n,v_n)(\frac{1}{\theta _1}u_n,\frac{1}{\theta _2}v_n) \\
&\quad +\int_{\mathbb{R}^N}(\frac{1}{q(x)}-\frac{1}{\theta _2})(| \nabla
v_n| ^{q(x)}+| v_n| ^{q(x)})\,dx
-\frac{l}{2}\int_{\mathbb{R}^N}[| u_n| ^{p(x)}+| v_n|^{q(x)}]\,dx-C \\
&\geq \frac{l}{2}\int_{\mathbb{R}^N}(| \nabla u_n| ^{p(x)}+| u_n|
^{p(x)})\,dx
+\frac{l}{2}\int_{\mathbb{R}^N}(| \nabla v_n| ^{q(x)}+| v_n|^{q(x)})\,dx \\
&\quad -\frac{1}{\theta _1}\| D_1J(u_n,v_n)\| _{\ast
,p(\cdot )}\| u_n\| _{p(\cdot )}
-\frac{1}{\theta _2}\| D_2J(u_n,v_n)\| _{\ast ,q(\cdot )}\|
v_n\| _{q(\cdot )}-C,
\end{align*}
where $l=\min \{(\frac{1}{p^{+}}-\frac{1}{\theta _1}),(\frac{1}{q^{+}}-
\frac{1}{\theta _2})\}$.

Without loss of generality, we  assume that $\| v_n\|
_{q(\cdot )}\leq \| u_n\| _{p(\cdot )}\to \infty$,
$n=1,2,\dots $. Therefore for large enough $n$, we have
\begin{equation*}
c+1\geq \frac{l}{2}\| u_n\| _{p(\cdot )}^{p^{-}}-(\frac{1
}{\theta _1}\| D_1J(u_n,v_n)\| _{\ast ,p(\cdot )}+
\frac{1}{\theta _2}\| D_2J(u_n,v_n)\| _{\ast
,q(\cdot )})\| u_n\| _{p(\cdot )}-C.
\end{equation*}
This is a contradiction. Thus $\{\| u_n\| _{p(\cdot )}\}$
and $\{\| v_n\| _{q(\cdot )}\}$ are bounded.
\end{proof}

\begin{lemma} \label{lem3.6}.
Suppose $F$ satisfies {\rm (A0)--(A3)}, $\{(u_n,v_n)\}$ satisfy
$J(u_n,v_n)\to c>0$,  where $c$ is the mountain pass level,
$J'(u_n,v_n)\to 0$, $\lambda $ is small enough, passing
to a subsequence still labeled by $n$, we have

(i) $\{(u_n,v_n)\}$ has a nontrivial weak limit $(u,v)\in X$ or
\begin{equation*}
\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx
+\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx\geq \delta >0;
\end{equation*}

(ii) If $c<J^{\infty }$, then $\{(u_n,v_n)\}$ has a nontrivial weak
limit.
\end{lemma}

\begin{proof}
 (i) It follows from Lemma \ref{lem3.5} that $\{(u_n,v_n)\}$ is
bounded in $X$. Without loss of generality, we may assume that
$(u_n,v_n)\rightharpoonup (u,v)$ in $X$.
If $(u,v)=(0,0)$, then Proposition \ref{prop2.6} implies
\begin{equation} \label{3.39}
\begin{gathered}
u_n\to 0\quad \text{in }L_{\rm loc}^{\alpha (\cdot )}(\mathbb{R}^N), p(\cdot )
\leq \alpha (\cdot )<p^{\ast }(\cdot ), \\
v_n\to 0\quad \text{in }L_{\rm loc}^{\beta (\cdot )}(\mathbb{R}^N), q(\cdot )
\leq \beta (\cdot )<q^{\ast }(\cdot ).
\end{gathered}
\end{equation}
Since $(u_n,v_n)\rightharpoonup (0,0)$, then Theorem \ref{thm2.9} implies
\begin{equation}
\int_{\mathbb{R}^N}\lambda \frac{a(x)}{\gamma (x)}| u_n| ^{\gamma
(x)}\,dx=o(1)
=\int_{\mathbb{R}^N}\lambda \frac{b(x)}{\delta (x)}| v_n| ^{\delta
(x)}\,dx.  \label{a.11}
\end{equation}
Recall that (A0)--(A1) imply (A0'). Then from (A0'), (A3) and \eqref{3.39}, 
it follows that
\begin{align*}
&\big| \int_{\mathbb{R}^N}(F_{u}(x,u_n,v_n)-\widetilde{F}_{u}(u_n,v_n))u_n\,dx|\\
&\leq \int_{| x| \geq R}| F_{u}(x,u_n,v_n)-
\widetilde{F}_{u}(u_n,v_n)| | u_n| \,dx \\
&\quad +C\int_{| x| \leq R}(| u_n|
^{p(x)}+| u_n| ^{\alpha (x)}+|
v_n| ^{q(x)}+| v_n| ^{\beta (x)})\,dx \\
&\leq \varepsilon (R)\int_{| x| \geq R}(|
u_n| ^{p(x)}+| u_n| ^{\alpha
(x)}+| v_n| ^{q(x)}+| v_n| ^{\beta
(x)})\,dx \\
&\quad +C\int_{| x| \leq R}(| u_n|
^{p(x)}+| u_n| ^{\alpha (x)}+|
v_n| ^{q(x)}+| v_n| ^{\beta (x)})\,dx,
\end{align*}
which implies
\begin{equation}
\int_{\mathbb{R}^N}F_{u}(x,u_n,v_n)u_n\,dx
=\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx+o(1)
\quad \text{as }n\to+\infty .  \label{3.40}
\end{equation}

Similar to the proof of \eqref{3.40}, we can verify
\begin{gather}
\int_{\mathbb{R}^N}F_{v}(x,u_n,v_n)v_n\,dx
=\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx+o(1)\quad \text{when }n\to
+\infty ,  \label{5.11}
\\
\int_{\mathbb{R}^N}F(x,u_n,v_n)\,dx
=\int_{\mathbb{R}^N}\widetilde{F}(u_n,v_n)\,dx+o(1)\quad
\text{as }n\to +\infty .
\label{3.41}
\end{gather}
Since $F(x,u,v)\geq 0$ and $J(u_n,v_n)\to c>0$, we have
\begin{equation}
\Phi (u_n,v_n)-\Psi _1(u_n,v_n)\geq J(u_n,v_n)\geq C_1>0,
\quad \text{for }n\backsimeq \infty ,  \label{10.1}
\end{equation}
which together with \eqref{a.11}-\eqref{3.41} and $J'(u_n,v_n)\to 0$ implies
\begin{equation}
\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx
+\int_{\mathbb{R} ^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx\geq \delta >0.
\label{3.47}
\end{equation}

