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\begin{document}

\title[Nonlocal fourth-order Kirchhoff systems]{Nonlocal fourth-order Kirchhoff systems with variable growth: low and high energy solutions}

\centerline{}
%--------------------------------------------------------------------------------------------------------------
\author[G.A. Afrouzi]{Ghasem A. Afrouzi}
\address{Ghasem A. Afrouzi,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto:
G.A. Afrouzi <afrouzi@umz.ac.ir>}{afrouzi@umz.ac.ir}}
%-----------------------------------------------------------------------------------------------------------
\author[M. Mirzapour]{M. Mirzapour}
\address{M. Mirzapour,
Farhangian university, Tehran, Iran} \email{\href{mailto: M.
Mirzapour <mirzapour@stu.umz.ac.ir>}{mirzapour@stu.umz.ac.ir}}
%--------------
\author[V.D. R\u adulescu]{Vicen\c tiu D. R\u adulescu$^\ast$}
\address{Vicen\c tiu D. R\u adulescu,
Institute of Mathematics ``Simion Stoilow" of the Romanian
Academy, P.O. Box 1-764, 014700 Bucharest, Romania \& Department
of Mathematics, University of Craiova,  Street A.I. Cuza No. 13,
200585 Craiova, Romania} \email{\href{mailto:
V.D. R\u adulescu <vicentiu.radulescu@imar.ro>}{vicentiu.radulescu@imar.ro}}
%--------------------------------------------------------------------------------------------------------------


\keywords{Variational  method; Nonlinear elliptic systems; $(p(x),q(x))$-Kirchhoff system; Nonlocal condition. \\
$^\ast$ Corresponding author. \\
\hspace*{.3cm} {\it 2000 Mathematics Subject Classifications}: 35J60, 35B30, 35B40, 58E05.}

%\date{\bf 13 November 2013}

\begin{abstract}

This paper is concerned with the existence and multiplicity of solutions for a class of nonlocal fourth-order \((p(x),q(x))\)-Kirchhoff systems. By means of a  variational analysis, we obtain conditions for the existence of infinitely many solutions with high (resp., low) energies. The arguments combine related critical point theory arguments with a careful analysis of the energy levels.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
%--------------------------------------------------------------------------------------------------------------


\section{Introduction}
In this paper, we study the following nonlocal elliptic system
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
\di -M_{1}\Big (\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}dx\Big)\Delta(|\Delta u|^{p(x)-2}\Delta u) = F_u (x,u,v)
 &\text{ in }\Omega,\\
\di -M_{2}\Big (\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}dx\Big)\Delta(|\Delta v|^{q(x)-2}\Delta v) = F_v (x,u,v)
 &\text{ in }\Omega,\\
u=v=\Delta u=\Delta v=0 & \text{ on } \partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega$ is a bounded domain in $\R^N$ with smooth boundary $\partial\Omega$, $p,\,q\in C(\overline\Omega)$, $p,\,q>1$ in $\overline\Omega$, and $M_1$, $ M_2$ are continuous functions. The  function $F$ satisfies Carath\'{e}odory conditions and is of class $C^{1}$ in $u,v\in \R$. The functions $F_u$, $F_v$ represent {\it source forces}, while $M_1$, $M_2$ are {\it Kirchhoff dissipative terms}.

For simplicity reasons, in the present paper we reduce to the case where $M_1= M_2=:M$. Notice that the results we establish in what follows remain valid for $M_1\neq M_2$ by adding some slight changes.

Boundary value problems like \eqref{e1.1} model several physical and biological systems where $u$ and $v$ describe a process depending on the average of itself, as for example, population densities. We refer the reader, for instance, to Alves and Figueiredo \cite{alves}, Autuori and Pucci \cite{AP1,AP2,AP3}, Autuori, Pucci and Salvatori \cite{APS}, Molica Bisci and R\u adulescu \cite{molica}, and Vasconcellos \cite{vasc}.

Problem \eqref{e1.1} is called a nonlocal problem because of the presence of the term $M$, which implies that the equation in \eqref{e1.1} is no longer a pointwise identity. This provokes some mathematical difficulties which make the study of such a problem particularly interesting. Nonlocal differential equations are also called Kirchhoff-type equations because Kirchhoff \cite{K} has investigated an equation of the form
\begin{align}\label{e1.2}
\rho\frac{\partial^{2}u}{\partial t^{2}}-\left(\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L \Big{|}\frac{\partial u}{\partial x}\Big{|}^{2}\,dx\right)\frac{\partial^{2}u}{\partial x^{2}}=0,
\end{align}
where $E$ is the Young modulus of the material, $\rho$ is the mass density, $L$ is the length of the string, $h$ is the area of the cross-section, and $\rho_0$ is the initial tension.
Eq. \eqref{e1.2} extends the classical D'Alembert's wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinguished feature of Eq. \eqref{e1.2} is that the equation contains a nonlocal coefficient $\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L \Big{|}\frac{\partial u}{\partial x}\Big{|}^{2}\,dx$, which depends on the average $\frac{1}{2L} \int_0^L \Big{|}\frac{\partial u}{\partial x}\Big{|}^{2}\,dx$, and hence the equation is no longer a pointwise identity.

We point out that the Kirchhoff model takes
into account the length changes of the string produced by transverse vibrations. We refer to Bernstein \cite{Ber} and Pohozaev \cite{Poh} as pioneering papers dedicated to Kirchhoff equations.
However, equation \eqref{e1.2} received much attention only after the paper by Lions \cite{Lions}, where an abstract
framework to the problem was proposed. D'Ancona and Spagnolo \cite{DS} considered Kirchhoff's equation as a quasi-linear hyperbolic Cauchy problem that describes the transverse oscillations of a stretched string.
For completeness we refer the reader to some recent interesting results obtained by Autuori and Pucci in \cite{AP1,AP2,AP3}
studying Kirchhoff equations by using different approaches. We also  recall that nonhomogeneous $p(x)$-Kirchhoff operators
have been used in the last decades to model various phenomena, see \cite{Harju,Zhikov}
and the references therein. Indeed, recently, there has been an increasing interest in
studying systems involving somehow nonhomogeneous
$p(x)$-Laplace operators,
motivated by the image restoration problem, by the modeling of {\it electro-rheological fluids} (sometimes referred to as
{\it smart fluids}), as well as the {\it thermo-convective
flows} of non-Newtonian fluids: details and further references can be found in \cite{Anton}.

The investigation of existence and multiplicity of solutions for problems with $p(x)$-biharmonic operators has drawn the attentions of many authors, see \cite{Amr,El,Kong,Lin1} for some recent work on this subject. Motivated by the above references, we establish the existence of infinitely many  low or high energy solutions for system \eqref{e1.1}.

Our paper is organized as follows. We first present some necessary preliminary results on variable exponent Sobolev spaces. Next, we give the main results about the existence of weak solutions. The final part of this paper is concerned with the existence of infinitely many low or high energy solutions.

\section{Functional setting}
In this section, we recall some definitions and basic properties of the variable exponent Lebesgue and Sobolev spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$, where $\Omega$ is a bounded domain in $\R^{N}$. Denote
$$
C_+(\overline{\Omega})=\{h(x);~h(x)\in C(\overline{\Omega}),~h(x)>1,\hspace{0.1cm}\textrm{ for all }\hspace{0.1cm} x\in \overline{\Omega}\}.
$$
For any $h\in C_+(\overline{\Omega})$, we define
$$
h^+=\max\{h(x);~x \in\overline{\Omega}\},\hspace{0.3cm}h^-=\min\{h(x);~x \in\overline{\Omega}\}.
$$
For any $p\in C_+(\overline{\Omega})$, we define the $\emph{variable exponent Lebesgue space}$
$$
L^{p(x)}(\Omega)=\Big\{u:\Omega\rightarrow\R;~ u\textrm{ is a measurable real-valued function and } \int_{\Omega}|u(x)|^{p(x)}dx<\infty\Big\},
$$
endowed with the  $\emph{Luxemburg norm}$
$$
|u|_{p(x )}=|u|_{L^{p(x)}(\Omega)}=\inf \Big\{\mu>0;~\int_{\Omega}|\frac{u(x)}{\mu}|^{p(x)}dx\leq 1\Big\}.
$$
Then $(L^{p(x)}(\Omega),|\cdot|_{p(x)})$ is a Banach space, cf. \cite{Kov}.

