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\title{ Small perturbations of nonlocal biharmonic problems with variable exponent and competing nonlinearities}
\author{ Khaled Kefi$\,^{1,2}$ and Vicen\c tiu D. R\u adulescu$\,^{3,4}$\footnote{Corresponding author} \\
$^1\,${\footnotesize Community College of Rafha, Northern Border University, Kingdom of Saudi Arabia}\\
$^2\,${\footnotesize Department of Mathematics, Faculty of Sciences Tunis El Manar, 1060 Tunis, Tunisia}\\
 {\footnotesize E-mail: {\tt khaled\_kefi@yahoo.fr} }\\
 $^3\,${\footnotesize Faculty of Applied Mathematics, AGH University of Science and Technology,} \\ {\footnotesize al. Mickiewicza 30, 30-059 Krak\'ow, Poland}\\
$^4\,${\footnotesize Department of Mathematics, University of Craiova, 
          200585 Craiova, Romania}\\
 {\footnotesize E-mail: {\tt vicentiu.radulescu@imar.ro} }}
\begin{document}
\baselineskip16pt \maketitle
\noindent{\small{\bf{\sc Abstract.}}
The present paper deals with the analysis of combined effects of an absorption term and a small perturbation of the reaction term in a $p(x)$-biharmonic Kirchhoff problem with Navier boundary condition.
The main result in this work establishes the existence of a continuous spectrum consisting in an interval.  The proofs combine variational methods with energy estimates.\\
{\bf 2010 Mathematics Subject Classification :} 35D05, 35D30, 35J58, 35J60, 35D30, 58E05.\\
\small{\bf Key words:}  $p(x)$-biharmonic Kirchhoff problem, Navier boundary condition, Ekeland
variational principle, generalized Sobolev spaces, continuous spectrum.}

\section{Introduction}
The qualitative analysis of nonlinear problems with one or several variable exponents started with the pioneering papers of Halsey \cite{halsey} and Zhikov \cite{zh2}, in strong relationship with the  behavior of strongly anisotropic materials. Their work is an important contribution to the refined mathematical analysis of nonlinear problems with one or more variable exponents, mainly because  it allows the understanding of some classes of nonlinear problems with possible lack of uniform convexity.
Nonlinear problems with this structure are motivated by numerous models in the applied sciences that are driven by partial differential equations with one or more variable exponents. In some circumstances, the standard analysis based on the theory of usual Lebesgue and Sobolev function  spaces,  $L^p$ and $W^{1,p}$, is not appropriate in the framework of material that involve non-homogeneities. For instance, both electro-rheological ``smart" fluids  and phenomena arising in image processing are described in a correct way by nonlinear models in which the exponent $p$ is not necessarily constant.
The variable exponent describe the geometry of a material which is allowed to change its hardening  exponent according to the point.
This leads to the analysis of variable exponents Lebesgue and Sobolev function spaces (denoted by $L^{p(x)}$ and $W^{1,p(x)}$), where $p$ is a real-valued (non-constant) function.  This is a common abstract framework in homogenization and nonlinear elasticity. We refer here to  the monograph by R\u adulescu and Repov\v s \cite{radrep}, which includes a thorough variational and topological analysis of several classes of problems with variable exponent (see also the survey paper R\u adulescu \cite{radnla} and the important contributions by Pucci {\it et al.} \cite{patrizia1, patrizia02, patrizia2}).

In this paper, we first discuss
the existence of a continuous spectrum consisting in an interval for the following nonlocal biharmonic problem with variable exponent:
\begin{equation}\label{p1}
\left\{\begin{array}{lll}M(t)(\Delta^{2}_{p(x)} u+a(x)|u|^{p(x)-2}u)&=\lambda
V_1(x)|u|^{q(x)-2}u,
\quad & \mbox{in}\ \Omega\\
%&u\not\equiv 0,\quad  &\mbox\in\Omega\\
&u=\Delta u=0,\quad   & \mbox{on}\ \partial\Omega,
\end{array}\right.
\end{equation}
where $\Delta^2_{p(x)} u = \Delta(|\Delta u|^{p(x)-2} \Delta u)$ is the
$p(x)$-biharmonic operator and $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$. We assume that $M(t)$ is a continuous function and $$t:=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)})dx,$$
 $\lambda$ is a positive parameter, $p, q$ are continuous functions
on $\overline{\Omega}$ and $a\in L^{\infty}(\Omega)$ such that $\displaystyle {\rm essinf}_{x\in\Omega}a(x)>0$.
We assume that $V_1$ is an potential in a generalized Lebesgue space
such that $V_1>0$ in an open set $\Omega_{0}\subset\subset\Omega$, where $|\Omega_{0}| >0$.

Next, we focus on the following perturbed problem
\begin{equation}\label{1}
\left\{\begin{array}{lll}M(t)(\Delta^{2}_{p(x)} u+a(x)|u|^{p(x)-2}u)&=\lambda (V_1(x)|u|^{q(x)-2}u- V_2(x) |u|^{\alpha(x)-2}u)
\quad & \mbox{in}\ \Omega\\
%&u\not\equiv 0,\quad  &\mbox\in\Omega\\
&u=\Delta u=0,\quad   & \mbox{on}\ \partial\Omega,
\end{array}\right.
\end{equation}
where $ \alpha$ is a continuous function on $\overline{\Omega}$ and
 $V_2$ is a nonnegative one in a generalized Lebesgue spaces.

Problems (\ref{p1}) and \eqref{1} are {\it nonlocal problems} because of the presence of the term $M$, which implies that the equations in (\ref{p1})and \eqref{1} are no longer pointwise. This provokes some mathematical difficulties which make the study of such a problem particulary interesting.

In 1883, Kirchhoff \cite{K} introduced a model given by the equation
 \begin{equation}\label{eq1.2}
 \rho \frac{\partial^2 u}{\partial t^2} - \Big(\frac{\rho_0}{h}
 + \frac{E}{2L} \int_0^L |\frac{\partial u}{\partial x}|^{2}dx\Big)
 \frac{\partial^2 u}{\partial x^2} = 0,
\end{equation}
which extends the classical d'Alembert's wave equation by
considering the effects of the changes in the length of the
strings during the vibrations. The parameters in the above equation have the following meanings: $L$ is the length of the string, $h$ is the area of the cross-section, $E$ is the young modulus of the material, $\rho$ is the mass density and $\rho_0$ is the initial tension. A feature of problem (\ref{eq1.2}) is that the equation contains a nonlocal coefficient $\frac{\rho_0}{h}
 + \frac{E}{2L} \int_0^L |\frac{\partial u}{\partial x}|^{2}dx$, which depends on the average $\frac{1}{2L} \int_0^L |\frac{\partial u}{\partial x}|^{2}dx.$
 Nonlocal effects also find various applications in biological systems.

 After the work of Lions \cite{LION}, various equations of Kirchhoff type have been investigated, see \cite{AR, DR}. Moreover, Kirchhoff-type equations involving $p$-Laplacian and $p(x)$-Laplacian have been studied in many papers; see, \cite{CO1, eleuteri, fuy, heidari, molica}. A parabolic version
of problem \eqref{eq1.2} can be used to describe the growth and movement of a particular species.
The movement, modeled by the integral term, is assumed to be dependent on the energy of the
entire system with $u$
being its population density. Alternatively, the movement of  particular species may be subject to the total population density within the domain (for instance, the
spreading of bacteria), which gives rise to nonlocal parabolic equations. We refer to \cite{puccicv} for details.

 We also mention that fourth-order elliptic equations arise in many domains like micro-electro-mechanical systems, surface diffusion on solids, thin film theory, flow in Hele-Shaw cells and phase field models of multiphasic systems, see \cite{SV,FW,M1}. Recent contributions concerning a fourth order elliptic problems with $p(x)$ biharmonic operators can be found in \cite{KV}.

In the present paper, we study problem (\ref{p1}) under the following assumptions:
\\
$(H_0)$ $M:\mathbb{R}\rightarrow [m_0,+\infty)$ is a continuous function, with $m_0>0$;\\
$(H_1)$ there exists $0<\theta<1$ such that $$\widehat{M}(t)\geq (1-\theta)M(t)t\ \mbox{for all}\ t\geq 0,\ \mbox{where}\ \ \widehat{M}(t)=\int_{0}^{t} M(s)ds;$$
$(H_2)$ $1<q(x)<p(x)< \frac{N}{2}<s_1(x)$, $\mbox{for all }x\in \overline{\Omega}$, $V_1\in L^{s_1(x)}(\Omega)$ and $V_1>0 \ \mbox{in} \ \Omega_{0}\subset\subset \Omega$, with $\mid\Omega_{0}\mid>0$.