(ii) By (A0), \eqref{b1} and \eqref{b2}, there exist
$t_n>0$ such that $(t_n^{1/\theta_1}u_n,t_n^{1/\theta _2} v_n)\in \mathcal{N}$;
that is,
\begin{equation} \label{3.42}
\begin{aligned}
&\frac{1}{\theta _1}\int_{\mathbb{R}^N}
\Big(| \nabla t_n^{1/\theta_1}u_n|
^{p(x)}+| t_n^{1/\theta_1}u_n| ^{p(x)}\Big)\,dx \\
&+\frac{1}{\theta _2}\int_{\mathbb{R}^N}\Big(| \nabla t_n^{1/\theta_2}v_n|
^{q(x)}+| t_n^{1/\theta_2} v_n| ^{q(x)}\Big)\,dx\\
&=\frac{1}{\theta _1}\int_{\mathbb{R}^N}\widetilde{F}_{u}
(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{\theta _2}}v_n)t_n^{1/\theta_1}u_n\,dx\\
&\quad +\frac{1}{\theta _2}
\int_{\mathbb{R}^N}\widetilde{F}_{v}(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{
\theta _2}}v_n)t_n^{1/\theta_2} v_n\,dx.
\end{aligned}
\end{equation}
Suppose $(u,v)$ is trivial, then \eqref{3.47} is valid. Noting that
$\{(u_n,v_n)\}$ is bounded in $X$. Obviously, there exist positive
constants $c_1$ and $c_2$ such that
\begin{equation}
c_1\leq t_n\leq c_2.  \label{5.13}
\end{equation}
From \eqref{a.11} and \eqref{5.13}, we have
\begin{equation}
\int_{\mathbb{R}^N}\lambda \frac{a(x)}{\gamma (x)}| 
t_n^{1/\theta _1}u_n| ^{\gamma (x)}\,dx=o(1)
=\int_{\mathbb{R}^N}\lambda \frac{b(x)}{\delta (x)}| t_n^{1/\theta_2}
v_n| ^{\delta (x)}\,dx.  \label{9.12}
\end{equation}
Since $J'(u_n,v_n)\to 0$ and $\{(u_n,v_n)\}$ is
bounded in $X$, it follows from \eqref{3.40} and \eqref{5.11} that
\begin{gather}
\begin{aligned}
&\int_{\mathbb{R}^N}(| \nabla u_n| ^{p(x)}+| u_n|^{p(x)})\,dx \\
&=\int_{\mathbb{R}^N}F_{u}(x,u_n,v_n)u_n\,dx+o(1)
=\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx+o(1),
\end{aligned} \label{3.43}\\
\begin{aligned}
&\int_{\mathbb{R}^N}(| \nabla v_n| ^{q(x)}+| v_n|^{q(x)})\,dx \\
&=\int_{\mathbb{R}^N}F_{v}(x,u_n,v_n)v_n\,dx+o(1)
=\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx+o(1).
\end{aligned}
\label{5.12}
\end{gather}
Obviously, there exist $\xi _n,\eta _n\in \mathbb{R}^N$ such that
\begin{gather*}
\int_{\mathbb{R} ^N}\Big(| \nabla t_n^{1/\theta_1}u_n|
^{p(x)}+| t_n^{1/\theta_1}u_n| ^{p(x)}\Big)\,dx
=t_n^{\frac{p(\xi _n)}{\theta _1}}\int_{\mathbb{R}^N}\Big(| \nabla u_n|
^{p(x)}+| u_n|^{p(x)}\Big)\,dx,
\\
\int_{\mathbb{R}^N}\Big(| \nabla t_n^{1/\theta_2}v_n|
^{q(x)}+| t_n^{1/\theta_2} v_n| ^{q(x)}\Big)\,dx
= t_n^{\frac{q(\eta _n)}{\theta _2}}
\int_{\mathbb{R}^N}(| \nabla v_n| ^{q(x)}+| v_n|^{q(x)})\,dx,
\end{gather*}
which together with \eqref{3.42}, \eqref{3.43} and \eqref{5.12} implies
\begin{align*}
&\frac{1}{\theta _1}t_n^{\frac{p(\xi _n)}{\theta _1}}
[\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx+o(1)]
+\frac{1}{\theta _2}t_n^{\frac{q(\eta _n)}{\theta _2}}
[\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx+o(1)] \\
&=\frac{1}{\theta _1}\int_{\mathbb{R}^N}\widetilde{F}_{u}
(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{
\theta _2}}v_n)t_n^{1/\theta_1}u_n\,dx
+\frac{1}{\theta _2}\int_{\mathbb{R}^N}\widetilde{F}_{v}
(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{\theta _2}}v_n)t_n^{1/\theta_2} v_n\,dx.
\end{align*}
Thus
\begin{equation}
\begin{aligned}
&\frac{1}{\theta _1}t_n^{\frac{p(\xi _n)}{\theta _1}}
\{\int_{\mathbb{R}^N}[\widetilde{F}_{u}(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{
\theta _2}}v_n)t_n^{\frac{1-p(\xi _n)}{\theta _1}}u_n-\widetilde{
F}_{u}(u_n,v_n)u_n]\,dx+o(1)\} \\
&+\frac{1}{\theta _2}t_n^{\frac{q(\eta _n)}{\theta _2}}
\{\int_{\mathbb{R}^N}[\widetilde{F}_{v}(t_n^{1/\theta_1}u_n,t_n^{\frac{1}{
\theta _2}}v_n)t_n^{\frac{1-q(\xi _n)}{\theta _2}}v_n-\widetilde{
F}_{v}(u_n,v_n)v_n]\,dx+o(1)\}\\
&=0.
\end{aligned} \label{3.44}
\end{equation}

From (A2), it is easy to see that
\[
\partial_2F(x,\tau ^{1/\theta_1}s,\tau ^{1/\theta_2}
t)s/| \tau | ^{\frac{\theta _1-1}{\theta _1}}\quad\text{and}\quad
\partial _{3}F(x,\tau ^{1/\theta_1}s,\tau ^{1/\theta_2} 
t)t/| \tau | ^{\frac{\theta _2-1}{\theta _2}}
\]
 are increasing about $\tau $ when $\tau >0$; obviously,
$\partial _1\widetilde{F}(\tau ^{1/\theta_1}s,\tau ^{1/\theta_2}
t)s/| \tau | ^{\frac{\theta _1-1}{\theta _1}}$ and
$\partial _2\widetilde{F}(\tau ^{1/\theta_1}s,
\tau ^{1/\theta _2}t)t/| \tau | 
^{\frac{\theta _2-1}{\theta_2}}$ are increasing when $\tau >0$. By (A1) and
\eqref{3.44}, we have

(1) If $t_n\geq 1$, then
\begin{equation}
\begin{aligned}
0&\leq \frac{1}{\theta _1}t_n^{\frac{p(\xi _n)}{\theta _1}}(t_n^{
\frac{\theta _1-p(\xi _n)}{\theta _1}}-1)\int_{\mathbb{R}^N}
 \widetilde{F}_{u}(u_n,v_n)u_n\,dx\\
&\quad +\frac{1}{\theta _2}t_n^{\frac{
q(\eta _n)}{\theta _2}}(t_n^{\frac{\theta _2-q(\xi _n)}{\theta _2
}}-1)\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx\leq o(1);
\end{aligned}  \label{3.45}
\end{equation}