\begin{proposition} [Fan and Zhao \cite{Fan2}]\label{prop2.1}
The space $(L^{p(x)}(\Omega),|\cdot|_{p(x)})$ is separable, uniformly convex, reflexive and its conjugate space is $L^{q(x)}(\Omega)$ where $q(x)$ is the conjugate function of $p(x)$, that is,
$$
\frac{1}{p(x)}+\frac{1}{q(x)}=1\quad\mbox{for all $x\in \Omega$.}
$$
 For $u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$, we have
$$
\Big{|}\int_{\Omega}uvdx\Big{|}\leq \Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)|u|_{p(x)}|v|_{q(x)}\leq 2|u|_{p(x)}|v|_{q(x)}.
$$
\end{proposition}
The Sobolev space with variable exponent $W^{k,p(x)}(\Omega)$ is defined as
$$
W^{k,p(x)}(\Omega)=\{ u\in L^{p(x)}(\Omega):~D^{\alpha}u\in L^{p(x)}(\Omega),~|\alpha|\leq k\},
$$
where $D^{\alpha}u=\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_1}\partial x_{2}^{\alpha_2}\dots \partial x_{N}^{\alpha_N}}u$, with $\alpha=(\alpha_1,\dots ,\alpha_N)$ is a multi-index and $|\alpha|=\sum_{i=1}^N \alpha_i$. The space $W^{k,p(x)}(\Omega)$ equipped with the norm
$$
\|u\|_{k,p(x)}=\sum_{|\alpha|\leq k}|D^{\alpha}u|_{p(x)},
$$
 becomes a separable and reflexive Banach space. For more details, we refer the reader to \cite{Fan2,Yao}. Denote
 for  $x\in \overline{\Omega}$ and $k\geq1$,
\begin{align*}
p^*(x)=\left\{
\begin{array}{ll}
\frac{Np(x)}{N-p(x)}      & \mbox { if }\hspace{0.3cm} p(x)<N,\\
+\infty       & \mbox { if }\hspace{0.3cm}p(x)\geq N,
\end{array}
\right.
\end{align*}
\begin{align*}
p_{k}^{*}(x)=\left\{
\begin{array}{ll}
\frac{Np(x)}{N-kp(x)}      & \mbox { if }\hspace{0.3cm} kp(x)<N,\\
+\infty       & \mbox { if }\hspace{0.3cm}kp(x)\geq N.
\end{array}
\right.
\end{align*}


\begin{proposition} [Fan and Zhao \cite{Fan2}]\label{prop2.2}
Let $p,r\in C_{+}(\overline{\Omega})$ such that $r(x)\leq p^{*}_{k}(x)$ for all $x\in \overline{\Omega}$. Then there is a continuous embedding
$$
W^{k,p(x)}(\Omega)\hookrightarrow L^{r(x)}(\Omega).
$$
If we replace $\leq$ with $<$, the embedding is compact.
\end{proposition}
We denote by $W_0^{k,p(x)}(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k,p(x)}(\Omega)$. Then the function space $\Big(W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega),\|u\|_{p(x)}\Big)$ is a separable and reflexive Banach space, where
$$
\|u\|_{p(x)}=\inf \Big{\{}\mu>0:\int_{\Omega}\Big{|}\frac{\Delta u(x)}{\mu}\Big{|}^{p(x)}\,dx\leq 1\Big{\}}.
$$
\begin{remark}
According to \cite{Fu}, the norm $\|\cdot\|_{2,p(x)}$ is equivalent to the norm $|\Delta \cdot|_{p(x)}$ in the space $W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)$. Consequently, the norms $\|\cdot\|_{2,p(x)}$, $\|\cdot\|_{p(x)}$ and $|\Delta \cdot|_{p(x)}$ are equivalent.
\end{remark}

Consider the functional
$$
\phi(u)=\int_{\Omega}|\Delta u|^{p(x)}\,dx.
$$
Then we have the following properties (see for example \cite[Proposition 3.2]{Amr}): if $u\in W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)$, then
\begin{gather}\label{mod1}
\|u\|_{p(x)}<1\;(=1;\,>1)\;\Leftrightarrow\;
\phi(u)<1\;(=1;\,>1);
\\
\label{mod2}
\|u\|_{p(x)}>1\;\Rightarrow\;
\|u\|_{p(x)}^{p^-}\leq\phi(u)
\leq\|u\|_{p(x)}^{p^+};
\\
\label{mod3}
\|u\|_{p(x)}<1\; \Rightarrow\;
\|u\|_{p(x)}^{p^+}\leq \phi(u)
\leq\|u\|_{p(x)}^{p^-};
\\
\label{mod4}
\|u\|_{p(x)}\to 0\;(\to\infty)\; \Leftrightarrow\;
\phi(u) \to 0\; (\to\infty).
\end{gather}
Note that the weak solutions of problem (\ref{e1.1}) are considered in the generalized Sobolev space
$$
X=\Big(W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)\Big)\times\Big(W^{2,q(x)}(\Omega)\cap W_0^{1,q(x)}(\Omega)\Big)
$$
equipped with the norm
$$\|(u,v)\|=\max\{\|u\|_{p(x)},\|v\|_{q(x)}\}.$$
We denote by $(u_n,v_n)\rightharpoonup (u,v)$ and $(u_n,v_n)\rightarrow (u,v)$ the weak convergence and strong convergence of $(u_n,v_n)$ to $(u,v)$ in $X$, respectively, denote by $c_i$ the positive constants. The dual space $X$ is denoted by $X^{*}$ and $\|\cdot\|_{*}$ stands for its norm. Therefore $$\|J'(u,v)\|_{*}=\|D_{1}J(u,v)\|_{*,p(x)}+\|D_{2}J(u,v)\|_{*,q(x)},$$
where $\|\cdot\|_{*,p(x)}$  (respectively $\|\cdot\|_{*,q(x)}$) is the norm of $(W^{2,p(x)}(\Omega)\cap W^{1,p(x)}(\Omega))^{*}$ (respectively $(W^{2,q(x)}(\Omega)\cap W^{1,q(x)}(\Omega))^{*}$).

Set
\begin{eqnarray*}
&&\hspace{-5cm}I_1(u)=\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx,\\
&&\hspace{-5cm}I_2(v)=\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx,\\
&&\hspace{-5cm}I(u,v)=I_1(u)+I_2(v),\\
&&\hspace{-5cm}\mathcal{F}(u,v)=\int_{\Omega}F(x,u,v)\,dx.
\end{eqnarray*}
Then
\begin{eqnarray*}
&&\hspace{-2cm}I'(u,v)(\varphi,\psi)=D_{1}I(u,v)(\varphi)+D_{2}I(u,v)(\psi),\\
&&\hspace{-2cm}\mathcal{F'}(u,v)(\varphi,\psi)=D_{1}\mathcal{F}(u,v)(\varphi)+D_{2}\mathcal{F}(u,v)(\psi),
\end{eqnarray*}
where
\begin{eqnarray*}
&&\hspace{-3cm}D_{1}I(u,v)(\varphi)=\int_{\Omega}|\Delta u|^{p(x)-2}\Delta u\Delta \varphi\, dx=I'_1(u)(\varphi),\\
&&\hspace{-3cm}D_{2}I(u,v)(\psi)=\int_{\Omega}|\Delta v|^{q(x)-2}\Delta v\Delta \psi \,dx=I'_2(v)(\psi),\\
&&\hspace{-3cm}D_{1}\mathcal{F}(u,v)(\varphi)=\int_{\Omega}\frac{\partial F}{\partial u} (x,u,v)\varphi \,dx,\\
&&\hspace{-3cm}D_{2}\mathcal{F}(u,v)(\psi)=\int_{\Omega}\frac{\partial F}{\partial v} (x,u,v)\psi \,dx.
\end{eqnarray*}
The Euler-Lagrange functional associated to problem \eqref{e1.1} is given by
$$
J(u,v)=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\Big)+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx\Big)
-\int_{\Omega}F(x,u,v)dx,
$$
where $\widehat{M}(t)=\int_0^t M(\tau)d\tau.$ Then $J\in C^{1}(X,\R)$ and
\begin{eqnarray}\label{e2.1}
J'(u,v)(\varphi,\psi)=D_{1}J(u,v)(\varphi)+D_{2}J(u,v)(\psi),
\end{eqnarray}
where
\begin{eqnarray*}
D_{1}J(u,v)(\varphi)=M\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\Big)\int_{\Omega}|\Delta u|^{p(x)-2}\Delta u\Delta \varphi\, dx
-D_{1}\mathcal{F}(u,v)(\varphi),\\
D_{2}J(u,v)(\psi)=M\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx\Big)\int_{\Omega}|\Delta v|^{q(x)-2}\Delta v\Delta \psi
\,dx-D_{2}\mathcal{F}(u,v)(\psi).
\end{eqnarray*}