We point out that Kefi \cite{KK} was the first to introduce assumption like $(H_2)$ to study problems involving Lebesgue and Sobolev spaces with variable exponents.

\section{Terminology and abstract setting}
To study $p(x)$-biharmonic problems, we need some results on the spaces $L^{p(x)}(\Omega)$, $W^{1,p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$; see \cite{harj,radnla,radrep} for details, complements and proofs.

 Let $$C_{+}(\overline{\Omega}):=\{h: h\in C(\overline{\Omega}), h(x)>1, \mbox{for all}\;\; x\in \overline{\Omega}\}.$$
For any $p\in C_{+}(\overline{\Omega})$, we denote $1< p^{-}:=\displaystyle\min_{x\in \overline{\Omega}}p(x) \leq  p^{+}=\displaystyle\max_{x\in \overline{\Omega}}p(x)<\infty$ and
 $$L^{p(x)}(\Omega)=\{u:\Omega\rightarrow \mathbb{R} \mbox{ measurable and } \int_{\Omega}|u(x)|^{p(x)}dx<\infty\}.$$
 The spaces $L^{p(x)}(\Omega)$ have been introduced by Orlicz \cite{orlicz}.

The space $L^{p(x)}(\Omega)$ is endowed with the  Luxemburg norm, which is defined by $$|u|_{p(x)}=\inf\left\{\mu>0:\int_{\Omega}\left|\frac{u(x)}{\mu}\right|^{p(x)}dx\leq 1\right\}.$$
Clearly, when $p(x)\equiv p$, the space $L^{p(x)}(\Omega)$ reduces to the classical Lebesgue space $L^p(\Omega)$ and the norm $|u|_{p(x)}$ reduces to the standard norm $\|u\|_{L^p}=(\int_{\Omega}|u|^p dx )^{\frac{1}{p}}$ in $L^p(\Omega)$.

For any positive integer $k$,  let $$W^{k,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): D^{\alpha}u\in L^{p(x)}(\Omega), |\alpha|\leq k\},$$ where $\alpha=(\alpha_1,\alpha_2,\ldots ,\alpha_n) $ is a multi-index, $|\alpha|=\sum_{i=1}^{n}\alpha_i$ and
$D^{\alpha}u=\frac{\partial^{|\alpha|}u}{\partial^{\alpha_1}x_1\ldots\partial^{\alpha_N}x_N}.$
Then $W^{k,p(x)}(\Omega)$ is a separable and reflexive Banach space equipped with the norm $$\|u\|_{k,p(x)}=\displaystyle\sum_{|\alpha|\leq k}|D^{\alpha}u|_{p(x)}.$$

The space $W_{0}^{k,p(x)}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k,p(x)}(\Omega).$

Let $L^{p'(x)}(\Omega)$ be the conjugate space of $L^{p(x)}(\Omega)$ with $\frac{1}{p}+\frac{1}{p'}=1$. Then the following H\"{o}lder-type inequality
\begin{equation}\label{7}
\left|\int_{\Omega}u v dx\right|\leq \left(\frac{1}{p^{-}}+\frac{1}{(p')^{-}}\right)|u|_{p(x)}|v|_{p'(x)}, \ \ \mbox{for all}\ u\in L^{p(x)}(\Omega), v\in L^{p'(x)}(\Omega)
\end{equation}
holds.
Moreover, if $h_1$, $h_2$ and
$h_3:\overline{\Omega}\rightarrow(1,\infty)$ are Lipschitz
continuous functions such that $1/h_{1}(x) + 1/h_{2}(x) + 1/h_{3}(x)
= 1$, then for any $u \in L^{h_{1}(x)}(\Omega), v\in
L^{h_{2}(x)}(\Omega)$ and $w\in L^{h_{3}(x)}(\Omega)$ the following
inequality holds (see \cite[Proposition 2.5]{FH}):
\begin{equation}\label{Hol1}
\left|\int_\Omega uvw\,dx\right|\leq\left(\frac{1}{h_{1}^-}+
\frac{1}{{h_{2}}^-}+\frac{1}{{h_{3}}^-}\right)|u|_{h_{1}(x)}|v|_{h_{2}(x)}|w|_{h_3(x)}.
\end{equation}
Inequality (\ref{7}) and its generalized version (\ref{Hol1}) are due to Orlicz \cite{orlicz}.

The modular on the space $L^{p(x)}(\Omega)$ is the map $\rho_{p(x)}:L^{p(x)}(\Omega)\rightarrow \mathbb{R}$ defined by $$\rho_{p(x)}(u):=\int_{\Omega}|u|^{p(x)}dx.$$

\begin{prop}\label{ll}(See \cite{R0})
For all $u, v\in L^{p(x)}(\Omega)$, we have
\begin{enumerate}
                       \item $|u|_{p(x)}<1\ (resp. =1, >1) \Leftrightarrow \rho_{p(x)}(u)<1\ (resp. =1, >1).$
                       \item $\min(|u|^{p^{-}}_{p(x)},|u|^{p^{+}}_{p(x)})\leq\rho_{p(x)}(u)\leq \max(|u|^{p^{-}}_{p(x)},|u|^{p^{+}}_{p(x)}).$
                       \item $\rho_{p(x)}(u-v)\rightarrow 0\Leftrightarrow |u-v|_{p(x)}\rightarrow 0.$
                     \end{enumerate}
\end{prop}
\begin{prop}\label{norm1}(See \cite{R7}) Let $p$ and $q$ be  measurable functions such
that $p\in L^{\infty}(\Omega)$, and $1\leq p(x)q(x)\leq\infty$, for a.e. $x\in\Omega$. Let $u\in L^{q(x)}(\Omega)$, $u\neq0$. Then
$$ \min(|u|^{p^{+}}_{p(x)q(x)},|u|^{p^{-}}_{p(x)q(x)})\leq||u|^{p(x)}|_{q(x)}\leq \max(|u|^{p^{-}}_{p(x)q(x)}, |u|^{p^{+}}_{p(x)q(x)}).$$
\end{prop}

  \begin{defin}\label{de1}
  Assume that spaces $E$, $F$ are Banach spaces, we define the norm on the space $X := E\cap F$
as $\|u\|_X = \|u\|_E + \|u\|_F$ .
\end{defin}

In order to discuss problems $\eqref{p1}$ and $\eqref{1}$, we need some properties of the space $X := W^{1,p(x)}_{0}(\Omega)\cap W^{2,p(x)}(\Omega)$.
From Definition \ref{de1}, we know that for any $u\in X$ we have $\|u\| = \|u\|_{1,p(x)}+\|u\|_{2,p(x)}$, thus  $\|u\| =|u|_{p(x)}+|\nabla u|_{p(x)}+\displaystyle\sum_{|\alpha|=2}|D^{\alpha}u|_{p(x)}.$
In Zang and Fu \cite{ZF}, the equivalence of the norms was proved, and it was even proved that the norm
$|\Delta u|_{p(x)}$ is equivalent to the norm $\|u\| $ (see \cite[Theorem 4.4]{ZF}).
 Note that $(X, \|.\|)$ is  a separable and
reflexive Banach space.