(2) If $t_n<1$, then
\begin{equation}
\begin{aligned}
0&\leq \frac{1}{\theta _1}t_n^{\frac{p(\xi _n)}{\theta _1}}(1-t_n^{
\frac{\theta _1-p(\xi _n)}{\theta _1}})\int_{\mathbb{R}^N}
\widetilde{F}_{u}(u_n,v_n)u_n\,dx\\
&\quad +\frac{1}{\theta _2}t_n^{\frac{q(\eta _n)}{\theta _2}}
(1-t_n^{\frac{\theta _2-q(\xi _n)}{\theta_2}})
\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx\leq o(1).
\end{aligned}  \label{3.46}
\end{equation}
From \eqref{3.47}, \eqref{5.13}, \eqref{3.45} and \eqref{3.46}, it follows
that
\begin{equation}
\lim_{n\to\infty} t_n=1.  \label{3.48}
\end{equation}
Together with \eqref{a.11}, \eqref{3.41} and the definition of 
$(u_n,v_n) $, we have
\begin{equation}
c=J(u_n,v_n)+o(1)=\widehat{J}(u_n,v_n)+o(1).  \label{3.49}
\end{equation}
From the bounded continuity of Nemytskii operator, we can see
\begin{equation}
\widehat{J}(u_n,v_n)=\widehat{J}(t_n^{1/\theta _1} u_n,t_n^{1/\theta_2} v_n)+o(1).  \label{3.50}
\end{equation}

Note that $(t_n^{1/\theta_1}u_n,t_n^{1/\theta_2} v_n)\in \mathcal{N}$. 
It follows from \eqref{3.48}, \eqref{3.49} and \eqref{3.50} that
\begin{equation*}
c=\widehat{J}(t_n^{1/\theta_1}u_n,t_n^{1/\theta_2}
v_n)+o(1)\geq J^{\infty }+o(1)\to J^{\infty }>c.
\end{equation*}
This is a contradiction.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
 From Lemmas \ref{lem3.4} and  \ref{lem3.5}, we know that
there exist a bounded PS sequence $\{(u_n,v_n)\}\subset X$ such that
\begin{equation*}
J(u_n,v_n)\to c>0\text{,\qquad }J'(u_n,v_n)\to 0,
\end{equation*}
where $c$ is the mountain pass level of $J$.
Moreover, from Proposition \ref{prop2.6} we have
\begin{equation}
\begin{gathered}
u_n\to u\quad \text{in }L_{\rm loc}^{\alpha (\cdot )}(\mathbb{R}^N), \quad
p(\cdot )\leq \alpha (\cdot )<<p^{\ast }(\cdot ),\\
v_n\to v\quad \text{in }L_{\rm loc}^{\beta (\cdot )}(\mathbb{R}^N), \quad
q(\cdot )\leq \beta (\cdot )<<q^{\ast }(\cdot ),
\end{gathered}\label{3.32}
\end{equation}
then
\begin{equation}
\text{$u_n\to u$  a.e. in $\mathbb{R}^N$, and 
$v_n\to v$ a.e. in $\mathbb{R}^N$.}  \label{b3}
\end{equation}

Since $c<J^{\infty }$, Lemma \ref{lem3.6} implies that $(u,v)$ is nontrivial. It only
remains to prove that $(u,v)$ is a solution for \eqref{eP}.
Since $J'(u_n,v_n)\to 0$ as $n\to \infty $, for
any $(\varphi ,\psi )\in X$, we have
\begin{align*}
&\int_{\mathbb{R}^N}(| \nabla u_n| ^{p(x)-2}\nabla u_n\nabla \varphi
+| u_n| ^{p(x)-2}u_n\varphi )\,dx \\
&-\int_{\mathbb{R}^N}\{\lambda a(x)| u_n| ^{\gamma
(x)-2}u_n+F_{u}(x,u_n,v_n)\}\varphi \,dx  \to 0\, 
\\
&\int_{\mathbb{R}^N}(| \nabla v_n| ^{q(x)-2}\nabla v_n\nabla \psi
+| v_n| ^{q(x)-2}v_n\psi )\,dx\\
&-\int_{\mathbb{R}^N}\{\lambda b(x)| v_n| ^{\delta
(x)-2}v_n+F_{v}(x,u_n,v_n)\}\psi \,dx \to 0.
\end{align*}
Since $\{\| u_n\| _{p(\cdot )}\}$ and 
$\{\|v_n\| _{q(\cdot )}\}$ are bounded, for any $(\varphi ,\psi )\in X$, 
it is easy to see that the following two groups are uniformly integrable
in $\mathbb{R}^N$,
\begin{gather*}
\{(| u_n| ^{p(x)-1}+| \lambda
a(x)| | u_n| ^{\gamma (x)-1}+|
F_{u}(x,u_n,v_n)| )\cdot | \varphi | \}, \\
\{(| v_n| ^{q(x)-1}+| \lambda
b(x)| | v_n| ^{\delta (x)-1}+|
F_{v}(x,u_n,v_n)| )\cdot | \psi | \}\,.
\end{gather*}
Combining this, \eqref{b3} and Vitali convergent theorem implies
\begin{align}
&\int_{\mathbb{R}^N}| u_n| ^{p(x)-2}u_n\varphi \,dx
-\int_{\mathbb{R}^N}\{\lambda a(x)| u_n| ^{\gamma
(x)-2}u_n+F_{u}(x,u_n,v_n)\}\varphi \,dx  \label{b5} \\
&\to \int_{\mathbb{R}^N}| u| ^{p(x)-2}u\varphi \,dx
-\int_{\mathbb{R}^N}\{\lambda a(x)| u| ^{\gamma(x)-2}u
+F_{u}(x,u,v)\}\varphi \,dx\quad \text{as }n\to \infty , 
\nonumber \\
&\int_{\mathbb{R}^N}| v_n| ^{q(x)-2}v_n\psi \,dx
-\int_{\mathbb{R}^N}\{\lambda b(x)| v_n| ^{\delta
(x)-2}v_n+F_{v}(x,u_n,v_n)\}\psi \,dx  \label{b6} \\
&\to \int_{\mathbb{R}^N}| v| ^{q(x)-2}v\psi \,dx
-\int_{\mathbb{R}^N}\{\lambda b(x)| v| ^{\delta
(x)-2}v+F_{v}(x,u,v)\}\psi \,dx\quad \text{as }n\to \infty . \nonumber 
\end{align}