Hereafter, $F(x,s,t)$ and $M(t)$ are always supposed to verify the following assumption:
\begin{enumerate}
\item[$(\mathbf{M_1})$] There exist $m_2\geq m_1>0$ and $\beta\geq\alpha>1$ such that for all $t\in \R^+$, $m_1t^{\alpha-1}\leq M(t)\leq m_2 t^{\beta-1}$.
\item[$(\mathbf{M_2})$] For all $t\in \R^+$, $\widehat{M}(t)\geq M(t)t$.
\item[$(\mathbf{F_1})$] For all $(x,s,t)\in \Omega\times \R^2$, we assume
$$
|F(x,s,t)|\leq c_1\Big(1+|s|^{\sigma_1(x)}+|t|^{\sigma_2(x)}+|s|^{\sigma_3(x)}|t|^{\sigma_4(x)}\Big),
$$
where $c_1$ is a positive constant, $(\sigma_1(x),\sigma_2(x),\sigma_3(x),\sigma_4(x))\in C_+(\overline{\Omega})^4$ such that
$$
\sigma_1(x)<p^*_2(x),\hspace{0.2cm}\sigma_2(x)<q^*_2(x),\hspace{0.2cm}\frac{2\sigma_3(x)}{p^*_2(x)}+\frac{2\sigma_4(x)}{q^*_2(x)}<1\hspace{0.2cm}
\textrm{in}\hspace{0.1cm}\overline{\Omega}.
$$
\item[$(\mathbf{F_2})$] There are $M>0,~\theta_1>\beta p^+$, $\theta_2>\beta q^+$ such that for all $x\in \Omega$ and all $(s,t)\in \R^2$ with $|s|^{\theta_1}+|t|^{\theta_2}\geq 2M,$ we have
$$
0<F(x,s,t)\leq \frac{s}{\theta_1}\frac{\partial F}{\partial s}(x,s,t)+\frac{t}{\theta_2}\frac{\partial F}{\partial t}(x,s,t),
$$
where $\beta$ comes from $(\mathbf{M_1})$ above.
\item[$(\mathbf{F_3})$] $F(x,s,t)=o(|s|^{\alpha p^+}+|t|^{\alpha q^{+}})$ as $(s,t)\rightarrow (0,0)$ uniformly with respect to $x\in \Omega$, where $\alpha$ comes from $(\mathbf{M_1})$.
\item[$(\mathbf{F_4})$] $F(x,-s,-t)=-F(x,s,t)$ for all $x\in \Omega$ and $(s,t)\in \R^2$.
\item[$(\mathbf{F_5})$] We have
$$F(x,s,t)\geq c_2(|s|^{\gamma_1(x)}+|t|^{\gamma_2(x)})\quad\mbox{as $(s,t)\rightarrow (0,0)$},$$
 where $(\gamma_1(x),\gamma_2(x))\in (C_+(\overline{\Omega}))^2$ such that $\gamma_1(x)<p^*_2(x)$, $\gamma_2(x)<q^*_2(x)$, $p^+<\gamma_1^-\leq \gamma_1^+<\beta p^-$, $q^+<\gamma_2^-\leq \gamma_2^+<\beta q^-$ for a.e. $x\in \Omega$, where $\beta$ comes from $(\mathbf{M_1})$.
\end{enumerate}

\begin{lemma} [El Amrouss et al. \cite{Amr}]\label{lem2.4}
We have the following assertions:
\begin{enumerate}
\item[$(\mathbf{1})$] $I_1'$ is a bounded homeomorphism and strictly monotone operator.
\item[$(\mathbf{2})$] $I_1'$ is a mapping of type $(S_{+})$, namely
$$
u_n\rightharpoonup u \hspace{0.2cm}\textrm{ and }\hspace{0.2cm}\limsup_{n\rightarrow+\infty}I_1'(u_n)
(u_n-u)\leq 0, \hspace{0.2cm}\textrm{ implies }\hspace{0.2cm} u_n\rightarrow u.
$$
\end{enumerate}
\end{lemma}
Since $X$ is a reflexive and separable Banach space, then $X^*$ is too. There exist
(see \cite{Zhao}) $\{e_j\}\subset X$ and $\{e_j^*\}\subset X^*$ such that
$$
X=\overline{\textrm{ span }\{e_j~:j=1,2,...\}},\hspace{0.5cm}X^*=\overline{\textrm{ span }\{e_j^*~:j=1,2,...\}},
$$
and
$$ \langle e_i,e_j^*\rangle=\left\{
\begin{array}{ll}
1        & \mbox { if }\hspace{0.3cm} i=j,\\
0       & \mbox { if }\hspace{0.3cm}i\neq j,
\end{array}
\right.
$$
where $\langle.,.\rangle$ denote the duality product between $X$ and $X^*$. We define
$$
X_j=\textrm{ span }\{e_j\},\hspace{0.5cm}Y_k=\bigoplus_{j=1}^{k}X_j,\hspace{0.5cm}Z_k=\overline{\bigoplus_{j=k}^{\infty}X_j}.
$$

A central role in our arguments will be played by the fountain theorem, which is due to Bartsch \cite{ref91}. This result is nicely presented in Willem \cite{Will} by using the quantitative deformation lemma. We also point out that the dual version of the fountain theorem is due to Bartsch and Willem, see \cite{Will}. Both the fountain theorem and its dual form are effective tools for studying the existence of infinitely many large or small energy solutions. It should be noted that the Palais-Smale condition plays an important role for these theorems and their applications.

\begin{lemma}[Fountain Theorem, see \cite{Will}]\label{lem3.30}
Assume
\begin{enumerate}
\item[$(\mathbf{A1})$] $X$ is a Banach space, $J\in C^{1}(X,\R)$ is an even functional.\\
Suppose that for every $k\in \mathbb{N}$, there exist $\rho_k>r_k>0$ such that
\item[$(\mathbf{A2})$] $\inf \{J(u)~:~u\in Z_k,~\|u\|=r_k\}\rightarrow+\infty$ as $k\rightarrow+\infty$.
\item[$(\mathbf{A3})$] $\max  \{J(u)~:~u\in Y_k,~\|u\|=\rho_k\}\leq0$.
\item[$(\mathbf{A4})$] $J$ satisfies the Palais-Smale condition for every $c>0.$
\end{enumerate}
Then $J$ has an unbounded sequence of critical points.
\end{lemma}

\begin{lemma}(Dual Fountain Theorem, see \cite{Will}).\label{lem3.50}
Assume $(\bf A1)$ is satisfied and there is $k_0>0$ so that, for each $k\geq k_0$, there exist $\rho_k>r_k>0$ such that
\begin{enumerate}
\item[$(\mathbf{B1})$] $a_k=\inf \{J(u)~:~u\in Z_k,~\|u\|=\rho_k\}\geq0$.
\item[$(\mathbf{B2})$] $b_k=\max  \{J(u)~:~u\in Y_k,~\|u\|=r_k\}<0$.
\item[$(\mathbf{B3})$] $d_k=\inf \{J(u)~:~u\in Z_k,~\|u\|\leq\rho_k\}\ \rightarrow0$ as $k\rightarrow+\infty$.
\item[$(\mathbf{B4})$] $J$ satisfies the $(PS)_{c}^{*}$ condition for every $c\in[d_{k_0},0).$
\end{enumerate}
Then $J$ has a sequence of negative critical values converging to $0$.
\end{lemma}

For every $a>1$, $u,v\in L^a(\Omega)$, we define
$$
|(u,v)|_a:=\max\{|u|_a,|v|_a\}.
$$
Set
\begin{equation}\label{e3.01}
a:=\max_{x\in \overline{\Omega}}\{\sigma_1(x),\sigma_2(x),2\sigma_3(x),2\sigma_4(x)\},
\end{equation}
\begin{equation}\label{e3.02}
b:=\min_{x\in \overline{\Omega}}\{\sigma_1(x),\sigma_2(x),2\sigma_3(x),2\sigma_4(x)\}.
\end{equation}
Then we have the following result.
\begin{lemma} [\cite{El}]\label{lem3.4}
Denote
\begin{equation}\label{e3.1}
\beta_{k}=\sup\{|(u,v)|_a;~\|(u,v)\|=1,~(u,v)\in Z_k\},
\end{equation}
\begin{equation}\label{e3.2}
\theta_{k}=\sup\{|(u,v)|_b;~\|(u,v)\|=1,~(u,v)\in Z_k\}.
\end{equation}
Then $\lim_{k\rightarrow\infty}\beta_k=\lim_{k\rightarrow\infty}\theta_k=0.$
\end{lemma}