Let $$\|u\|_a:=\inf\left\{\mu>0:\displaystyle\int_{\Omega}\left(|\frac{\Delta u}{\mu}|^{p(x)}+a(x)|\frac{u}{\mu}|^{p(x)}\right)dx\leq 1 \right\}\quad\mbox{for}\ \ u\in X.$$
Since $a\in L^{\infty}(\Omega)$ and ${\rm essinf}_{x\in \Omega} a>0$, we deduce that  $\|u\|_a$ is equivalent to the norms $\|u\|$ and $|\Delta u|_{p(x)}$ in $X$. In our paper, we will use the norm $\|u\|_a$
and the modular is defined as
$\rho_{p(x)} : X \rightarrow \mathbb{R}$ by $$\rho_{p(x)}(u)=\int_{\Omega}|\Delta u|^{p(x)}+a(x)|u|^{p(x)}dx,$$ which satisfies the same properties as Proposition \ref{1}. Accordingly, we have the following property.

\begin{prop}\label{am}
For all $u\in L^{p(x)}(\Omega)$, we have
\begin{enumerate}
                       \item $\|u\|_{a}<1\ (resp. =1, >1) \Leftrightarrow \rho_{p(x)}(u)<1\ (resp. =1, >1).$
                       \item $\min(\|u\|_{a}^{p^{-}},\|u\|_{a}^{p^{+}})\leq\rho_{p(x)}(u)\leq \max(\|u\|_{a}^{p^{-}},\|u\|_{a}^{p^{+}}).$
                       \item $\|u_n\|_a \rightarrow 0\ (\mbox{respectively},\rightarrow\infty)\Leftrightarrow \rho_{p(x)}(u_n)\rightarrow 0\ (\mbox{respectively},\rightarrow\infty).$
                     \end{enumerate}
\end{prop}
\begin{prop}\label{111}
Let $L(u)=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)})dx$, then
\begin{enumerate}
                       \item $L:X\rightarrow\mathbb{R}$ is sequentially weakly lower semi continuous, $L\in C^1(X,\mathbb{R})$.
                       \item  The mapping $L':X\rightarrow X^*$ is a strictly monotone, bounded homeomorphism and is of type $(S_+)$, that is, if $u_n\rightharpoonup u$ and $\displaystyle\limsup_{n\rightarrow +\infty} L'(u_n)(u_n-u)\leq 0$, then $u_n\rightarrow u$.

                     \end{enumerate}
\end{prop}

We recall that the critical Sobolev exponent is defined as follows:
 $$
\left\{
\begin{array}{l}
p^*(x)=\displaystyle\frac{Np(x)}{N-2p(x)},\quad p(x)<\frac{N}{2},\\
p^*(x)=\displaystyle+\infty, \hspace{0.8cm}\quad\quad p(x)\geq \frac{N}{2}. \\
\end{array}
\right.
$$

We point out that if
 $q \in C^+(\overline{\Omega})$ and $q(x) < p^*(x)$ for all $x\in\overline\Omega$, then $X$ is
continuously and compactly embedded in $L^{q(x)}(\Omega)$.


The  Lebesgue and Sobolev
spaces with variable exponents coincide with the usual Lebesgue and Sobolev spaces provided that $p$ is constant. According to \cite[pp. 8-9]{radrep}, the function spaces $L^{p(x)}$ and $W^{1,p(x)}$ have some non-usual properties,
such as:

(i) If $p\geq 1$ is a real number, then the following co-area formula 
$$\int_\Omega |u(x)|^pdx=p\int_0^\infty t^{p-1}\,|\{x\in\Omega ;\ |u(x)|>t\}|\,dt$$
has  no analogue in the framework of variable exponents (namely, if $p:\overline\Omega\rightarrow [1,\infty)$ is a nonconstant smooth function).

(ii) Spaces $L^{p(x)}$ do {\it not} satisfy the {\it mean continuity property}. More exactly, if $p$ is nonconstant and continuous in an open ball $B$, then there is some $u\in L^{p(x)}(B)$ such that $u(x+h)\not\in L^{p(x)}(B)$ for every $h\in{\mathbb R}^N$ with arbitrary small norm.

(iii) Function spaces with variable exponent
 are {\it never} invariant with respect to translations.  The convolution is also limited. For instance,  the classical Young inequality
$$| f*g|_{p(x)}\leq C\, | f|_{p(x)}\, \| g\|_{L^1}$$
remains true if and only if
$p$ is constant.

\section{Auxiliary properties and main result for problem \eqref{p1}}
Throughout this section, the letters $ c, c_i, i=1,2,....$ denote positive constants which may change from line to line.
Let $s'_{1}(x)$ denote the conjugate exponent of the
function $s_1(x)$ and set $r_1(x):=\frac{s_1(x) q(x)}{s_1(x)-q(x)}$.
Then we have the following embedding property.

\begin{rem}\label{r1}
Assume that assumption $(H_2)$ is fulfilled, then $\max(r_1(x),s{'}_{1}(x)q(x))<p^{*}(x)$, for all $x\in \overline{\Omega}$, consequently
the embeddings $X\hookrightarrow L^{s'_{1}(x)
q(x)}(\Omega)$ and $X\hookrightarrow
L^{r_1(x)}(\Omega)$ are compact and continuous.
\end{rem}

\begin{defin}\label{d1}
We say that $\lambda\in \mathbb{R}$ is an eigenvalue of problem
\eqref{p1}, if there exists  $u\in X\setminus\{0\}$  such that $\Delta u=0$ on $\partial\Omega$ and
$$M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a|u|^{p(x)}dx)\Big)\ \int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a|u|^{p(x)-2}uv) dx=\lambda\int_{\Omega} V_1|u|^{q(x)-2}u v dx,$$
for any $v\in X$. If $\lambda$
is an eigenvalue of problem $(\ref{p1})$, then the corresponding
$u\in X\setminus\{0\}$ is a weak solution of problem
$(\ref{p1})$.
\end{defin}

\smallskip
The first main result of this part is the following.
\begin{teo}\label{thm1}
Assume that hypotheses $(H_0)$, $(H_1)$, $(H_2)$ are fulfilled. Then there exists
$\lambda^{*}>0$, such that any $\lambda\in (0,\lambda^{*})$ is an
eigenvalue of problem \eqref{p1}.
\end{teo}

 In order to describe the variational framework associated to \eqref{p1}, we define the functionals $\Phi,\, J:X\to \mathbb{R}$ defined as follows:
$$\Phi(u)=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+a(x)|u|^{p(x)})dx\Big)\  \ \mbox{and} \ J(u)=\int_{\Omega}\frac{V_1(x)}{q(x)}|u|^{q(x)}dx.$$

By Proposition \ref{norm1} and Remark \ref{r1}, $J$ is well defined and for all $u\in X$
$$\displaystyle{|J(u)| \leq \frac{1}{q^{-}} |V_1|_{s_1(x)}||u|^{q(x)}|_{s'_{1}(x)}}\leq \left\{
\begin{array}{l}
\frac{1}{q^{-}} |V_1|_{s_1(x)}|u|^{q^{-}}_{s'_{1}(x)q(x)},\quad \mbox{if}\ |u|_{s'_{1}(x)q(x)}\leq 1,\\
\frac{1}{q^{-}} |V_1|_{s_1(x)}|u|^{q^{+}}_{s'{1}(x)q(x)}, \quad\mbox{if}\  |u|_{s'_{1}(x)q(x)}>1 . \\
\end{array}
\right.$$

The Euler-Lagrange functional corresponding to problem
\eqref{p1} is $\Psi_{\lambda}:X\to \mathbb{R}$ and is defined by
$$
\Psi_\lambda(u):=\Phi(u)-\lambda J(u).
$$

\subsection{Proof of Theorem \ref{thm1}}
We start with the following auxiliary property.

\begin{prop}\label{pr00}
Assume that hypotheses $(H_0)$, $(H_1)$ and $(H_2)$ are fulfilled. Then $\Psi_\lambda\in C^{1}(X,\mathbb{R})$ is weakly lower semi-continuous
and $u\in X$ is a critical point of $\Psi_\lambda$ if and only if
$u$ is a weak solution of problem \eqref{p1}.
\end{prop}
\begin{proof}
To show that $\Psi_\lambda\in C^{1}(X,\mathbb{R})$, we establish that for
all $\varphi\in X$,
$$\lim_{{t\to 0^{+}}}{\frac{\Psi_\lambda(u+t\varphi)-\Psi_\lambda(u)}{t}}=\langle d\Psi_\lambda(u), \varphi\rangle,$$
and $d\Psi_\lambda:X\rightarrow X^{*}$ continuous, where we denote
by $X^{*}$ the dual space of $X$.