Thus, to prove that $(u,v)$ is a weak solution of \eqref{eP}, we only need
to prove that for any $(\varphi ,\psi )\in X$ there holds
\begin{equation} \label{3.33}
\begin{gathered}
\int_{\mathbb{R}^N}| \nabla u_n| ^{p(x)-2}\nabla u_n\nabla \varphi
\,dx\to \int_{\mathbb{R}^N}| \nabla u| ^{p(x)-2}\nabla u\nabla \varphi \,dx,
\quad n\to +\infty ,  
\\
\int_{\mathbb{R}^N}| \nabla v_n| ^{q(x)-2}\nabla v_n\nabla \psi
\,dx\to \int_{\mathbb{R}^N}| \nabla v| ^{q(x)-2}\nabla v\nabla \psi \,dx,
\quad n\to +\infty .
\end{gathered}
\end{equation}
Choose $\phi \in C_0^{\infty }(\mathbb{R}^N)$ with $0\leq \phi \leq 1$, we have
\begin{equation}
\int_{\mathbb{R}^N}\phi (| \nabla u_n| ^{p(x)-2}\nabla
u_n-| \nabla w| ^{p(x)-2}\nabla w)(\nabla u_n-\nabla
w)\,dx\geq 0, \forall w\in X_1.  \label{3.34}
\end{equation}
Since $\{(u_n,v_n)\}$ is bounded in $X$ and $J'(u_n,v_n)\to 0$, we have
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}[| \nabla u_n| ^{p(x)-2}\nabla u_n\nabla (\phi
(u_n-w))+| u_n| ^{p(x)-2}u_n\phi (u_n-w)]\,dx \\
&=\int_{\mathbb{R}^N}[\lambda a(x)| u_n| 
^{\gamma(x)-2}u_n+F_{u}(x,u_n,v_n)]\phi (u_n-w)\,dx+o(1), 
\end{aligned} \label{3.35}
\end{equation}
for all $w\in X_1$.
It follows from \eqref{3.34} and \eqref{3.35} that
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R} ^N}\{[\lambda a(x)| u_n| ^{\gamma
(x)-2}u_n+F_{u}(x,u_n,v_n)]-| u_n|
^{p(x)-2}u_n\}\phi (u_n-w)\,dx \\
&-\int_{\mathbb{R} ^N}(u_n-w)| \nabla u_n| ^{p(x)-2}\nabla u_n\nabla
\phi \,dx \\
&-\int_{\mathbb{R} ^N}\phi | \nabla w| ^{p(x)-2}\nabla w(\nabla
u_n-\nabla w)\,dx+o(1)\geq 0,\text{ }\forall w\in X_1.
\end{aligned} \label{b9}
\end{equation}
Note that $(u_n,v_n)$ is bounded in $X$, we may assume
\begin{gather}
(u_n,v_n) \rightharpoonup (u,v)\quad \text{in }X,   \notag \\
\nabla u_n \rightharpoonup \nabla u\quad \text{in }(L^{p(\cdot )}(
\mathbb{R}^N))^N,  \label{b7} \\
\nabla v_n \rightharpoonup \nabla v\quad \text{in }(L^{q(\cdot )}(\mathbb{R}^N))^N 
 \notag \\
| \nabla u_n| ^{p(x)-2}\nabla u_n \rightharpoonup T \quad
\text{in }(L^{p^0(\cdot )}(\mathbb{R}^N))^N,  \label{b8} \\
| \nabla v_n| ^{q(x)-2}\nabla v_n \rightharpoonup S\quad
\text{in }(L^{q^0(\cdot )}(\mathbb{R}^N))^N.  \notag
\end{gather}

Note that $\phi $ has compact support, letting $n\to +\infty $,
according to \eqref{b5}, \eqref{b9}, \eqref{b7} and \eqref{b8}, we obtain
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}\{[\lambda a(x)| u| ^{\gamma
(x)-2}u+F_{u}(x,u,v)]-| u| ^{p(x)-2}u\}\phi (u-w)\,dx\\
&-\int_{\mathbb{R}^N}(u-w)T\nabla \phi \,dx 
-\int_{\mathbb{R} ^N}\phi | \nabla w| ^{p(x)-2}\nabla w(\nabla u-\nabla
w)\,dx\geq 0,
\end{aligned} \label{3.36}
\end{equation}
for all $w\in X_1$.
On the other hand
$| \nabla u_n| ^{p(x)-2}\nabla u_n\rightharpoonup T$
$J'(u_n,v_n)\to 0$,
which implies that
\begin{equation}
\int_{\mathbb{R}^N}(T\nabla \varphi +| u| ^{p(x)-2}u\varphi )\,dx
=\int_{\mathbb{R}^N}[\lambda a(x)| u| ^{\gamma
(x)-2}u+F_{u}(x,u,v)]\varphi \,dx.  \label{3.37}
\end{equation}
Set $\varphi =\phi (u-w)$, it follows from \eqref{3.36} and \eqref{3.37}
that
\begin{equation}
\int_{\mathbb{R}^N}\phi (T-| \nabla w| ^{p(x)-2}\nabla w)(\nabla
u-\nabla w)\,dx\geq 0,\quad \forall w\in X_1.  \label{3.38}
\end{equation}
Set $w=u-\varepsilon \xi $, where $\xi \in X_1$, $\varepsilon >0$. From 
\eqref{3.38} we have
\begin{equation*}
\int_{\mathbb{R}^N}\phi (T-| \nabla u| ^{p(x)-2}\nabla u)\nabla \xi
\,dx\geq 0,\quad \forall \xi \in X_1,
\end{equation*}
then
\begin{equation*}
\int_{\mathbb{R}^N}\phi (T-| \nabla u| ^{p(x)-2}\nabla u)\nabla \xi
\,dx=0,\quad \forall \xi \in X_1,
\end{equation*}
it is easy to see that
\begin{equation*}
\int_{\mathbb{R}^N}(T-| \nabla u| ^{p(x)-2}\nabla u)\nabla \xi
\,dx=0,\forall \xi \in X_1.
\end{equation*}
Thus \eqref{3.33} is valid. Therefore
\begin{align*}
&\int_{\mathbb{R}^N}(| \nabla u| ^{p(x)-2}\nabla u\nabla \varphi
+| u| ^{p(x)-2}u\varphi )\,dx\\
&=\int_{\mathbb{R}^N}[\lambda a(x)| u| ^{\gamma
(x)-2}u+F_{u}(x,u,v)]\varphi \,dx, \forall \varphi \in X_1.
\end{align*}
Similarly, we have
\begin{align*}
&\int_{\mathbb{R}^N}(| \nabla v| ^{q(x)-2}\nabla v\nabla \psi
+| v| ^{q(x)-2}v\psi )\,dx\\
&=\int_{\mathbb{R}^N}[\lambda b(x)| v| ^{\delta (x)-2}v+F_{v}(x,u,v)]\psi
\,dx, \forall \psi \in X_2.
\end{align*}
Thus $(u,v)$ is a solution of \eqref{eP}.
\end{proof}

\subsection{Proof of Theorem \ref{thm3.2}}

Motivated by the property of translation invariant for $p$-Laplacian, we get
a sufficient condition for $c<J^{\infty }$. To prove Theorem \ref{thm3.2},
we need the following Lemma.

\begin{lemma} \label{lem3.7}
 If $F$ satisfies {\rm (A0)--(A2)}, then for any ($u,v)\in
X\backslash \{(0,0)\}$, there exists a unique $t(u,v)>0$ such that
\begin{itemize}
\item[(1)] $\widehat{J}((t(u,v))^{1/\theta_1}u,(t(u,v))^{\frac{1}{\theta _2}} v)
=\underset{s\in [ 0,+\infty )}{\max }\widehat{J}(s^{ \frac{1}{\theta _1}}
u,s^{1/\theta_2} v)$,

\item[(2)] $((t(u,v))^{1/\theta_1}u,(t(u,v))^{1/\theta_2}v)\in \mathcal{N}$,

\item[(3)] The operator $(u,v)\mapsto t(u,v)$ is continuous from $X\backslash
\{(0,0)\}$ to $(0,+\infty )$, and the operator 
$(u,v)\mapsto ((t(u,v))^{ \frac{1}{\theta _1}}u,(t(u,v))^{1/\theta_2} v)$ is a
homeomorphism from the unit sphere in $X$ to $\mathcal{N}$.
\end{itemize}
\end{lemma}

\begin{proof} 
For any $(u,v)\in X\backslash \{(0,0)\}$, define
\begin{equation*}
g(t)=\widehat{J}(t^{1/\theta_1}u,t^{1/\theta_2}v),\quad \forall t\in [ 0,+\infty ).
\end{equation*}

(1) Similar to the proof of \eqref{3.42}, we have $g(t)>0$ as $t>0$ is
small enough, and $g(t)<0$ as $t\to +\infty $. Obviously, $g$ is
continuous, then $g$ attains it's maximum in $(0,+\infty )$.