We conclude this preliminary section by recalling the definition of the {\it localized Palais-Smale condition}, which was introduced by Brezis and Nirenberg \cite{brecpam}.

\begin{definition}\label{def3.6}
We say that $J$ satisfies the $(PS)_{c}^{*}$ condition (with respect to $(Y_n)$), if any sequence $\{u_{n_{j}}\}\subset X$ such that $n_{j}\rightarrow+\infty$, $u_{n_{j}}\in Y_{n_{j}}$, $J(u_{n_{j}})\rightarrow c$ and $(J|_{Y_{n_{j}}})'(u_{n_{j}})\rightarrow 0$, contain a subsequence converging to a critical point of $J$.
\end{definition}


\section{Existence results in a non-symmetric setting}
In this section we establish two existence results under general hypotheses on the potential~$F$.

\begin{theorem}\label{theo3.1}
Assume that $M$ satisfies $(\mathbf{M_1})$ and
$$
|F(x,s,t)|\leq c_3(1+|s|^{\alpha_1}+|t|^{\beta_1}),
$$
where $\alpha_1,\beta_1$ are two constants with $1\leq \alpha_1<\min\{\alpha p^-,\alpha q^-\}$, $1\leq \beta_1<\min\{\alpha p^-,\alpha q^-\}$ $(\alpha$ comes from $(\mathbf{M_1}))$. Then problem \eqref{e1.1} has a nontrivial weak solution.
\end{theorem}
\begin{proof}
In view of $(\mathbf{M_1})$, the functional $J$ is weakly lower semi-continuous. In the following, we will prove that $J$ is coercive, that is, $J(u,v)\rightarrow+\infty$ as $\|(u,v)\|\rightarrow+\infty$. From $(\mathbf{M_1})$ we have $\widehat{M}(t)\geq \frac{m_1}{\alpha}t^{\alpha}$. For $(u,v)\in X$ such that $\|(u,v)\|\rightarrow+\infty$, we obtain
\begin{align*}
\begin{split}
J(u,v)&=\di\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\Big)+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx\Big)-\int_{\Omega}F(x,u,v)dx\\
&\geq\di\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\Big)^{\alpha}+\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx\Big)^{\alpha}-c_3\int_{\Omega}|u|^{\alpha_1}\,dx\\
&\hspace{0.3cm}-c_3\int_{\Omega}|v|^{\beta_1}dx-c_3|\Omega|\\
&\geq \di\frac{m_1}{\alpha(p^+)^\alpha}\|u\|_{p(x)}^{\alpha p^-}+\frac{m_1}{\alpha(q^+)^\alpha}\|v\|_{q(x)}^{\alpha q^-}-c_4\|u\|_{p(x)}^{\alpha _1}-c_5\|v\|_{q(x)}^{\beta_1}-c_3|\Omega|\,,
\end{split}
\end{align*}
where $|\Omega|$ denote the measure of $\Omega$. Without loss of generality, we may assume $\|u\|_{p(x)}\geq \|v\|_{q(x)}$.

If $\|v\|_{q(x)}>1$ we have
$$
J(u,v)\geq\frac{m_1}{\alpha(p^+)^\alpha}\|u\|_{p(x)}^{\alpha p^-}+\frac{m_1}{\alpha(q^+)^\alpha}\|v\|_{q(x)}^{\alpha q^-}-c_4\|u\|_{p(x)}^{\alpha _1}-c_5\|v\|_{q(x)}^{\beta_1}-c_3|\Omega|\,.
$$

If $\|v\|_{q(x)}<1$ we have
$$
J(u,v)\geq \frac{m_1}{\alpha(p^+)^\alpha}\|u\|_{p(x)}^{\alpha p^-}-c_4\|u\|_{p(x)}^{\alpha _1}-c_6.
$$
By the assumptions on $\alpha_1$ and $\beta_1$, we deduce the coercivity of $J$ and hence $J$ attains its minimum on $X$, which yields a solution of  problem \eqref{e1.1}.
\end{proof}



\begin{lemma}\label{lem3.2}
Let $(u_n,v_n)$ be a Palais-Smale sequence for the Euler-Lagrange functional $J.$ If conditions $(\mathbf{M_1})$,  $(\mathbf{M_2})$,  $(\mathbf{F_2})$ 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 $\|J'(u_n,v_n)\|_*\rightarrow0$ as $n\rightarrow+\infty$. Thus, there is a positive constant $c_7$ such that
\begin{align*}
\begin{split}
c_7&\geq\di J(u_n,v_n)\\
&=\di \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{q(x)}\,dx\Big)-\int_{\Omega}F(x,u_n,v_n)\,dx\\
&\geq\di M\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx-\int_{\Omega}\frac{u_n}{\theta_1}\frac{\partial F}{\partial u}(x,u_n,v_n)\,dx\\
&\hspace{0.3cm}+M\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{q(x)}\,dx\Big)\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{p(x)}\,dx-\int_{\Omega}\frac{v_n}{\theta_2}\frac{\partial F}{\partial v}(x,u_n,v_n)\,dx-c_8,\\
\end{split}
\end{align*}
where $c_8$ is some positive constant. Therefore
\begin{align*}
\begin{split}
c_7&\geq \di J(u_n,v_n)\\
&\geq\di (\frac{1}{p^{+}}-\frac{1}{\theta_1})M\Big(\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)\int_{\Omega}|\Delta u_n|^{p(x)}\,dx-\frac{1}{\theta_1}D_1J(u_n,v_n)(u_n)\\
&\hspace{0.3cm}\di+(\frac{1}{q^{+}}-\frac{1}{\theta_2})M\Big{(}\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{q(x)}dx\Big{)}\int_{\Omega}|\Delta v_n|^{q(x)}\,dx-\frac{1}{\theta_2}D_2J(u_n,v_n)(v_n)-c_8\\
&\geq \di \frac{m_1}{(p^+)^{\alpha-1}}(\frac{1}{p^{+}}-\frac{1}{\theta_1})\Big(\int_{\Omega}|\Delta u_n|^{p(x)}\,dx\Big)^{\alpha}+\frac{m_1}{(q^+)^{\alpha-1}}(\frac{1}{q^{+}}-\frac{1}{\theta_2})\Big(\int_{\Omega}|\Delta v_n|^{q(x)}\,dx\Big)^{\alpha}\\
&\hspace{0.3cm}\di-\frac{1}{\theta_1}\|D_1J(u_n,v_n)\|_{*,p(x)}\|u_n\|-\frac{1}{\theta_2}\|D_2J(u_n,v_n)\|_{*,q(x)}\|v_n\|-c_8.
\end{split}
\end{align*}
Now, we suppose that the sequence $(u_n,v_n)$ is not bounded. Without loss of generality, we may assume $\|u_n\|_{p(x)}\geq\|v_n\|_{q(x)}$.\\
Therefore, for $n$ large enough that $\|u_n\|_{p(x)}>1$, we obtain
$$c_7\geq \frac{m_1}{(p^+)^{\alpha-1}}(\frac{1}{p^{+}}-\frac{1}{\theta_1})\|u_n\|_{p(x)}^{\alpha p^{-}}-\Big{(}\frac{1}{\theta_1}\|D_1J(u_n,v_n)\|_{*,p}
+\frac{1}{\theta_2}\|D_2J(u_n,v_n)\|_{*,q}\Big{)}\|u_n\|_{p(x)}.$$
But this cannot hold true since $\alpha p^->p^{-}>1$. Hence, $(u_n,v_n)$ is bounded.
\end{proof}