For all $\varphi\in X$ we have
\begin{eqnarray*}
\lim_{{t\to 0^{+}}}{\frac{J(u+t\varphi)-J(u)}{t}}
&=&\frac{d}{dt}J(u+t\varphi)|_{t=0}
=\frac{d}{dt}\int_{\Omega}\frac{V_1(x)}{q(x)}|u+t\varphi|^{q(x)}dx|_{t=0}\\
&=&\int_{\Omega}\frac{\partial}{\partial
t}\left(\frac{V_1(x)}{q(x)}|u+t\varphi|^{q(x)}\right)|_{t=0}dx\\
&=&\int_{\Omega}V_1(x)|u+t\varphi|^{q(x)-1}sgn(u+t\varphi)\varphi|_{t=0}dx\\
&=&\int_{\Omega}V_1(x)|u+t\varphi|^{q(x)-2}(u+t\varphi)\varphi|_{t=0}dx
=\int_{\Omega}V_1(x)|u|^{q(x)-2}u\varphi dx\\
&=&\langle d J(u), \varphi\rangle.
\end{eqnarray*}
The differentiation under the integral is allowed, since  for all $|t|<1$ we have:
$$|V_1(x)|u+t\varphi|^{q(x)-2}(u+t\varphi)\varphi|\leq |V_1(x)|(|u|+|\varphi|)^{q(x)-1}|\varphi|\in L^{1}(\Omega)
.$$
Since $u,\varphi\in X$ we have
$$|u|,|\varphi|\in X\hookrightarrow
L^{q(x)}(\Omega) \ \mbox{and}\ \ |\varphi|\in X\hookrightarrow
L^{r_1(x)}(\Omega).$$ Due to the fact that $V_1\in
L^{s_1(x)}(\Omega)$, the conclusion is an immediate consequence of
the generalized H\"older inequality (\ref{Hol1}).

Next, we show that for all $u\in X$,  $dJ(u)$ is in $
X^{*}$. We first observe that $dJ(u)$ is
linear.
Since there is a continuous embedding $X\hookrightarrow
L^{r_1(x)}(\Omega)$, we have
\begin{equation}\label{4}
\begin{gathered}
 $$|v|_{{r_1(x)}}\leq c \|v\|_a,\quad \mbox{for all } v\in X.$$
\end{gathered}
\end{equation}
 Using inequalities \eqref{Hol1} and \eqref{4} we
obtain
\begin{eqnarray*}
|\langle
dJ(u),\varphi\rangle|&=&|\int_{\Omega}V_1(x)|u|^{q(x)-2}u\varphi dx|\\
&\leq& \int_{\Omega}|V_1(x)||u|^{q(x)-1}|\varphi| dx\\ &\leq&
|V_1|_{s_1(x)}||u|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}|\varphi|_{r_1(x)}\\
&\leq& c |V_1|_{s_1(x)}||u|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}} \|\varphi\|_a.
\end{eqnarray*}
Thus, there exists $c_1:= c|V_1(x)|
||u|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}>0$ such that $$|\langle
dJ(u),\varphi\rangle|\leq c_1 \|\varphi\|_a.$$ Using the linearity of
$dJ(u)$ and the above inequality we deduce that $dJ(u)\in
X^{*}$. For the Fr\'echet differentiability we
need the following auxiliary property.

\begin{lemma}[See \cite{KM}]\label{l0}
The map $u\in L^{q(x)}(\Omega)\mapsto |u|^{q(x)-2}u\in
L^{\frac{q(x)}{q(x)-1}}(\Omega)$ is continuous.
\end{lemma}
We conclude that $J$ is Fr\'echet differentiable.

The functional $\Phi$ is well defined, is continuously G\^ateaux differentiable and its G\^ateaux derivative is given by
$$\langle d\Phi(u),v\rangle=M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)}dx)\Big)\ \int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a(x)|u|^{p(x)-2}uv) dx,$$
for all $u,v\in X$.

We deduce that $\Psi_{\lambda}\in C^{1}(X,\RR)$ because $\Phi,\,J
\in C^{1}(X,\RR)$. Moreover
\begin{eqnarray*}
\langle d\Psi_{\lambda}(u),v\rangle=M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)}dx)\Big)\ \int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a(x)|u|^{p(x)-2}uv) dx\\
-\lambda\int_{\Omega}V_1(x)|u|^{q(x)-2}u v dx
\end{eqnarray*}
 for all $u,v \in X$.

Let $u$ be a critical point of $\Psi_{\lambda}$. Then we have
$d\Psi_{\lambda}(u)=0_{X^{*}}$, that is,
$$\langle d\Psi_{\lambda}(u),v\rangle=0,\quad \mbox{ for all } v\in X.$$
Therefore
$$\begin{array}{ll}
&\di M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)}dx)\Big)\, \int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a(x)|u|^{p(x)-2}uv) dx=\\
&\di\lambda\int_{\Omega}V_1(x) |u|^{q(x)-2}u v dx,\end{array}$$
for all $v\in X$.
It follows that $u$ is a weak solution of problem \eqref{p1}.

Now we assume that $u$ is a weak solution of  \eqref{p1}.
By Definition \ref{d1} we deduce that $\langle d\Psi_{\lambda}(u),v\rangle=0$, for all $v\in X.$ We
obtain $d\Psi_{\lambda}(u)=0_{X^{*}}$, hence $u$ is a critical point
of $\Psi_{\lambda}$. This completes the proof of Proposition
\ref{pr00}.
\end{proof}

The following property shows the existence of a {\it mountain} for $\Psi_\lambda$ near the origin.
\begin{lemma}\label{l5}
Suppose that the  hypotheses of Theorem \ref{thm1} are fulfilled. Then for all
$\rho \in (0,1)$, there exist $\lambda^*>0$ and $b>0$ such that for
all $u\in X$ with $\|u\|_a=\rho$
$$\Psi_{\lambda}(u)\geq b>0 \quad \quad \mbox{for all} \quad \quad \lambda \in (0,\lambda^*).$$
\end{lemma}

\begin{proof}
Since the embedding $X\hookrightarrow L^{s'_{1}(x)q(x)}(\Omega)$ is
continuous, we have
\begin{eqnarray}\label{ineg1}
|u|_{s'_{1}(x)q(x)}\leq c_2\|u\|_a,\ \mbox{for all}\ u\in X.
\end{eqnarray}
Let us assume
that $\|u\|_a<\min(1,1/c_2)$, where $c_2$ is the positive constant of inequality (\ref{ineg1}). It follows that $|u|_{s'_{1}(x)q(x)}<1$. Moreover, by hypothesis $(H_0)$, we have $\widehat{M}(t)\geq m_0 t$. Consequently, by combining H\"older's inequality
(\ref{7}), Proposition \ref{am} and inequality (\ref{ineg1}), we deduce that for all $u \in X$
with $\|u\|_a=\rho $,
\begin{eqnarray*}
\Psi_{\lambda}(u)&=&\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta
u|^{p(x)}+a(x)|u|^{p(x)})dx\Big)- \frac{\lambda}{q^{-}}{\int_{\Omega}V_1(x)|
u|^{q(x)}dx}\\
&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda }{q^{-}} |V_1|_{s_1(x)}||u|^{q(x)}|_{s'(x)}\\
&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda }{q^{-}} |V_1|_{s_1(x)}|u|^{q^{-}}_{s'(x)q(x)}\\
&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda }{q^{-}} |V_1|_{s_1(x)}c_{2}^{q^{-}}\|u\|_{a}^{q^{-}}\\
&=&\frac{m_0}{p^{+}}\rho^{p^{+}}-\frac{\lambda }{q^{-}}
c_{2}^{q^{-}}|V_1|_{s_1(x)}\rho^{q^{-}}=\rho^{q^-}
(\frac{m_0}{p^{+}}\rho^{p^{+}-q^{-}}-\frac{\lambda }{q^{-}} c_{2}^{q^{-}} |V_1|_{s_1(x)}).\\
\end{eqnarray*}
By the above inequality, we remark that if we define
\begin{eqnarray}\label{L16}
\displaystyle \lambda^*=\frac
{m_0\rho^{p^{+}-q^{-}}}{2p^+}\frac{q^{-}}{c_{2}^{q^-}|V_1|_{s_1(x)}},
\end{eqnarray}
then for any $\lambda \in (0,\lambda^*)$ and $u \in X$ with
$\|u\|_a=\rho$ there exists $ b>0$ such that
$$\Psi_{\lambda}(u)\geq b>0.$$
The proof of Lemma \ref{l5} is complete.
\end{proof}

The following result asserts the existence of a {\it valley} for $\Psi_\lambda$ near the origin.
\begin{lemma}\label{l6}
There exists $\phi \in X\setminus\{0\}$ such that $\phi \geq0$ and
$\Psi_{\lambda}(t\phi)<0$, for $t>0$ small enough.
\end{lemma}

\begin{proof}
By hypothesis $(H_1)$, there exists $t_0>0$ such that
    for all $t>t_0$, we have $$\widehat{M}(t)\leq\frac{\widehat{M}(t_0)}{t_0^{\frac{1}{1-\theta}}}t^{\frac{1}{1-\theta}}=c_3 t^{\frac{1}{1-\theta}}.$$
    Moreover, by $(H_2)$, we have $q(x)<p(x)$, for all  $x\in
\overline{\Omega}_0$.