(2) From (A2), it is not hard to check that 
$(t^{1/\theta_1}u,t^{1/\theta_2} v)\in \mathcal{N}$ if
and only if $tg'(t)=0$; that is,
\begin{align*}
&\int_{\mathbb{R}^N}[| \nabla u| ^{p(x)}\frac{1}{\theta _1}
t^{\frac{p(x)}{\theta _1}}\,dx
+| u| ^{p(x)}\frac{1}{\theta _1}t^{\frac{p(x)}{\theta _1}}]\,dx\\
&+\int_{\mathbb{R}^N}[| \nabla v| ^{q(x)}\frac{1}{\theta _2}t^{\frac{
q(x)}{\theta _2}}\,dx+| v| ^{q(x)}\frac{1}{\theta _2}t^{
\frac{q(x)}{\theta _1}}]\,dx \\
&= \int_{\Omega }\widetilde{F}_1(t^{1/\theta_1}u,t^{1/\theta _2} 
v)\frac{1}{\theta _1}t^{1/\theta _1} u\,dx
+\int_{\Omega }\widetilde{F}_2(t^{1/\theta_1}u,t^{1/\theta _2} v)
\frac{1}{\theta _2}t^{1/\theta_2} v\,dx,
\end{align*}
which can be rearranged as
\begin{align*}
&\int_{\mathbb{R}^N}\frac{1}{\theta _1}[| \nabla u| ^{p(x)}t^{\frac{
p(x)}{\theta _1}-1}\,dx+| u| ^{p(x)}t^{\frac{p(x)}{\theta
_1}-1}]\,dx\\
&+\int_{\mathbb{R} ^N}\frac{1}{\theta _2}[| \nabla v| ^{q(x)}
t^{\frac{q(x)}{\theta _2}-1}\,dx+| v| ^{q(x)}t^{\frac{q(x)}{\theta
_1}-1}]\,dx \\
&= \int_{\Omega }\frac{1}{\theta _1}\frac{\widetilde{F}_1
(t^{1/\theta _1}u,t^{1/\theta_2} v)}{t^{\frac{\theta _1-1}{\theta _1}}}u\,dx
+\int_{\Omega }\frac{1}{\theta _2}\frac{\widetilde{F}_2
(t^{1/\theta _1}u,t^{1/\theta_2} v)}{t^{\frac{\theta _2-1}{\theta _2}}}v\,dx.
\end{align*}

It follows from \eqref{b1} and \eqref{b2} that the left hand is strictly
decreasing with respect to $t$, while the right hand is increasing. 
Thus $g'(t)=0$ has a unique solution $t(u,v)$ such that 
$((t(u,v))^{1/\theta _1}u,(t(u,v))^{1/\theta_2} v)\in \mathcal{N}$.

We claim that $g(t)$ is increasing on $[0,t(u,v)]$, and  decreasing
on $[t(u,v),+\infty )$.
Denote $(u_{\ast },v_{\ast })=((t(u,v))^{1/\theta_1}u,(t(u,v))
^{1/\theta _2}v)$. 
Define $\rho (t)=\widehat{J}(t^{1/\theta_1}u_{\ast },t^{1/\theta_2} v_{\ast })$.
 We only need to prove
that $\rho (t)$ is increasing on $[0,1]$, and $\rho (t)$ is decreasing on 
$[1,+\infty )$. From (1), it is easy to see that there exists $t_{\#}>0$
such that
\begin{equation*}
\rho (t_{\#})=\max_{t\geq 0}\widehat{J}(t^{1/\theta _1}u_{\ast },
t^{1/\theta_2} v_{\ast }),
\end{equation*}
therefore $\rho '(t_{\#})=0$.

Suppose $t>1$. By (A2), we have
\begin{align*}
&\rho '(t) \\
&=\int_{\mathbb{R}^N}\frac{1}{\theta _1}t^{\frac{p(x)}{\theta _1}-1}
(| \nabla u_{\ast }| ^{p(x)}+| u_{\ast }|^{p(x)})\,dx
+\int_{\mathbb{R}^N}\frac{1}{\theta _2}t^{\frac{q(x)}{\theta _2}-1}(| \nabla
v_{\ast }| ^{q(x)}+| v_{\ast }| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}^N}\widetilde{F}_1(t^{1/\theta_1}u_{\ast },
t^{1/\theta_2} v_{\ast })\frac{1}{\theta _1}t^{\frac{1}{\theta _1}-1}u_{\ast
}\,dx
-\int_{\mathbb{R}^N}\widetilde{F}_2(t^{1/\theta_1}u_{\ast },
t^{1/\theta_2} v_{\ast })\frac{1}{\theta _2}t^{\frac{1}{\theta_2}-1}v_{\ast }\,dx
\\
&<\int_{\mathbb{R}^N}\frac{1}{\theta _1}(| \nabla u_{\ast }|
^{p(x)}+| u_{\ast }| ^{p(x)})\,dx
+\int_{\mathbb{R}^N}\frac{1}{\theta _2}(| \nabla v_{\ast }|
^{q(x)}+| v_{\ast }| ^{q(x)})\,dx \\
&\quad -\int_{\mathbb{R}^N}\widetilde{F}_1(u_{\ast },v_{\ast })
\frac{1}{\theta _1}u_{\ast}\,dx
-\int_{\mathbb{R} ^N}\widetilde{F}_2(u_{\ast },v_{\ast })
\frac{1}{\theta _2}v_{\ast }\,dx
\\
&=\widehat{J}'(u_{\ast },v_{\ast })(\frac{1}{\theta _1}u_{\ast },
\frac{1}{\theta _2}v_{\ast })=0.
\end{align*}
Thus $\rho (t)$ is strictly decreasing when $t>1$.

Suppose $t<1$. Similarly, we have
\begin{equation*}
\rho '(t)>\widehat{J}'(u_{\ast },v_{\ast })(\frac{1}{
\theta _1}u_{\ast },\frac{1}{\theta _2}v_{\ast })=0.
\end{equation*}
Thus $\rho (t)$ is strictly increasing when $t<1$.
Therefore $g(t)$ is increasing on $[0,t(u,v)]$ and decreasing on $
[t(u,v),+\infty )$.

(3) We only need to proof that $t(\cdot ,\cdot )$ is continuously.
Let $(u_{m},v_{m})\to (u,v)$ in $X$, then 
$\widehat{J}(t^{1/\theta _1}u_{m},t^{1/\theta_2} 
v_{m})\to \widehat{J}(t^{\frac{1}{\theta _1}}u,t^{1/\theta_2} v)$.
 We choose a constant $t_0$ large enough such that 
$\widehat{J}(t_0^{\frac{1}{\theta _1}}u,t_0^{1/\theta_2} v)<0$, 
then there exists a $M>0$ such that 
\[
\widehat{J}(t_0^{1/\theta_1}u_{m},t_0^{1/\theta_2}v_{m})<0
\]
for any $m>M$. Therefore $t(u_{m},v_{m})<t_0$ when $m>M$, then
$\{t(u_{m},v_{m})\}$ has a convergent subsequence 
$\{t(u_{m_{j}},v_{m_{j}})\}$ satisfying $t(u_{m_{j}},v_{m_{j}})\to t_{\ast }$. Thus
\begin{equation*}
\widehat{J}((t(u_{m_{j}},v_{m_{j}}))^{1/\theta _1}
u_{m_{j}},(t(u_{m_{j}},v_{m_{j}}))^{1/\theta_2}
v_{m_{j}})\to \widehat{J}(t_{\ast }^{1/\theta_1} u,t_{\ast }^{1/\theta_2} v).
\end{equation*}