\begin{lemma}\label{lem3.3}
Let $(u_n,v_n)$ be a bounded Palais-Smale sequence for the Euler-Lagrange functional $J.$ If conditions $(\mathbf{M_1})$,  $(\mathbf{M_2})$, $(\mathbf{F_1})$,  $(\mathbf{F_2})$ 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  $J$. Then there is a subsequence still denoted by $(u_n,v_n)$ which converges weakly in $X$. Without loss of generality, we assume that $(u_n,v_n)\rightharpoonup (u,v)$, then $J'(u_n,v_n)(u_n-u,v_n-v)\rightarrow0$. We obtain
\begin{align*}
\begin{split}
&\di J'(u_n,v_n)(u_n-u,v_n-v)\\
&=\di M\Big(\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)\int_{\Omega}|\Delta u_n|^{p(x)-2}\Delta u_n (\Delta u_n-\Delta u)\,dx\\
&\hspace{0.3cm}+\di M\Big(\int_\Omega\frac{1}{q(x)}|\Delta v_n|^{q(x)}dx\Big)\int_{\Omega}|\Delta v_n|^{q(x)-2}\Delta v_n (\Delta v_n-\Delta v)\,dx\\
&\hspace{0.5cm}-\di \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 \rightarrow0.
\end{split}
\end{align*}
On the other hand, let $\tilde{\sigma_1}(x)$ and $\tilde{\sigma_2}(x)$ be two continuous and positive functions on $\overline{\Omega}$ such that for all $x\in \overline{\Omega}$
$$
\frac{2\sigma_3(x)+\tilde{\sigma_1}(x)}{p^*_2(x)}+\frac{2\sigma_4(x)+\tilde{\sigma_2}(x)}{q^*_2(x)}=1.
$$
Using the Young inequality, we obtain
$$
|s|^{\sigma_3(x)}|t|^{\sigma_4(x)}\leq |s|^{\frac{\sigma_3(x)p^*_2(x)}{2\sigma_3(x)+\tilde{\sigma_1}(x)}}+|t|^{\frac{\sigma_4(x)q^*_2(x)}{2\sigma_4(x)+\tilde{\sigma_2}(x)}}=|s|^{\sigma_5(x)}
+|t|^{\sigma_6(x)},
$$
where $\sigma_5(x):=\frac{\sigma_3(x)p^*_2(x)}{2\sigma_3(x)+\tilde{\sigma_1}(x)}<p^*_2(x)$ and $\sigma_6(x):=\frac{\sigma_4(x)q^*_2(x)}{2\sigma_4(x)+\tilde{\sigma_2}(x)}<q^*_2(x)$. From $(\mathbf{F_1})$, we obtain  $\sigma_6(x),\sigma_7(x)\in C_+(\overline{\Omega})$ with $\sigma_6(x)<p^*_2(x)$, $\sigma_7(x)<q^*_2(x)$ in $\overline{\Omega}$ such that
$$
|F(x,s,t)|\leq c_9(1+|s|^{\sigma_6(x)}+|t|^{\sigma_7(x)}).
$$
>From this inequality, Propositions \eqref{prop2.1} and \eqref{prop2.2}, we deduce that
$$
\int_{\Omega}\frac{\partial F}{\partial u}(x,u_n,v_n)(u_n-u)\,dx\rightarrow0
$$
and
$$
\int_{\Omega}\frac{\partial F}{\partial v}(x,u_n,v_n)(v_n-v)\,dx\rightarrow0.
$$
Therefore
$$
M\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\Big)\int_{\Omega}|\Delta u_n|^{p(x)-2}\Delta u_n(\Delta u_n-\Delta u)\,dx\rightarrow0,
$$
$$
M\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{q(x)}dx\Big)\int_{\Omega}|\Delta v_n|^{q(x)-2}\Delta v_n(\Delta v_n-\Delta v)\,dx\rightarrow0.
$$
Since $(u_n,v_n)$ is bounded in $X$, passing to a subsequence, if necessary, we may assume that when $n\rightarrow +\infty$
$$
\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\rightarrow t_0\geq 0
$$
and
$$
\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{p(x)}\,dx\rightarrow t_1\geq 0.
$$
If $t_0=0=t_1$ then $(u_n,v_n)$ converges strongly to $(0,0)$ and the proof is finished. Otherwise, since the function $M$ is continuous, when $n\rightarrow+\infty$ we find
$$
M\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)\rightarrow M(t_0)
$$
and
$$
M\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{p(x)}\,dx\Big)\rightarrow M(t_1).
$$
Thus, by $(\mathbf{M_1})$, for sufficiently large $n$, we have
\begin{equation}\label{e3.10}
0<c_9\leq M\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\Big)\leq c_{10},
\end{equation}
\begin{equation}\label{e3.20}
0<c_{11}\leq M\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v_n|^{q(x)}\,dx\Big)\leq c_{12}.
\end{equation}
>From \eqref{e3.10} and \eqref{e3.20}, we deduce that
$$
\int_{\Omega}|\Delta u_n|^{p(x)-2}\Delta u_n(\Delta u_n-\Delta u)\,dx\rightarrow0,
$$
$$
\int_{\Omega}|\Delta v_n|^{q(x)-2}\Delta v_n(\Delta v_n-\Delta v)\,dx\rightarrow0.
$$
Using Lemma \ref{lem2.4}, we have $u_n\rightarrow u$ in $W^{2,p(x)}(\Omega) \cap W_0^{1,p(x)}(\Omega)$ and $v_n\rightarrow v$ in $W^{2,q(x)}(\Omega) \cap W_0^{1,q(x)}(\Omega)$, which implies that $(u_n,v_n)\rightarrow(u,v)$ in $X$.
\end{proof}

\begin{theorem}\label{theo3.4}
Assume that $M$ satisfies $(\mathbf{M_{1}})$, $(\mathbf{M_2})$ and $F$ verifies $(\mathbf{F_{1}})-(\mathbf{F_3})$ and
\begin{enumerate}
\item[$(\mathbf{F_6})$] $\sigma_1^-,2\sigma_3^->\alpha p^+$ and $\sigma_2^-,2\sigma_4^->\alpha q^+$.
\end{enumerate}
Then problem \eqref{e1.1} has a nontrivial weak solution.
\end{theorem}
\begin{proof}
Let us show that $J$ satisfies the conditions of mountain pass theorem (see \cite{amb,Will}).
As pointed out by Brezis and Browder \cite{brebro}, the mountain pass theorem
``extends ideas already present in Poincar\'e and Birkhoff".