In the sequel, we denote
$$q_{0}^{-}:=\inf_{\Omega_0}q(x)\quad \mbox{and}\quad
p_{0}^{-}:=\inf_{\Omega_0}p(x).$$
 Let $\epsilon_0$ be such that
$q_{0}^{-}+\epsilon_0<p_{0}^{-}$. Since $q\in
C(\overline{\Omega}_0)$, there exists an open set
$\Omega_1\subset\Omega_0$ such that $|q(x)-q_{0}^{-}|<\epsilon_0,\
\mbox{for all}\ x\in \Omega_1$. It follows that
$q(x)\leq q_{0}^{-}+\epsilon_0<p_{0}^{-},\ \mbox{for all}\ x\in \Omega_1$.

  Let $\phi \in C_{0}^{\infty}(\Omega)$ be such that ${\rm supp}\,(\phi)\subset \Omega_1\subset\Omega_0$, $\phi=1$ in a subset $\Omega'_{1}\subset {\rm supp}\,(\phi)$, and $0\leq\phi\leq 1$ in $\Omega_{1}$. It follows that
\begin{eqnarray*}
\Psi_{\lambda}(t\phi)&=&\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta
(t\phi)|^{p(x)}+a(x)|t\phi|^{p(x)})dx\Big)-\lambda\int_{\Omega}\frac{V_1(x)}{q(x)}|
t\phi|^{q(x)}dx\\
&\leq&c_3\Big(\int_{\Omega_{0}}\frac{t^{p(x)}}{p(x)}\big(|\Delta\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}-\lambda\int_{\Omega_1}\frac{V_1(x)}{q(x)}t^{q(x)}V(x)|\phi|^{q(x)}dx\\
&\leq&\displaystyle
\frac{t^{\frac{p_{0}^-}{1-\theta}}}{(p_{0}^-)^{\frac{1}{1-\theta}}}\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}-\frac{\lambda t^{q_{0}^{-}+\epsilon_0}}{q_{0}^{+}}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx\\
&\leq&\displaystyle c_4 t^{p_{0}^{-}}\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}-\frac{\lambda t^{q_{0}^{-}+\epsilon_0}}{q_{0}^{+}}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx.\\
\end{eqnarray*}
Therefore
$$\displaystyle \Psi_{\lambda}(t\phi)<0$$
for $\displaystyle t<\delta^{1/(p_{0}^{-}-q_{0}^{-}-\epsilon_0)}$
with
$$\displaystyle 0<\delta<\mbox{min}\left\{1,\frac{\frac{\lambda}{c_4 q_{0}^+}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx}{\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}}\right\}.$$ Finally, we point out that
$$\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx>0.$$
Indeed, supposing the contrary we have
$\int_{\Omega}(|\Delta\phi|^{p(x)}+a(x)|u|^{p(x)})dx=0$.
By Proposition \ref{am}, we deduce that $\|\phi\|_a=0$ and consequently $\phi=0$ in $\Omega$, a contradiction.
The proof of Lemma \ref{l6} is complete.
\end{proof}

{\it\bf{ Proof of Theorem \ref{thm1} completed.}} Let $\lambda^*>0$ be defined as in
\eqref{L16} and $\lambda \in (0,\lambda^*)$. By Lemma \ref{l5} it
follows that on the boundary of the ball centered at the origin and
of radius $\rho$ in $X$, denoted by $B_{\rho}(0)$, we have
\begin{equation}\label{L17}
\inf_{\partial B_{\rho}(0)}\Psi_{\lambda}>0.
\end{equation}
On the other hand, by Lemma \ref{l6}, there exists $\phi \in X$ such
that $\Psi_{\lambda}(t\phi)<0$ for all $t>0$ small enough. Moreover,
by hypothesis $(H_0)$, H\"older's inequality
(\ref{7}), Proposition \ref{am} and inequality (\ref{ineg1}), we deduce that for any $u
\in B_{\rho}(0)$ we have
\begin{equation*}
\Psi_{\lambda}(u)\geq
\displaystyle\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda}{q^{-}}c_{2}^{q-}|V_1|_{s_1(x)}\|u\|_{a}^{q^{-}}{{\bf
.}}
\end{equation*}
 It follows that
$$-\infty<\underline{c}:=\inf_{\overline{B_{\rho}(0)}}\Psi_{\lambda}<0.$$
Let $ 0<\epsilon<\inf_{\partial
B_{\rho}(0)}\Psi_{\lambda}-\inf_{B_{\rho}(0)}\Psi_{\lambda}$. Due to
the above information, the functional $\Psi_{\lambda}:\overline{B_{\rho}(0)}\longrightarrow \mathbb{R}$ is
lower bounded on $\overline{B_{\rho}(0)}$ and $\Psi_{\lambda}\in
C^{1}(\overline{B_{\rho}(0)},\RR)$. Thus, by Ekeland's variational
principle \cite{EK}, there exists $u_\epsilon \in \overline{B_{\rho}(0)}$ such
that
\begin{equation*}
\left\{
\begin{array}{lll}
\displaystyle \underline{c} \leq \Psi_{\lambda}(u_\epsilon)  \leq \underline{c}+\epsilon \\
\displaystyle 0<
\Psi_{\lambda}(u)-\Psi_{\lambda}(u_\epsilon)+\epsilon \cdot
\parallel u-u_\epsilon \parallel_a, \quad u\neq u_\epsilon.
\end{array}
\right.
\end{equation*}
Since
$$\Psi_{\lambda}(u_\epsilon)\leq\inf_{\overline{B_{\rho}(0)}}\Psi_{\lambda}+\epsilon \leq \inf_{B_{\rho}(0)}\Psi_{\lambda}
+\epsilon <\inf_{\partial B_{\rho}(0)}\Psi_{\lambda},$$ we deduce
that $u_\epsilon \in {B_{\rho}(0)}.$ Now, we define
$I_{\lambda}:\overline{B_{\rho}(0)}\longrightarrow\mathbb{R}$ by
$I_{\lambda}(u)= \Psi_{\lambda}(u)+\epsilon \cdot \parallel
u-u_\epsilon \parallel_a.$ It is clear that $u_\epsilon$ is a minimum
point of $I_{\lambda}$ and thus
$$\frac{I_{\lambda}(u_\epsilon+t\cdot v)-I_{\lambda}(u_\epsilon)}{t}\geq 0$$
for small $t>0$ and any $v \in B_{1}(0)$. The above relation yields
$$\frac{\Psi_{\lambda}(u_\epsilon+t\cdot v)-\Psi_{\lambda}(u_\epsilon)}{t} + \epsilon\cdot\parallel v\parallel_a\geq 0$$
Letting $t\rightarrow 0$ it follows that
$<d\Psi_{\lambda}(u_\epsilon),v>+\epsilon\cdot\parallel v \parallel_a
\geq0 $
and we infer that $\parallel d\Psi_{\lambda}(u_\epsilon)\parallel_a \leq\epsilon $.\\
We deduce that there exists a sequence $\{w_n\}\subset B_{\rho}(0)$
such that
\begin{equation}\label{L170}
  \Psi_{\lambda}(w_n) \longrightarrow \underline{c}\quad \mbox{and} \quad  d\Psi_{\lambda}(w_n)\longrightarrow 0_{X^*}.
\end{equation}

The sequence $\{w_n\}$ is bounded in $X$. Thus, there exists $w$
in $X$ such that, up to a subsequence, $\{w_n\}$ converges weakly to
$w$ in $X$. Since $r_1(x)<p^*(x)$ for all $x\in
\overline{\Omega}$ we deduce that there exists a compact embedding
$E\hookrightarrow L^{r_1(x)}(\Omega)$ and consequently $\{w_n\}$
converges strongly in $L^{r_1(x)}(\Omega)$. In order to have strong
convergence, we need the following auxiliary result.