From (1) we know that
\begin{align*}
&\widehat{J}((t(u_{m_{j}},v_{m_{j}}))^{\frac{1}{\theta _1}
}u_{m_{j}},(t(u_{m_{j}},v_{m_{j}}))^{1/\theta_2} v_{m_{j}})\\
&\geq \widehat{J}((t(u,v))^{1/\theta_1}u_{m_{j}},(t(u,v))
^{1/\theta _2}v_{m_{j}}),
\end{align*}
and hence letting $j\to \infty $, we obtain
\begin{equation*}
\widehat{J}(t_{\ast }^{1/\theta_1}u,t_{\ast }^{1/\theta_2} v)
\geq \widehat{J}((t(u,v))^{1/\theta_1}u,(t(u,v))^{1/\theta _2}v).
\end{equation*}
From (1), we have $t_{\ast }=t(u,v)$. Thus $t(u,v)$ is
continuous.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.2}]
 Let $\{ (u_n,v_n)\} \subset \mathcal{N}$ be a minimizing sequences 
of $\widehat{J}$, that is
\begin{equation*}
\lim_{n\to+\infty}\widehat{J}(u_n,v_n)=J^{\infty }>0.
\end{equation*}
Similar to the proof of Lemma \ref{lem3.5}, we can see $\{(u_n,v_n)\}$ is bounded
in $X$. Thus there exists a positive constant $\kappa >1$ such that
\begin{equation}
\int_{\mathbb{R}^N}(| u_n| ^{p(x)}+| v_n|
^{q(x)})\,dx\leq \kappa ,n=1,2,\dots .  \label{5.3}
\end{equation}
We claim that for any fixed $\delta >0$ and$p(\cdot )<<\alpha
(\cdot )<<p^{\ast }(\cdot )$, 
$q(\cdot )<<\beta (\cdot )<<q^{\ast }(\cdot )$, there exist a 
$\varepsilon _{o}>0$ such that
\begin{equation}
\sup_{y\in \mathbb{R}^N} \int_{B(y,\delta )}| u_n| ^{\alpha (x)}\,dx+
\sup_{y\in\mathbb{R}^N} \int_{B(y,\delta )}| v_n| ^{\beta
(x)}\,dx\geq 2\varepsilon _{o},\quad n=1,2,\dots .  \label{3.52}
\end{equation}
Indeed, suppose otherwise. Then it follows from Proposition \ref{prop2.7} that
\begin{gather}
u_n \to 0\quad \text{in }L^{\alpha (\cdot )}(\mathbb{R}^N),\;
 \forall p(\cdot )<<\alpha (\cdot )<<p^{\ast }(\cdot ), \label{5.1} \\
v_n \to 0\quad \text{in }L^{\beta (\cdot )}(\mathbb{R}^N),\;
 \forall q(\cdot )<<\beta (\cdot )<<q^{\ast }(\cdot ).
\label{5.2}
\end{gather}
We claim that
\begin{equation}
\int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx\to 0,\quad 
n\to +\infty.  \label{b4}
\end{equation}

For any $\varepsilon >0$, (A0)--(A1) imply
\begin{equation*}
| \widetilde{F}_{u}(u_n,v_n)u_n| \leq \frac{
\varepsilon }{2\kappa }(| u_n| ^{p(x)}+|
v_n| ^{q(x)})+C(\varepsilon )(| u_n|
^{\alpha (x)}+| v_n| ^{\beta (x)}),
\end{equation*}
and
\begin{equation}
\begin{aligned}
&\big| \int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx\big| \\
&\leq \frac{\varepsilon}{2\kappa }\int_{\mathbb{R}^N}(| u_n| ^{p(x)}+| v_n|
^{q(x)})\,dx+C(\varepsilon )
\int_{\mathbb{R}^N}(| u_n| ^{\alpha (x)}+| v_n| ^{\beta (x)})\,dx.
\end{aligned}  \label{5.4}
\end{equation}
Combining \eqref{5.1} and \eqref{5.2}, there exist $N_0>0$ such that
\begin{equation}
C(\varepsilon )\int_{\mathbb{R}^N}(| u_n| ^{\alpha (x)}+| v_n|
^{\beta (x)})\,dx\leq \frac{\varepsilon }{2},\quad n\geq N_0.  \label{5.5}
\end{equation}
From \eqref{5.3}, \eqref{5.4} and \eqref{5.5}, we have
\begin{equation*}
\big| \int_{\mathbb{R}^N}\widetilde{F}_{u}(u_n,v_n)u_n\,dx\big|
 \leq \varepsilon,\forall n\geq N_0.
\end{equation*}
Thus \eqref{b4} is valid.
Similarly, we can get
\begin{equation}
\int_{\mathbb{R}^N}\widetilde{F}_{v}(u_n,v_n)v_n\,dx\to 0,\quad
n\to +\infty.  \label{b10}
\end{equation}
Note that $(u_n,v_n)\in \mathcal{N}$. It follows from \eqref{b4}, 
\eqref{b10} and $\| (u_n,v_n)\| \to 0$ that
\begin{equation*}
\widehat{J}(u_n,v_n)\to 0.
\end{equation*}
This is a contradiction to $\lim_{n\to+\infty}\widehat{J}
(u_n,v_n)=J^{\infty }>0$. Thus \eqref{3.52} is valid.

From \eqref{3.52}, without loss of generality, we  assume that
\begin{equation*}
\sup_{y\in\mathbb{R}^N} \int_{B(y,\delta )}| u_n| ^{\alpha
(x)}\,dx\geq \varepsilon _{o},\quad n=1,2,\dots
\end{equation*}
We may assume that
\begin{gather*}
\int_{B(y_n,\delta )}| u_n| ^{\alpha (x)}\,dx 
\geq \frac{1}{2}\sup_{y\in\mathbb{R}^N} 
\int_{B(y,\delta )}| u_n| ^{\alpha (x)}\,dx,
\\
\int_{B(\eta _n,\delta )}| v_n| ^{\beta (x)}\,dx \geq 
\frac{1}{2}\sup_{y\in\mathbb{R}^N} 
\int_{B(y,\delta )}| v_n| ^{\beta (x)}\,dx.
\end{gather*}
From  $p(\cdot )$ begin periodic, for any $y_n$,$\eta _n$, there exist
 $x_n,\xi _n\in Q(x_{o},A)$ such that
\begin{gather*}
p(x) = p(y_n-x_n+x), \forall x\in \mathbb{R}^N,  \\
q(x) = q(\eta _n-\xi _n+x), \forall x\in \mathbb{R}^N.
\end{gather*}
Now, we  consider $J(t^{1/\theta_1}u_n(y_n-x_n+x),
t^{1/\theta _2}v_n(\eta _n-\xi _n+x))$.
Denote $J_1(u,v)=\Phi (u,v)-\int_{\mathbb{R}^N}F(x,u,v)\,dx$. 
It follows from (A2) that 
$F(x,t^{\frac{1}{\theta _1}}u,t^{1/\theta_2} v)/t$ is increasing with respect
to $t$. 
Suppose $t\in (0,1)$, it follows from \eqref{b2} that
\begin{equation}
\begin{aligned}
&J_1(t^{1/\theta_1} u_n(y_n-x_n+x),t^{1/\theta _2} v_n(\eta _n-\xi _n+x))  \\
&=\Phi (t^{1/\theta_1} u_n,t^{1/\theta_2} v_n)
-\int_{\mathbb{R}^N}F(x,t^{1/\theta_1}u_n(y_n-x_n+x),t^{1/\theta_2} v_n(\eta _n
-\xi _n+x))\,dx   \\
&\geq t^{\max\{\frac{p^{+}}{\theta _1},\frac{q^{+}}{\theta _2}\}}
\Phi (u_n,v_n)
-t\int_{\mathbb{R}^N}F(x,u_n(y_n-x_n+x),v_n(\eta _n-\xi _n+x))\,dx  \\
&= t\{t^{\max \{\frac{p^{+}}{\theta _1},\frac{q^{+}}{\theta _2}\}-1}\Phi
(u_n,v_n)-\int_{\mathbb{R}^N}F(x,u_n(y_n-x_n+x),v_n(\eta _n-\xi _n+x))\,dx\}.
\end{aligned} \label{10.11}
\end{equation}
From \eqref{3.52} and the boundedness of $\{(u_n,v_n)\}$, we can see
that there exists positive constants $C_1,C_2$ such that
\begin{equation}
C_1\leq \| (u_n,v_n)\| \leq C_2.  \label{a*}
\end{equation}