By Lemmas \eqref{lem3.2} and \eqref{lem3.3}, $J$ satisfies Palais-Smale condition in $X$.\\
For $\|(u,v)\|<1$, using the Young inequality, the fact $\frac{2\sigma_3(x)}{p^*_2(x)}+\frac{2\sigma_4(x)}{q^*_2(x)}<1$ in $\overline{\Omega}$, Proposition \ref{prop2.2} and \eqref{mod3}, we obtain
$$
\int_{\Omega}|u|^{\sigma_3(x)}|v|^{\sigma_4(x)}\,dx\leq \frac{1}{2}\int_{\Omega}|u|^{2\sigma_3(x)}\,dx+\frac{1}{2}\int_{\Omega}|v|^{2\sigma_4(x)}\,dx\leq c_{13}(\|u\|_{p(x)}^{2\sigma_3^-}+\|v\|_{q(x)}^{2\sigma_4^-}).
$$
Since $\alpha p^+<p^*_2(x)$ and $\alpha q^+<q^*_2(x)$, we deduce $W^{2,p(x)}(\Omega) \cap W_0^{1,p(x)}(\Omega)\hookrightarrow L^{\alpha p^+}(\Omega)$ and $W^{2,q(x)}(\Omega)\cap W_0^{1,q(x)}(\Omega)\hookrightarrow L^{\alpha q^+}(\Omega)$. Then there exist $c_{14},c_{15}>0$ such that
$$
|u|_{\alpha p^+}\leq c_{14}\|u\|_{p(x)}\hspace{0.3cm}\textrm{for}\hspace{0.2cm}u\in W^{2,p(x)}(\Omega) \cap W_0^{1,p(x)}(\Omega),
$$
$$
|v|_{\alpha q^+}\leq c_{15}\|v\|_{q(x)}\hspace{0.3cm}\textrm{for}\hspace{0.2cm}v\in W^{2,q(x)}(\Omega) \cap W_0^{1,q(x)}(\Omega),
$$
where $|\cdot|_r$ denote the norm on $L^{r}(\Omega)$. Let $\epsilon>0$ be small enough such that $\epsilon c_{14}^{\alpha p^+}\leq \frac{m_1}{2\alpha (p^+)^{\alpha}}$ and $\epsilon c_{15}^{\alpha q^+}\leq \frac{m_1}{2\alpha (q^+)^{\alpha}}$. By the assumptions $(\mathbf{F_{1}})$ and $(\mathbf{F_3})$, we have
$$
|F(x,s,t)|\leq \epsilon (|s|^{\alpha p^+}+|t|^{\alpha q^+})+c(\epsilon)(|s|^{\sigma_1(x)}+|t|^{\sigma_2(x)}+|s|^{\sigma_3(x)}|t|^{\sigma_4(x)})
$$
for all $(x,s,t)\in \Omega\times \R^2$. In view of $(\mathbf{M_{1}})$ and the above inequality, for $\|(u,v)\|$ sufficiently small, noting Proposition \ref{prop2.2}, we have
\begin{align*}
\begin{split}
J(u,v)&\geq \di\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\Big)^\alpha+\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta v|^{q(x)}\,dx\Big)^\alpha\\
&\hspace{0.3cm}\di-\epsilon\int_{\Omega}|u|^{\alpha p^+}\,dx-\epsilon\int_{\Omega}|v|^{\alpha q^+}\,dx- c(\epsilon)
\int_{\Omega}(|u|^{\sigma_1(x)}+|v|^{\sigma_2(x)}+|u|^{\sigma_3(x)}|v|^{\sigma_4(x)})\,dx\\
&\geq\di \frac{m_1}{\alpha (p^{+})^\alpha}\|u\|_{p(x)}^{\alpha p^+}-\epsilon c_{14}^{\alpha p^+}\|u\|_{p(x)}^{\alpha p^+}+\frac{m_1}{\alpha (q^{+})^\alpha}\|v\|_{q(x)}^{\alpha q^+}-\epsilon c_{15}^{\alpha q^+}\|v\|_{q(x)}^{\alpha q^+}\\
&\hspace{0.3cm}-\di c(\epsilon)(\|u\|_{p(x)}^{\sigma_{1}^{-}}+\|v\|_{q(x)}^{\sigma_{2}^{-}}+c_{13}\|u\|_{p(x)}
^{2\sigma_{3}^{-}}+c_{13}\|v\|_{q(x)}^{2\sigma_{4}^{-}})\\
&\geq\di \frac{m_1}{2\alpha (p^{+})^\alpha}\|u\|_{p(x)}^{\alpha p^+}+\frac{m_1}{2\alpha (q^{+})^\alpha}\|v\|_{ q(x)}^{\alpha q^+}-c(\epsilon)(\|u\|_{p(x)}^{\sigma_{1}^{-}}+\|v\|_{q(x)}^{\sigma_{2}^{-}}+c_{13}\|u\|_{p(x)}
^{2\sigma_{3}^{-}}\\
&\hspace{0.3cm}\di +c_{13}\|v\|_{q(x)}^{2\sigma_{4}^{-}}).
\end{split}
\end{align*}
Since $\sigma_{1}^-,2\sigma_{3}^->\alpha p^+$ and $\sigma_{2}^-,2\sigma_{4}^->\alpha q^+$, there exist $r>0$ and $\delta>0$ such that $J(u,v)\geq \delta>0$ for every $(u,v)\in X$ satisfying $\|(u,v)\|=r.$\\
On the other hand, our assumption $(\mathbf{F_{2}})$ implies the following assertion: for every $x\in \overline{\Omega}$, $s,t\in \R$, the inequality
\begin{equation}\label{e4.3}
F(x,s,t)\geq c_{16}(|s|^{\theta_1}+|t|^{\theta_2}-1)
\end{equation}
holds, see \cite{El}. For $(\widetilde{u},\widetilde{v})\in X\backslash \{(0,0)\}$ and $t>1$, by $(\mathbf{M_1})$ we have
\begin{align*}
\begin{split}
J(t\widetilde{u},t\widetilde{v})&=\di \widehat{M}\Big{(}\int_{\Omega}\frac{1}{p(x)}\Big{(}|t\Delta \widetilde{u}|^{p(x)}\,dx\Big{)}+\widehat{M}\Big{(}\int_{\Omega}\frac{1}{q(x)}
|t\Delta \widetilde{v}|^{q(x)}\,dx\Big{)}-\int_{\Omega}F(x,t\widetilde{u},t\widetilde{v})\,dx\\
&\leq \di \frac{m_2}{\beta}\Big{(}\int_{\Omega}\frac{1}{p(x)}|t\Delta \widetilde{u}|^{p(x)}\,dx\Big{)}^{\beta}-c_{16}\int_{\Omega}|t \widetilde
{u}|^{\theta_1}dx\\
&\hspace{0.3cm}+\di \frac{m_2}{\beta}\Big{(}\int_{\Omega}\frac{1}{q(x)}|t\Delta \widetilde{v}|^{q(x)}\,dx\Big{)}^\beta-c_{16}\int_{\Omega}|t\widetilde
{v}|^{\theta_2}\,dx-c_{17}\\
&\leq\di  \frac{m_2}{\beta (p^-)^\beta }t^{\beta p^+}\Big{(}\int_{\Omega}|\Delta \widetilde{u}|^{p(x)}\,dx\Big{)}^{\beta}-c_{16}t^{\theta_1}\int_{\Omega}|\widetilde
{u}|^{\theta_1}\,dx\\
&\hspace{0.3cm}\di +\frac{m_2}{\beta (q^-)^\beta }t^{\beta q^+}\Big{(}\int_{\Omega}(|\Delta \widetilde{v}|^{q(x)}\,dx\Big{)}^{\beta}-c_{16}t^{\theta_2}\int_{\Omega}|\widetilde
{v}|^{\theta_2}\,dx-c_{17}\\
&\rightarrow -\infty,\hspace{0.2cm}\textrm{ as }\hspace{0.2cm}t\rightarrow+\infty,
\end{split}
\end{align*}
due to $\theta_1>\beta p^+$ and $\theta_2>\beta q^+$. Since $J(0,0)=0$, considering Lemmas \ref{lem3.2} and \ref{lem3.3}, we conclude that $J$ satisfies the conditions of mountain pass lemma. So $J$ admits at least one nontrivial critical point.
\end{proof}


\section{Infinitely many low or high energy solutions}
In this section we establish two multiplicity results, provided that the potential $F$ has a suitable symmetry. The first property
shows the existence of a sequence of {\it high energy solutions} while the second result deals with the existence of a sequence of solutions
with negative energies that converge to zero (that is, {\it small energy solutions}).

\begin{theorem}\label{theo4.1}
 Assume that $M$ satisfies $(\mathbf{M_1})$, $(\mathbf{M_2})$ and $F$ fulfills hypotheses $(\mathbf{F_1})$, $(\mathbf{F_2})$, $(\mathbf{F_4})$, and $(\mathbf{F_6})$. Then problem \eqref{e1.1} has a sequence of weak solutions $(\pm u_k)$ such that $J(\pm u_k)\rightarrow+\infty$ as $k\rightarrow\infty$.
\end{theorem}

\begin{theorem}\label{theo4.2}
Assume that $M$ satisfies $(\mathbf{M_1})$, $(\mathbf{M_2})$ and $F$ fulfills hypotheses  $(\mathbf{F_1})$, $(\mathbf{F_2})$-$(\mathbf{F_5})$, and
\begin{enumerate}
\item[$(\mathbf{F_7})$] $\alpha p^+>\sigma_1^-,2\sigma_3^-$ and $\alpha q^+>\sigma_2^-,2\sigma_4^-$.
\end{enumerate}
Then problem \eqref{e1.1} has a sequence of weak solutions $(\pm v_k)$ such that $J(\pm v_k)<0$, $J(\pm v_k)\rightarrow0$ as $k\rightarrow\infty$.
\end{theorem}