\begin{prop}\label{lim} We have
$$ \displaystyle\lim_{n\to
\infty} \int_{\Omega}V_1(x)|w_{n}|^{q(x)-2}w_{n}(w_{n}-w)dx=0.$$
\end{prop}

\begin{proof}
Using H\"older's inequality (\ref{7}) we have
\begin{eqnarray*}
\displaystyle\int_{\Omega}V_1(x)|w_{n}|^{q(x)-2}w_{n}(w_{n}-w)dx &\leq& |V_1|_{s_1(x)}||w_n|^{q(x)-2}w_n(w_n-w)|_{s'_{1}(x)}\\
 &\leq& |V_1|_{s_1(x)}|||w_n|^{q(x)-2}w_n|_{\frac{q(x)}{q(x)-1}}|w_n-w|_{r_1(x)}.
\end{eqnarray*}
If $||w_n|^{q(x)-2}w_n|_{\frac{q(x)}{q(x)-1}}>1$, by Proposition
\ref{norm1}, we have\\
$$||w_n|^{q(x)-2}w_n|_{\frac{q(x)}{q(x)-1}}\leq
|w_n|^{q^{+}}_{q(x)}.$$ Using now the compact embedding $X\hookrightarrow
L^{q(x)}(\Omega)$, we conclude the proof.
\end{proof}

Since $d\Psi_{\lambda}(w_n)\to  0$ and $\{w_n\}$ is bounded in $X$ we
have
\begin{eqnarray*}
|\langle d\Psi_{\lambda}(w_n),w_n-w\rangle|&\leq&
|\langle d\Psi_{\lambda}(w_n),w_n\rangle|+|\langle d\Psi_{\lambda}(w_n),w\rangle|\\
&\leq&\|d\Psi_{\lambda}(w_n)\|_a\| w_n\|_a+ \| d\Psi_{\lambda}(w_n)\|_a \|
w\|_a.
\end{eqnarray*}
Moreover, by Proposition \ref{lim}, we have
$$
\lim_{n\to \infty}\langle d\Psi_{\lambda}(w_n),w_n-w\rangle=0.
$$
Therefore
\begin{eqnarray*}
\displaystyle\lim_{n\to \infty}M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta w_n|^{p(x)}+a(x)|w_n|^{p(x)})dx\Big)\ \int_{\Omega}\Big(|\Delta w_n|^{p(x)-2}\Delta w_n (\Delta w_n-\Delta w)+\\a(x)|w_n|^{p(x)-2}w_n(w_n-w)\Big)dx=0.
\end{eqnarray*}
Combining hypothesis $(H_0)$ and Proposition \ref{111},  we deduce that
$\{w_n\}$ converges strongly to $w$ in $X$.
Since $\Psi_{\lambda}\in C^1(X,\mathbb{R})$, we conclude that
\begin{equation}\label{191}
 d\Psi_{\lambda}(w_n)\rightarrow d\Psi_{\lambda}(w),\ \mbox{as}\ n\rightarrow \infty.
\end{equation}

Now, relations \eqref{L170} and \eqref{191} yield
\begin{equation}\label{critica}
\Psi_{\lambda}(w)=\underline{c}<0\ \mbox{ and}\ \ d\Psi_{\lambda}(w)=0.
\end{equation}

In order to show that $w$ is a
solution of problem $(\ref{p1})$, it remains to show that $\Delta w=0$ on $\partial\Omega$.
Due to relation \eqref{critica}, $w\in X\setminus\{0\}$ is a critical point of $\Psi_\lambda$, so
\begin{equation}\label{star1}M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega m(x)vdx\quad\mbox{for all}\ v\in X,
\end{equation}
where
$$m(x)=\lambda V_1(x)|w|^{q(x)-2}w-M(t)a(x)|w|^{p(x)-2}w.$$
Relation (\ref{star1}) implies that
\begin{equation}\label{1star1}M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega m(x)vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).\end{equation}

Let $\zeta$ be the unique solution of the problem
$$\left\{\begin{array}{lll}
&\Delta\zeta =m(x)\quad&\mbox{in}\ \Omega\\
&\zeta =0\quad&\mbox{on}\ \partial\Omega.\end{array}\right.
$$
Relation (\ref{1star1}) yields
$$M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega (\Delta\zeta)vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).$$Using the Green formula we have
$$\int_\Omega (\Delta\zeta)vdx=\int_\Omega \zeta\Delta vdx.$$
Therefore
\begin{equation}\label{2star1}
M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega\zeta \Delta vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).\end{equation}

On the other hand, for all $\tilde{w}\in C^\infty_0(\Omega)$ there exists a unique $v\in C^\infty_0(\Omega)$ such that $\Delta v=\tilde{w}$ in $\Omega$. Thus, relation (\ref{2star1}) can be rewritten as
$$\int_\Omega (M(t)|\Delta w|^{p(x)-2}\Delta w-\zeta)\tilde{w}dx=0\quad\mbox{for all}\ \tilde{w}\in C^\infty_0(\Omega).$$
Applying the fundamental lemma of the calculus of variations, we deduce that
$$M(t)|\Delta w|^{p(x)-2}\Delta w-\zeta=0\quad\mbox{in}\ \Omega.$$
According to assertion $(H_0)$ and since $\zeta=0$ on $\partial\Omega$, we conclude that $\Delta w=0$ on $\partial\Omega$.
Thus, $w$ is a  nontrivial weak solution of problem
\eqref{p1} such that $\Delta w=0$. Since $\Psi_{\lambda}(|w|)=\Psi_{\lambda}(w)$
then problem \eqref{p1} has a non-negative solution. The proof is now complete.\quad$\square$

%{\sc \bf{Proof of Theorem \ref{thm}.}} Using condition $(H_0)$ and H$\ddot{o}$lder inequality (\ref{7}) for %$\|u\|_a>1$, we have
%\begin{eqnarray*}
%\Psi_{\lambda}(u)&=&\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta
%u|^{p(x)}+a(x)|u|^{p(x)})dx\Big)- \frac{\lambda}{q^{-}}{\int_{\Omega}V(x)|
%u|^{q(x)}dx}\\
%&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{-}}-\frac{\lambda }{q^{-}} |V|_{s(x)}||u|^{q(x)}|_{s'(x)}\\
%&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{-}}-\frac{\lambda }{q^{-}} |V|_{s(x)}|u|^{q^{-}}_{s'(x)q(x)}\\
%&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{-}}-\frac{\lambda }{q^{-}} |V|_{s(x)}C^{q^{-}}\|u\|_{a}^{q^{-}}\rightarrow %+\infty,\ \mbox{as} \ \|u\|_a\rightarrow +\infty.\\
%\end{eqnarray*}
%As a conclusion, since $\Psi_{\lambda}$ is weakly lower semi-continuous then it has a global minimizer which is %solution of problem (\ref{1}), moreover lemma \ref{l6} ensures that this solution is not trivial, which ends the %proof.\quad$\square$



\section{Main result for problem \eqref{1}}
In what follows assume we the following hypothesis:\\
$(H_3)$ $1<q(x)<\alpha(x)<p(x)< \frac{N}{2}<\min(s_1(x),s_2(x))$, $\mbox{for all }x\in \overline{\Omega}$, where $s_2\in C(\overline\Omega)$ and \\ $V_2\in L^{s_2(x)}(\Omega)$ such that $V_2\geq0$ in $\Omega$.

 Let
$s_{2}'(x)$ denote the conjugate exponent of the function
 $s_2(x)$ and
$r_2(x):=\frac{s_2(x) \alpha(x)}{s_2(x)-\alpha(x)}$. Then the following embedding properties hold.

\begin{rem}\label{r}
Under assumption $(H_3)$, we have $\displaystyle\max(r_2(x)s_2'(x)\alpha(x))<p^{*}(x),\ \mbox{for all}\ x\in
\overline{\Omega}$. Consequently, the embeddings
$X\hookrightarrow L^{s_2'(x)
\alpha(x)}(\Omega)$ and $X\hookrightarrow
L^{r_2(x)}(\Omega)$ are compact and continuous.
\end{rem}


\begin{defin}\label{d11}
We say that $\lambda\in \mathbb{R}$ is an eigenvalue of problem
\eqref{1}, if there exists  $u\in X\setminus\{0\}$  such that $\Delta u=0$ on $\partial\Omega$ and
$$M(t)\int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a(x)|u|^{p(x)-2}uv) dx=\lambda\int_{\Omega}(V_1(x)|u|^{q(x)-2}- V_2(x) |u|^{\alpha(x)-2})uvdx,$$
for any $v\in X$. If $\lambda$
is an eigenvalue of problem $(\ref{1})$, then the corresponding
$u\in X\setminus\{0\}$ is a weak solution of problem
$(\ref{1})$.
\end{defin}