Since $\theta _1>p^{+}$ and $\theta _2>q^{+}$, there exists a fixed 
$ t_{\ast }\in (0,1)$ such that, for any $n=1,2,\dots $, we have
\begin{equation}
\begin{aligned}
&t_{\ast }^{\max \{\frac{p^{+}}{\theta _1},\frac{q^{+}}{\theta _2}
\}-1}\Phi (u_n,v_n)
-\int_{\mathbb{R}^N}F(x,u_n(y_n-x_n+x),v_n(\eta _n-\xi _n+x))\,dx\\
&\geq \frac{1}{2} \Phi (u_n,v_n)\geq C_{3}>0. 
\end{aligned} \label{10.12}
\end{equation}
From \eqref{10.11} and \eqref{10.12}, we obtain
\begin{equation*}
J_1(t_{\ast }^{1/\theta_1}u_n(y_n-x_n+x),t_{\ast }
^{1/\theta _2}v_n(\eta _n-\xi _n+x))\geq t_{\ast
}C_{3}>0,\quad n=1,2,\dots .
\end{equation*}
Suppose $\lambda $ is small enough. From the above inequality, we have
\begin{equation}
J(t_{\ast }^{1/\theta_1}u_n(y_n-x_n+x),t_{\ast }^{\frac{1}{
\theta _2}}v_n(\eta _n-\xi _n+x))\geq \frac{1}{2}t_{\ast
}C_{3}>0,\quad n=1,2,\dots .  \label{10.13}
\end{equation}
Obviously, $J(0,0)=0$ and 
$J(t^{1/\theta _1}u_n(y_n-x_n+x),t^{1/\theta_2} v_n(\eta _n-\xi
_n+x))\to -\infty $ as $t\to +\infty $. Thus there exist
 $t_n\in (0,+\infty )$ such that
\begin{equation}
\begin{aligned}
&J(t_n^{1/\theta_1}u_n(y_n-x_n+x),t_n^{1/\theta_2} v_n(\eta _n-\xi _n+x))\\
&=\max_{t\geq 0} J(t^{\frac{1}{ \theta _1}}u_n(y_n-x_n+x),t^{1/\theta_2} v_n(\eta
_n-\xi _n+x))>0.
\end{aligned}  \label{3.53}
\end{equation}
It follows from \eqref{10.13} and the boundedness of $\{(u_n,v_n)\}$
that there exist a positive constant $\epsilon$ such that
\begin{equation}
t_n\geq \epsilon , \quad n=1,2,\dots . \label{3.54b}
\end{equation}
Denote 
\[
g(t)=\widehat{J}(t^{1/\theta_1}u_n(y_n-x_n+x),t^{
\frac{1}{\theta _2}}v_n(\eta _n-\xi _n+x)).
\]
Since $\{ (u_n,v_n)\} \subset \mathcal{N}$, Lemma \ref{lem3.7} implies
\begin{equation}
\begin{aligned}
&\widehat{J}(u_n(y_n-x_n+x),v_n(\eta _n-\xi _n+x))\\
&=\max_{t\geq 0} \widehat{J}(t^{1/\theta_1}u_n(y_n-x_n+x),
t^{1/\theta _2}v_n(\eta _n-\xi _n+x)).
\end{aligned}  \label{3.55}
\end{equation}
Suppose $\lambda $ is small enough. From \eqref{3.51}, \eqref{3.53}, 
\eqref{3.54b} and \eqref{3.55}, we have
\begin{align*}
&\max_{t\geq 0} J(t^{1/\theta _1}u_n(y_n-x_n+x),t^{1/\theta_2} 
v_n(\eta _n-\xi _n+x))
\\
&= J(t_n^{1/\theta_1}u_n(y_n-x_n+x),
t_n^{1/\theta _2}v_n(\eta _n-\xi _n+x)) \\
&\leq \widehat{J}(t_n^{1/\theta_1}u_n(y_n-x_n+x),t_n^{1/\theta _2}
v_n(\eta _n-\xi _n+x))  \\
&\quad+\int_{\mathbb{R}^N}\lambda [ \frac{| a(x)| }{\gamma (x)}
| t_n^{1/\theta_1}u_n(y_n-x_n+x)|
^{\gamma (x)}+\frac{| b(x)| }{\delta (x)}|
t_n^{1/\theta_2} v_n(\eta _n-\xi _n+x)| ^{\delta (x)}]\,dx \\
&\quad -\frac{\tau }{2\alpha _{o}^{+}}\int_{B(x_n,\delta )}| t_n^{
\frac{1}{\theta _1}}u_n(y_n-x_n+x)| ^{\alpha _{o}(x)}\,dx \\
&\quad - \frac{\tau }{2\beta _{o}^{+}}\int_{B(\xi _n,\delta )}| t_n^{
\frac{1}{\theta _2}}v_n(\eta _n-\xi _n+x)| ^{\beta_{o}(x)}\,dx \\
&\leq \widehat{J}(t_n^{1/\theta_1}u_n,t_n^{\frac{1
}{\theta _2}}v_n)-\frac{\tau \varepsilon _0\epsilon ^{\max \{\alpha
_{o}^{+},\beta _{o}^{+}\}}}{2\max \{\alpha _{o}^{+},\beta _{o}^{+}\}} \\
&\quad+\int_{\mathbb{R}^N}\lambda [ \frac{| a(x)| }{\gamma (x)}
| t_n^{1/\theta_1}u_n(y_n-x_n+x)|
^{\gamma (x)}+\frac{| b(x)| }{\delta (x)}|
t_n^{1/\theta_2} v_n(\eta _n-\xi _n+x)| ^{\delta(x)}]\,dx \\
&\leq \widehat{J}(u_n,v_n)-\frac{\tau \varepsilon _0\epsilon ^{\max
\{\alpha _{o}^{+},\beta _{o}^{+}\}}}{4\max \{\alpha _{o}^{+},\beta _{o}^{+}\}
}<J^{\infty }\quad \text{where }\zeta \in Q(x_{o},A)\,.
\end{align*}
This completes the proof.
\end{proof}

\subsection{Proof of Theorem \ref{thm3.3}}

 According to the \cite[Theorems 2.2 and 3.2]{19},  $u$ and $v$ are locally bounded.
From \cite[Theorem 1.2]{12},  $u$ and $v$ are locally $C^{1,\alpha }$ continuous. 
Similar to the proof of \cite[Proposition 2.5]{15}, we  obtain 
that $u,v\in C^{1,\alpha }(\mathbb{R}^N)$, 
$u(x)\to 0$, 
$|\nabla u(x)|\to 0$, $v(x)\to 0 $ and $|\nabla v(x)|\to 0$ as 
$|x|\to \infty $.