\subsection*{Proof of Theorem \ref{theo4.1}}
According to $(\mathbf{F_4})$ and Lemmas \ref{lem3.2} and \ref{lem3.3}, $J$ is an even functional and satisfies the Palais-Smale condition. We prove that if $k$ is large enough, then there exist $\rho_k>r_k>0$ such that $(\mathbf{A_2})$ and $(\mathbf{A_3})$ are fulfilled. Thus, the assertion of conclusion can be obtained from fountain theorem.
\\
$\bf(A_2):$ For any $(u_k,v_k)\in Z_k$, $\|u_k\|_{p(x)}\geq1$, $\|v_k\|_{q(x)}\geq1$ and $\|(u_k,v_k)\|=r_k$ ($r_k$ will be specified below), we have
\begin{align*}
\begin{split}
J(u_k,v_k)&=\di\widehat{M}\Big{(}\int_{\Omega}\frac{1}{p(x)}| \Delta u_k|^{p(x)}\,dx\Big{)}+\widehat{M}\Big{(}\int_{\Omega}\frac{1}{q(x)}| \Delta v_k|^{q(x)}\,dx\Big{)}-\int_{\Omega}F(x,u_k,v_k)\,dx\\
&\geq \di\frac{m_1}{\alpha (p^+)^{\alpha}}\Big(\int_{\Omega}|\Delta u_k|^{p(x)}\,dx\Big{)}^{\alpha}+\frac{m_1}{\alpha (q^+)^{\alpha}}\Big(\int_{\Omega}|\Delta v_k|^{q(x)}\,dx\Big{)}^{\alpha}\\
&\hspace{0.3cm}\di-c_1\int_{\Omega}(1+|u_k|^{\sigma_1(x)}+|v_k|^{\sigma_2(x)}+|u_k|^{\sigma_3(x)}|v_k|^{\sigma_4(x)})\,dx\\
&\geq \di\frac{m_1}{\alpha (p^+)^{\alpha}}\|u_k\|_{p(x)}^{\alpha p^-}+\frac{m_1}{\alpha (q^+)^{\alpha}}\|v_k\|_{q(x)}^{\alpha q^-}-c_1|u_k|_{\sigma_1(x)}^{\sigma_1(\xi_1^k)}-c_1|v_k|_{\sigma_2(x)}^{\sigma_2(\xi_2^k)}\\
&\hspace{0.3cm}\di-c_{18}|u_k|_{2\sigma_3(x)}^{2\sigma_3(\eta_1^k)}
-c_{18}|v_k|_{2\sigma_4(x)}^{2\sigma_4(\eta_2^k)}-c_1|\Omega|,
\end{split}
\end{align*}
where $\xi_1^k,\xi_2^k,\eta_1^k,\eta_2^k\in \Omega$. Therefore
\begin{align*}
\begin{split}
&\di J(u_k,v_k)\\
&\di \geq \frac{m_1}{\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\min\{\alpha p^-,\alpha q^-\}}
-c_1|u_k|_a^{\sigma_1(\xi_1^k)}-c_1|v_k|_a^{\sigma_2(\xi_2^k)}\\
&\hspace{0.3cm}\di-c_{18}|u_k|_a^{2\sigma_3(\eta_1^k)}-c_{18}|v_k|_a^{2\sigma_4(\eta_2^k)}-c_1|\Omega|\\
&\di \geq \frac{m_1}{\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\min\{\alpha p^-,\alpha q^-\}}-c_1(\beta_k\|(u_k,v_k)\|)^{\sigma_1(\xi_1^k)}
\\
&\hspace{0.3cm}\di-c_1(\beta_k\|(u_k,v_k)\|)^{\sigma_2(\xi_2^k)}-c_{18}(\beta_k\|(u_k,v_k)\|)^{2\sigma_3(\eta_1^k)}-c_{18}(\beta_k\|(u_k,v_k)\|)^{2\sigma_4
(\eta_2^k)}
-c_1|\Omega|\\
&\di \geq\frac{m_1}{\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\min\{\alpha p^-,\alpha q^-\}}-c_{19}\beta_k^b\|(u_k,v_k)\|^a-c_1|\Omega|,
\end{split}
\end{align*}
where $a,b$ are defined in \eqref{e3.01} and \eqref{e3.02}. At this stage, we fix $r_k$ as follows:
$$
r_k:=\Big(\frac{m_1}{2c_{19}\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}\beta_k^b}\Big)^{\frac{1}{a-\min\{\alpha p^-,\alpha q^-\}}}\rightarrow+\infty\hspace{0.2cm}\textrm{ as }\hspace{0.2cm}k\rightarrow+\infty.
$$
Consequently, if $\|(u_k,v_k)\|=r_k$ then
$$
J(u_k,v_k)\geq \frac{m_1}{2\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\min\{\alpha p^-,\alpha q^-\}}-c_1|\Omega|\rightarrow+\infty\hspace{0.2cm}\textrm{ as }\hspace{0.2cm}k\rightarrow+\infty.
$$
\\
$\bf(A_3):$ From $(\mathbf{F_2})$, we have $F(x,s,t)\geq c_{16}(|s|^{\theta_1}+|t|^{\theta_2}-1)$ for every $x\in \Omega$ and $s,t\in \R$. Therefore, for any $(u,v)\in Y_k$ with $\|(u,v)\|=1$ and $1<\rho_k=t_k$ with $t_k\rightarrow+\infty$, using $(\mathbf{M_1})$ we have
\begin{align*}
\begin{split}
&\di J(t_ku,t_kv)\\
&\di =\widehat{M}\Big{(}\int_{\Omega}\frac{1}{p(x)}| \Delta (t_ku)|^{p(x)}\,dx\Big{)}+\widehat{M}\Big{(}\int_{\Omega}\frac{1}{q(x)}|\Delta (t_kv)|^{q(x)}\,dx\Big{)}-\int_{\Omega}F(x,t_ku,t_kv)\,dx\\
&\leq\frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{p(x)}|\Delta (t_ku)|^{p(x)}\,dx\Big)^{\beta}+\frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{q(x)}|\Delta (t_kv)|^{q(x)}\,dx\Big)^{\beta}\\
&\hspace{0.3cm}\di-c_{16}\int_{\Omega}|t_ku|^{\theta_1}\,dx-c_{16}\int_{\Omega}|t_kv|^{\theta_2}\,dx+c_{20}\\
&\leq\frac{m_2}{\beta(p^-)^{\beta}}t_k^{\beta p^+}\Big(\int_{\Omega}|\Delta u|^{p(x)}\,dx\Big)^{\beta}-c_{16}t_k^{\theta_1}\int_{\Omega}|u|^{\theta_1}\,dx\\
&\hspace{0.3cm}\di+\frac{m_2}{\beta(q^-)^{\beta}}t_k^{\beta q^+}\Big(\int_{\Omega}|\Delta v|^{q(x)}\,dx\Big)^{\beta}-c_{16}t_k^{\theta_2}\int_{\Omega}|v|^{\theta_2}\,dx+c_{20}.
\end{split}
\end{align*}
By $\theta_1>\beta p^+,~\theta_2>\beta q^+$ and dim$\,Y_k<\infty$, we deduce that $J(u_k,v_k)\rightarrow-\infty$ as $\|(t_ku,t_kv)\|\rightarrow+\infty$ for $(u,v)\in Y_k$. The
conclusion of Theorem \ref{theo4.1} is reached by the fountain theorem.\qed