\smallskip
The main result on this part of the paper is the following.
\begin{teo}\label{thm}
Assume that hypotheses $(H_0)$, $(H_1)$ and $(H_3)$ are fulfilled. Then there exists
$\lambda^{*}>0$, such that any $\lambda\in (0,\lambda^{*})$ is an
eigenvalue of problem \eqref{1}.
\end{teo}

 In order to describe the variational framework associated to \eqref{1}, we define the functional $\gamma :X\to \mathbb{R}$  as follows:
$$ \gamma(u)=\int_{\Omega}\frac{V_2(x)}{\alpha(x)}|u|^{\alpha(x)}dx.$$

By Proposition \ref{norm1} and Remark \ref{r}, $\gamma$ is well defined and for all $u\in X$
$$|\gamma(u)|\leq\frac{1}{\alpha^-}|V_2|_{s_2(x)}||u|^{\alpha(x)}|_{s_{2}'(x)}\leq\left\{
\begin{array}{l}
\frac{1}{\alpha^{-}} |V_2|_{s(x)}|u|^{\alpha^{-}}_{s_{2}'(x)\alpha(x)},\quad \mbox{if}\ |u|_{s_{2}'(x)\alpha(x)}\leq 1,\\
\frac{1}{\alpha^{-}} |V_2|_{s(x)}|u|^{\alpha^{+}}_{s_{2}'(x)\alpha(x)}, \quad\mbox{if}\  |u|_{s_{2}'(x)\alpha(x)}>1 . \\
\end{array}
\right.$$

The Euler-Lagrange functional corresponding to problem
\eqref{1}  is $\Psi_{\lambda}:X\to \mathbb{R}$ and is defined by
$$
\Psi_\lambda(u):=\Phi(u)-\lambda J(u)+\lambda \gamma(u).
$$
According to Proposition \ref{111}, one has $\Phi\in C^1(X,\mathbb{R})$. Moreover, under assumption $(H_3)$ and Proposition 2 in \cite{MK} one has $J, \gamma\in C^1(X,\mathbb{R})$, so $\Psi_\lambda\in C^1(X,\mathbb{R})$ and
\begin{eqnarray*}
\langle d\Psi_{\lambda}(u),v\rangle=M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)}dx)\Big)\ \int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u \Delta v+a(x)|u|^{p(x)-2}uv) dx\\
-\lambda\int_{\Omega}V_1(x)|u|^{q(x)-2}u v dx+\lambda\int_{\Omega}V_2(x)|u|^{\alpha(x)-2}u v dx,\\
\end{eqnarray*}
 for all $u,v \in X$.






\subsection{Proof of Theorem \ref{thm}}
The following property shows the existence of a {\it mountain} for $\Psi_\lambda$ near the origin.
\begin{lemma}\label{l55}
Suppose that the  hypotheses of Theorem \ref{thm} are fulfilled. Then for all
$\rho \in (0,1)$, there exists $\lambda^*>0$ and $b>0$ such that for
all $u\in X$ with $\|u\|_a=\rho$
$$\Psi_{\lambda}(u)\geq b>0 \quad \quad \mbox{for all} \quad \quad \lambda \in (0,\lambda^*).$$
\end{lemma}

\begin{proof}
Let us assume
that $\|u\|_a<\min(1,1/c_2)$, where $c_2$ is the positive constant of inequality (\ref{ineg1}). It follows that $|u|_{s'_{1}(x)q(x)}<1$. Since $V_2\geq 0$ on $\Omega$, we use the same steps as the proof of Lemma \ref{l55} to deduce
 that for all $u \in X$
with $\|u\|_a=\rho $,
\begin{eqnarray*}
\Psi_{\lambda}(u)&\geq&\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta
u|^{p(x)}+a(x)|u|^{p(x)})dx\Big)- \frac{\lambda}{q^{-}}{\int_{\Omega}V_1(x)|
u|^{q(x)}dx}+\frac{\lambda}{\alpha^{+}}{\int_{\Omega}V_2(x)|u|^{\alpha(x)} dx}\\
&\geq&\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda }{q^{-}} |V_1|_{s_1(x)}||u|^{q(x)}|_{s_{1}'(x)}\\
&=&\frac{m_0}{p^{+}}\rho^{p^{+}}-\frac{\lambda }{q^{-}}
c_{2}^{q^{-}}|V_1|_{s_1(x)}\rho^{q^{-}}=\rho^{q^-}
(\frac{m_0}{p^{+}}\rho^{p^{+}-q^{-}}-\frac{\lambda }{q^{-}} c_{2}^{q^{-}} |V_1|_{s_1(x)}).\\
\end{eqnarray*}
By the above inequality, we remark that if we define
\begin{eqnarray}\label{L166}
\displaystyle \lambda^*=\frac
{m_0\rho^{p^{+}-q^{-}}}{2p^+}\frac{q^{-}}{c_{2}^{q^-}|V_1|_{s_1(x)}},
\end{eqnarray}
then for any $\lambda \in (0,\lambda^*)$ and $u \in X$ with
$\|u\|_a=\rho$ there exists $ b>0$ such that
$$\Psi_{\lambda}(u)\geq b>0.$$
The proof of Lemma \ref{l55} is complete.
\end{proof}

The following result asserts the existence of a {\it valley} for $\Psi_\lambda$ near the origin.
\begin{lemma}\label{l66}
There exists $\phi \in X\setminus\{0\}$ such that $\phi \geq0$ and
$\Psi_{\lambda}(t \phi)<0$, for $t>0$ small enough.
\end{lemma}

\begin{proof}
By assumption $(H_3)$, one has $q(x)<\alpha(x)$, for all  $x\in
\overline{\Omega}_0$.
In the sequel, we denote
$$q_{0}^{-}:=\inf_{\Omega_0}q(x),\
p_{0}^{-}:=\inf_{\Omega_0}p(x) \quad\mbox{and}\quad \alpha_{0}^{-}:=\inf_{\Omega_0}\alpha(x).$$
 Let $\epsilon_0$ be such that
$q_{0}^{-}+\epsilon_0<\alpha_{0}^{-}$. Since $q\in
C(\overline{\Omega}_0)$, there exists an open set
$\Omega_1\subset\Omega_0$ such that $|q(x)-q_{0}^{-}|<\epsilon_0,\
\mbox{for all}\ x\in \Omega_1$. It follows that
$q(x)\leq q_{0}^{-}+\epsilon_0<\alpha_{0}^{-},\ \mbox{for all}\ x\in \Omega_1$.

  Let $\phi \in C_{0}^{\infty}(\Omega)$ be such that ${\rm supp}\,(\phi)\subset \Omega_1\subset\Omega_0$, $\phi=1$ in a subset $\Omega'_{1}\subset {\rm supp}\,(\phi)$, and $0\leq\phi\leq 1$ in $\Omega_{1}$. It follows that
\begin{eqnarray*}
\Psi_{\lambda}(t\phi)&=&\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta
(t\phi)|^{p(x)}+a(x)|t\phi|^{p(x)})dx\Big)-\lambda\int_{\Omega}\frac{V_1(x)}{q(x)}|
t\phi|^{q(x)}dx\\
&+&\lambda\int_{\Omega}\frac{t^{\alpha(x)}}{\alpha(x)}V_2(x)|\phi|^{\alpha(x)} dx\\
&\leq&c_3\Big(\int_{\Omega_{0}}\frac{t^{p(x)}}{p(x)}\left(|\Delta\phi|^{p(x)}+
a(x)|\phi|^{p(x)})dx\right)^{\frac{1}{1-\theta}}-\lambda\int_{\Omega_1}\frac{V_1(x)}{q(x)}t^{q(x)}|\phi|^{q(x)}dx\\
&+&\lambda\int_{\Omega_0}\frac{t^{\alpha(x)}}{\alpha(x)}V_2(x)|\phi|^{\alpha(x)} dx\\
&\leq&\displaystyle
c_3\frac{t^{\frac{p_{0}^-}{1-\theta}}}{(p_{0}^-)^{\frac{1}{1-\theta}}}\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}-\frac{\lambda t^{q_{0}^{-}+\epsilon_0}}{q_{0}^{+}}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx\\
&+&\lambda\frac{t^{\alpha_{0}^{-}}}{\alpha_{0}^{-}}\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx\\
&\leq&\displaystyle c_3\frac{ t^{p_{0}^{-}}}{(p_{0}^{-})^{\frac{1}{1-\theta}}}\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}-\frac{\lambda t^{q_{0}^{-}+\epsilon_0}}{q_{0}^{+}}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx\\
&+&\lambda\frac{t^{\alpha_{0}^{-}}}{\alpha_{0}^{-}}\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx\\
&\leq&\displaystyle\frac{\max(c_3,1)t^{\alpha_{0}^{-}}}{\min(\alpha_{0}^{-},(p_{0}^{-})^{\frac{1}{1-\theta}})} \Big[\int_{\Omega_0}\Big(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}+\lambda\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx\Big]\\
&-&\frac{\lambda t^{q_{0}^{-}+\epsilon_0}}{q_{0}^{+}}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx\\
\end{eqnarray*}
Therefore
$$\displaystyle \Psi_{\lambda}(t\phi)<0$$
for $\displaystyle t<\delta^{1/(\alpha_{0}^{-}-q_{0}^{-}-\epsilon_0)}$
with
$$\displaystyle 0<\delta<\mbox{min}\left\{1,\frac{\frac{\lambda\min(\alpha_{0}^{-},(p_{0}^{-})^{\frac{1}{1-\theta}})}{\max(c_3,1) q_{0}^+}\int_{\Omega_1}V_1(x)|\phi|^{q(x)}dx}{\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}+\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx}\right\}.$$ Finally, we point out that
$$\Big(\int_{\Omega_0}(|\Delta
\phi|^{p(x)}+a(x)|\phi|^{p(x)})dx\Big)^{\frac{1}{1-\theta}}+\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx>0.$$
Indeed, supposing the contrary we have
$\Big(\int_{\Omega}(|\Delta\phi|^{p(x)}+a(x)|u|^{p(x)}dx)\Big)^{\frac{1}{1-\theta}}+\int_{\Omega_0}V_2(x)|\phi|^{\alpha(x)} dx=0$.
By Proposition \ref{am}, we deduce that $\|\phi\|_a=0$ and consequently $\phi=0$ in $\Omega$, a contradiction.
The proof of Lemma \ref{l66} is complete.
\end{proof}