\textbf{Note 1.} Let us consider the existence of solutions for the 
system
\begin{gather*}
\begin{aligned}
&-\operatorname{div}| \nabla u_i| ^{p_i(x)-2}\nabla
u_i)+| u_i| ^{p_i(x)-2}u_i\\
&=\lambda a_i(x)| u_i| ^{\gamma _i(x)-2}u_i+F_{u_i}(x,u_1,\dots ,u_n)
\quad \text{in }\mathbb{R}^N,
\end{aligned} \\
u_i\in W^{1,p_i(\cdot )}(\mathbb{R}^N),
\end{gather*}
$i=1,\dots ,n$, where $u=(u_1,\dots ,u_n)$, suppose $\lambda $ is
small enough, then the system has a nontrivial solution if it satisfies the
following assumptions:
\begin{itemize}
\item[(H0)] $p_i(\cdot )$ are Lipschitz continuous, 
$1<<p_i(\cdot )<<N$, $1<<\gamma _i(\cdot )<<p_i(\cdot )$, 
$a_i(\cdot )\in L^{\frac{p_i(\cdot )}{p_i(\cdot )
-\gamma _i(\cdot )}}(\mathbb{R}^N)$, 
$F\in C^1(\mathbb{R}^N\times\mathbb{R}^{n},\mathbb{R})$ and satisfies
\begin{align*}
&|F_{u_i}(x,u_1,\dots ,u_n)|\\
&\leq C(|u_i|^{p_i(x)-1}+|u_i|^{\alpha _i(x)-1}
+\sum_{1\leq j\leq n,\, j\neq i} 
[|u_{j}|^{p_{j}(x)/\alpha _i^0(x)}+|u_{j}|^{\alpha
_{j}(x)/\alpha _i^0(x)}]),
\end{align*}
where $F_{u_i}=\frac{\partial }{\partial u_i}F$, 
$\alpha _i\in C(\mathbb{R}^N)$, and
$p_i(\cdot )\leq \alpha _i(\cdot )<<p_i^{\ast }(\cdot )$, 
where
\begin{equation*}
p_i^{\ast }(x)=\begin{cases}
Np_i(x)/(N-p_i(x)) , &p_i(x)<N, \\
\infty , &p(x)\geq N,
\end{cases}
\end{equation*}

\item[(H1)] $F\in C^1(\mathbb{R}^N\times\mathbb{R}^{n})$ and satisfies 
the following conditions
\begin{gather*}
0\leq s_iF_{s_i}(x,s_1,\dots ,s_n),\quad \forall (x,s_1,\dots,s_n)\in
\mathbb{R}^N\times\mathbb{R}^{n},\;i=1,\dots ,n, \\
0<F(x,s_1,\dots ,s_n)\leq \underset{1\leq i\leq n}{\sum }
\frac{1}{\theta _i}s_iF_{s_i}(x,s_1,\dots ,s_n),\quad 
\forall (x,s_1,\dots,s_n)\in\mathbb{R}^N\times\mathbb{R}^{n};
\end{gather*}

\item[(H2)] For any $(s,t)\in (\mathbb{R}\times\mathbb{R})$, 
$F_{s_i}(x,\tau ^{1/\theta_1}s_1,\dots ,
\tau ^{1/\theta _n}s_n)/\tau ^{\frac{\theta _i-1}{\theta _i}}$
 ($i=1,\dots ,n$) are increasing respect to $\tau >0$;

\item[(H3)] There is a measurable function 
$\widetilde{F}(s_1,\dots ,s_n)$ such that
\begin{equation*}
\lim_{| x| \to +\infty } F(x,s_1,\dots ,s_n)=\widetilde{F}(s_1,\dots ,s_n)
\end{equation*}
for bounded $\sum_{1\leq i\leq n} | s_i| $ uniformly,
\[
| \widetilde{F}(s_1,\dots ,s_n)| +|
\sum_{1\leq i\leq n} s_i\widetilde{F}_{s_i}(s_1,\dots ,s_n)| 
\leq C\sum_{1\leq i\leq n} (|s_i| ^{p_i^{+}}+| s_i| ^{\alpha
_i^{-}}),
\]
for all $(s_1,\dots ,s_n)\in\mathbb{R}^{n}$,
and
\begin{gather*}
| F(x,s_1,\dots ,s_n)-\widetilde{F}(s_1,\dots ,s_n)| 
\leq
\varepsilon (R)\underset{1\leq i\leq n}{\sum }(| s_i|
^{p_i(x)}+| s_i| ^{p_i^{\ast }(x)})\quad \text{when }| x| \geq R,
\\
\begin{aligned}
&| F_{s_i}(x,s_1,\dots ,s_n)-\widetilde{F}_{s_i}(s_1,\dots,s_n)| \\
&\leq \varepsilon (R)\{| s_i|^{p_i(x)-1}+| s_i| ^{p_i^{\ast }(x)-1} \\
&\quad +\underset{1\leq j\leq n,j\neq i}{\sum }[| s_{j}|
^{p_{j}(x)(p_i^{\ast }(x)-1)/p_i^{\ast }(x)}+| s_{j}|
^{p_{j}^{\ast }(x)(p_i^{\ast }(x)-1)/p_i^{\ast }(x)}]\}\quad
\text{when }| x| \geq R,
\end{aligned}
\end{gather*}
where $\varepsilon (R)$ satisfies
$\lim_{R\to +\infty } \varepsilon (R)=0$.


\item[(H4)] 
\begin{gather*}
F(x,s_1,\dots ,s_n)\geq \widetilde{F}(s_1,\dots ,s_n),\quad 
 \forall (x,s_1,\dots ,s_n)\in \mathbb{R}^N\times (\mathbb{R}^{+})^{n}, \\
\begin{aligned}
&F(x,s_1,\dots ,s_n)\\
&\geq \widetilde{F}(s_1,\dots ,s_n)+\underset{
1\leq i\leq n}{\sum }\tau s_i^{\alpha _i^{\#}(x)-1},\quad
\forall (x,s_1,\dots ,s_n)\in B(Q(x_{o},A),\delta )\times (\mathbb{R}^{+})^{n},
\end{aligned}
\end{gather*}
where $p_i(\cdot )<<\alpha _i^{\#}(\cdot )<<p_i^{\ast }(\cdot )$.
\end{itemize}

\subsection*{Acknowledgments} 
The authors would like to thank Professor Julio G. Dix for
his suggestions, and the anonymous referees for their
valuable comments.

This research was partly supported by the National
Natural Science Foundation of China (11326161, 10971087),
and by the key projects of Science and Technology Research of
the Henan Education Department (14A110011).

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\end{document}