\subsection*{Proof of Theorem \ref{theo4.2}}
>From $(\mathbf{F_4})$, we know that $J$ satisfies $(\mathbf{A_1})$, the assertion of conclusion can be obtained from the dual fountain theorem.\\
$\bf(B_1)$: For any $(u_k,v_k)\in Z_k$, $\|u_k\|_{p(x)}<1$, $\|v_k\|_{q(x)}<1$ and $\|(u_k,v_v)\|=\rho_k$ ($\rho_k$ will be specified below), we have
\begin{align*}
\begin{split}
&\di J(u_k,v_k)\\
&\geq \di\frac{m_1}{\alpha (p^+)^{\alpha}}\|u_k\|_{p(x)}^{\alpha p^+}+\frac{m_1}{\alpha (q^+)^{\alpha}}\|v_k\|_{q(x)}^{\alpha q^+}-\epsilon c_{14}^{\alpha p^+}
\|u_k\|_{p(x)}^{\alpha p^+}-\epsilon c_{15}^{\alpha q^+}\|v_k\|_{q(x)}^{\alpha q^+}\\
&\hspace{0.3cm}\di -c(\epsilon)|u_k|_{\sigma_1(x)}^{\sigma_1(\xi_3^k)}-c(\epsilon)|v_k|_{\sigma_2(x)}^{\sigma_2(\xi_4^k)}-c(\epsilon)|u_k|_{2\sigma_3(x)}^{2\sigma_3(\eta_3^k)}
-c(\epsilon)|v_k|_{2\sigma_4(x)}^{\sigma_4(\eta_4^k)},
\end{split}
\end{align*}
where $\xi_3^k,\xi_4^k,\eta_3^k,\eta_4^k\in \Omega$. Therefore
\begin{align}\label{e5.1}
\begin{split}
&J(u_k,v_k)\\
&\geq \frac{m_1}{\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\max\{\alpha p^+,\alpha q^+\}}-c(\epsilon)|u_k|_b^{\sigma_1(\xi_3^k)}
-c(\epsilon)|v_k|_b^{\sigma_2(\xi_4^k)}\\
&\hspace{0.3cm}-c(\epsilon)|u_k|_b^{2\sigma_3(\eta_3^k)}-c(\epsilon)|v_k|_b^{\sigma_4(\eta_4^k)}\\
&\geq \frac{m_1}{\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\max\{\alpha p^+,\alpha q^+\}}-c(\epsilon)(\theta_k\|(u_k,v_k)\|)^{\sigma_1(\xi_3^k)}\\
&\hspace{0.3cm}-c(\epsilon)(\theta_k\|(u_k,v_k)\|)^{\sigma_2(\xi_4^k)}-c(\epsilon)(\theta_k\|(u_k,v_k)\|)^{2\sigma_3(\eta_3^k)}-c(\epsilon)(\theta_k\|(u_k,v_k)\|)
^{2\sigma_4(\eta_4^k)}\\
&\geq \frac{m_1}{\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\}}\|(u_k,v_k)\|^{\max\{\alpha p^+,\alpha q^+\}}-c_{21}\theta_k^b\|(u_k,v_k)\|^b.
\end{split}
\end{align}
Choose $\rho_k=\Big(2c_{21}\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\} \theta_{k}^{b} m_1^{-1}\Big)^{\frac{1}{\max\{\alpha p^+,\alpha q^+\}-b}}$. Then
$$
J(u_k,v_k)\geq \frac{m_1}{2\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}}\rho_k^{\max\{\alpha p^+,\alpha q^+\}}-\frac{m_1}{2\max\{\alpha (p^+)^\alpha,\alpha(q^+)^\alpha\}}\rho_k^{\max\{\alpha p^+,\alpha q^+\}}=0.
$$
Since $\alpha p^+>\sigma_1(x),2\sigma_3(x)$ and $\alpha q^+>\sigma_2(x),2\sigma_4(x) $, we have $\rho_k\rightarrow0$ as $k\rightarrow+\infty$.\\
$\bf(B_2):$ For $(u,v)\in Y_k$, $\|(u,v)\|=1$ and $0<t_k<\rho_k<1$, using $(\mathbf{F_1})$, $(\mathbf{F_5})$  we obtain
\begin{align*}
\begin{split}
&\di J(t_ku,t_kv)\\
&\di \leq\frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{p(x)}|t_k\Delta u|^{p(x)}\,dx\Big)^\beta+\frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{q(x)}|t_k\Delta v|^{q(x)}\,dx\Big)^\beta\\
&\hspace{0.3cm}\di -c_2\int_{\Omega}|t_ku|^{\gamma_1(x)}\,dx-c_2\int_{\Omega}|t_kv|^{\gamma_2(x)}\,dx\\
&\di \leq\frac{m_2}{\beta(p^-)^\beta}t_k^{\beta p^-}\Big(\int_{\Omega}|\Delta u|^{p(x)}\,dx\Big)^\beta-c_2 t_k^{\gamma_1^+}\int_{\Omega}|u|^{\gamma_1(x)}\,dx\\
&\hspace{0.3cm}\di +\frac{m_2}{\beta(q^-)^\beta}t_k^{\beta q^-}\Big(\int_{\Omega}|\Delta v|^{q(x)}\,dx\Big)^\beta-c_2 t_k^{\gamma_2^+}\int_{\Omega}|u|^{\gamma_2(x)}\,dx.
\end{split}
\end{align*}
Conditions $\gamma_1^+<\beta p ^-$ and $\gamma_2^+<\beta q^-$ imply that there exists  $r_k\in (0,\rho_k)$ such that $J(t_ku,t_kv)<0$ when $t_k=r_k$. Hence, we deduce that
$$
b_k:=\max_{(u,v)\in Y_k,~\|(u,v)\|=r_k}J(u,v)<0.
$$\\
$\bf(B_3):$ Because $Y_k\cap Z_k\neq \emptyset$ and $r_k<\rho_k$, we have
$$
d_k=\inf \{J(u)~:~u\in Z_k,~\|u\|\leq\rho_k\}\ \leq b_k=\max  \{J(u)~:~u\in Y_k,~\|u\|=r_k\}<0.
$$
>From \eqref{e5.1}, for $(u',v')\in Z_k$, $\|(u',v')\|=1$, $0\leq t\leq\rho_k$ and $(u,v)=(tu',tv')$, we have
$$
J(u,v)=J(tu',tv')\geq\frac{m_1}{\max\{\alpha(p^+)^\alpha,\alpha(q^+)^\alpha\}}t^{\max\{\alpha p^+,\alpha q^+\}}-c_{21}t^b\theta_k^b\geq-c_{21}t^b\theta_k^b,
$$
hence, $d_k\rightarrow0$, that is, $\bf(B_3)$ is satisfied.

Finally, we verify the $(PS)_{c}^{*}$ condition. Suppose $(u_{n_{j}},v_{n_{j}})\subset X$ such that $n_{j}\rightarrow +\infty$, $(u_{n_{j}},v_{n_{j}})\in Y_{n_{j}}$ and $(J|_{Y_{n_{j}}})'(u_{n_{j}},v_{n_{j}})\rightarrow 0$. Similar to the process in the proof of Lemma \ref{lem3.2}, we deduce the boundedness of $\|(u_{n_{j}},v_{n_{j}})\|$.
Going if necessary to a subsequence, we can assume $(u_{n_{j}},v_{n_{j}})\rightharpoonup (u,v)$ in $X$. As $X=\overline{\cup_{n_j}Y_{n_{j}}}$, we can choose $(u'_{n_{j}},v'_{n_{j}})$ such that $(u'_{n_{j}},v'_{n_{j}})\rightarrow (u,v)$. Hence
\begin{align*}
\begin{split}
&\di \lim_{n_j\rightarrow+\infty} J'(u_{n_{j}},v_{n_{j}})(u_{n_{j}}-u,v_{n_{j}}-v)\\
&=\di \lim_{n_j\rightarrow+\infty} J'(u_{n_{j}},v_{n_{j}})(u_{n_{j}}-u'_{n_{j}},v_{n_{j}}-v'_{n_{j}})+\lim_{n_j\rightarrow+\infty} J'(u_{n_{j}},v_{n_{j}})(u'_{n_{j}}-u,v'_{n_{j}}-v)\\
&=\di\lim_{n_j\rightarrow+\infty}(J|_{Y_{n_{j}}})'(u_{n_{j}},v_{n_{j}})(u_{n_{j}}-u'_{n_{j}},v_{n_{j}}-v'_{n_{j}})\\
&=\di 0.
\end{split}
\end{align*}
Similar to the process of verifying the Palais-Smale condition in the proof of Lemma \ref{lem3.3}, we conclude $u_{n_{j}}\rightarrow u$ in $W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)$, and $v_{n_{j}}\rightarrow v$ in $W^{2,q(x)}(\Omega)\cap W_0^{1,q(x)}(\Omega)$, which implies that $(u_{n_{j}},v_{n_{j}})\rightarrow(u,v)$ in $X$. Furthermore, we have $J'(u_{n_{j}},v_{n_{j}})\rightarrow J'(u,v)$.

Let us prove that $J'(u,v)=0$. Taking $(\omega_k,\omega'_k)\in Y_k$, notice that when $n_j\geq k$ we have
\begin{align*}
\begin{split}
J'(u,v)(\omega_k,\omega'_k)&=\di( J'(u,v)-J'(u_{n_{j}},v_{n_{j}}))(\omega_k,\omega'_k)+ J'(u_{n_{j}},v_{n_{j}})(\omega_k,\omega'_k)\\
&=\di( J'(u,v)-J'(u_{n_{j}},v_{n_{j}}))(\omega_k,\omega'_k)+(J|_{Y_{n_{j}}})'(u_{n_{j}},v_{n_{j}})(\omega_k,\omega'_k).
\end{split}
\end{align*}
Going to the limit we obtain
$$
 J'(u,v)(\omega_k,\omega'_k)=0,\hspace{0.3cm}\textrm{for all}\hspace{0.2cm} (\omega_k,\omega'_k)\in Y_k,
$$
so $J'(u,v)=0$, this show that $J$ satisfies the $(PS)_{c}^{*}$ condition for every $c\in \R$.
The conclusion of Theorem \ref{theo4.2} is reached by the dual fountain theorem.\qed

\medskip
 \noindent \textbf{Acknowledgements.}  V. R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.



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