{\it {\bf Proof of Theorem \ref{thm} completed.}} Let $\lambda^*>0$ be defined as in
\eqref{L166} and $\lambda \in (0,\lambda^*)$. By Lemma \ref{l55} it
follows that on the boundary of the ball centered at the origin and
of radius $\rho$ in $X$, denoted by $B_{\rho}(0)$, we have
\begin{equation}\label{L177}
\inf_{\partial B_{\rho}(0)}\Psi_{\lambda}>0.
\end{equation}
On the other hand, by Lemma \ref{l66}, there exists $\phi \in X$ such
that $\Psi_{\lambda}(t\phi)<0$ for all $t>0$ small enough. Moreover,
by hypothesis $(H_0)$, H\"older's inequality
(\ref{7}), Proposition \ref{am} and inequality (\ref{ineg1}), we deduce that for any $u
\in B_{\rho}(0)$ we have
\begin{equation*}
\Psi_{\lambda}(u)\geq
\displaystyle\frac{m_0}{p^{+}}\|u\|_{a}^{p^{+}}-\frac{\lambda}{q^{-}}c_{2}^{q-}|V_1|_{s_1(x)}\|u\|_{a}^{q^{-}}{{\bf
.}}
\end{equation*}
 It follows that
$$-\infty<\underline{c}:=\inf_{\overline{B_{\rho}(0)}}\Psi_{\lambda}<0.$$
Let $ 0<\epsilon<\inf_{\partial
B_{\rho}(0)}\Psi_{\lambda}-\inf_{B_{\rho}(0)}\Psi_{\lambda}$. Due to
the above information, the functional $\Psi_{\lambda}:\overline{B_{\rho}(0)}\longrightarrow \mathbb{R}$ is
lower bounded on $\overline{B_{\rho}(0)}$ and $\Psi_{\lambda}\in
C^{1}(\overline{B_{\rho}(0)},\RR)$. Thus, by Ekeland's variational
principle, there exists $u_\epsilon \in \overline{B_{\rho}(0)}$ such
that
\begin{equation*}
\left\{
\begin{array}{lll}
\displaystyle \underline{c} \leq \Psi_{\lambda}(u_\epsilon)  \leq \underline{c}+\epsilon \\
\displaystyle 0<
\Psi_{\lambda}(u)-\Psi_{\lambda}(u_\epsilon)+\epsilon \cdot
\parallel u-u_\epsilon \parallel_a, \quad u\neq u_\epsilon.
\end{array}
\right.
\end{equation*}
Since
$$\Psi_{\lambda}(u_\epsilon)\leq\inf_{\overline{B_{\rho}(0)}}\Psi_{\lambda}+\epsilon \leq \inf_{B_{\rho}(0)}\Psi_{\lambda}
+\epsilon <\inf_{\partial B_{\rho}(0)}\Psi_{\lambda},$$ we deduce
that $u_\epsilon \in {B_{\rho}(0)}.$ Now, we define
$I_{\lambda}:\overline{B_{\rho}(0)}\longrightarrow\mathbb{R}$ by
$I_{\lambda}(u)= \Psi_{\lambda}(u)+\epsilon \cdot \parallel
u-u_\epsilon \parallel_a.$ It is clear that $u_\epsilon$ is a minimum
point of $I_{\lambda}$ and thus
$$\frac{I_{\lambda}(u_\epsilon+t\cdot v)-I_{\lambda}(u_\epsilon)}{t}\geq 0$$
for small $t>0$ and any $v \in B_{1}(0)$. The above relation yields
$$\frac{\Psi_{\lambda}(u_\epsilon+t\cdot v)-\Psi_{\lambda}(u_\epsilon)}{t} + \epsilon\cdot\parallel v\parallel_a\geq 0$$
Letting $t\rightarrow 0$ it follows that
$<d\Psi_{\lambda}(u_\epsilon),v>+\epsilon\cdot\parallel v \parallel_a
\geq0 $
and we infer that $\parallel d\Psi_{\lambda}(u_\epsilon)\parallel_a \leq\epsilon $.\\
We deduce that there exists a sequence $\{w_n\}\subset B_{\rho}(0)$
such that
\begin{equation}\label{L1770}
  \Psi_{\lambda}(w_n) \longrightarrow \underline{c}\quad \mbox{and} \quad  d\Psi_{\lambda}(w_n)\longrightarrow 0_{X^*}.
\end{equation}

The sequence $\{w_n\}$ is bounded in $X$. Thus, there exists $w$
in $X$ such that, up to a subsequence, $\{w_n\}$ converges weakly to
$w$ in $X$.
Since $\max(r_1(x), r_2(x))<p^{\ast}(x)$, for all $x\in\overline{\Omega}$,
then $X$ is compactly embedded in $L^{r_1(x)}(\Omega)$ and
$L^{r_2(x)}(\Omega)$.
In order to establish the strong convergence of $\{u_n\}$ on $X$, we use proposition \ref{lim} and the following auxiliary property.
\begin{prop}\label{lim1}
We have\\
%$(i)$ $\displaystyle\lim_{n\to
%\infty} \int_{\Omega}V_1(x)|u_{n}|^{q(x)-2}u_{n}(u_{n}-u)dx=0.$\\
 $\displaystyle\lim_{n\to
\infty} \int_{\Omega}V_2(x)|u_{n}|^{\alpha(x)-2}u_{n}(u_{n}-u
)dx=0.$

\end{prop}

\begin{proof}
Using H\"older's inequality (\ref{7}) we have
$$\displaystyle
\int_{\Omega}V_2(x)|u_{n}|^{\alpha(x)-2}u_{n}(u_{n}-u)dx\leq
|V_2|_{s_{2}(x)}||u_n|^{\alpha(x)-2}u_n|_{\frac{\alpha(x)}{\alpha(x)-1}}|u_n-u|_{r_2(x)},$$

Next, by Proposition \ref{norm1},
 if $$\displaystyle ||u_n|^{\alpha(x)-2}u_n|_{\frac{\alpha(x)}{\alpha(x)-1}}>1$$
 then $$\displaystyle
||u_n|^{\alpha(x)-2}u_n|_{\frac{\alpha(x)}{\alpha(x)-1}}\leq |u_n|^{\alpha^+}_{\alpha(x)}.$$
Using the compact embedding $X\hookrightarrow L^{\alpha(x)}(\Omega)$, we conclude the proof of the proposition.
\end{proof}
Since $d\Psi_{\lambda}(w_n)\to  0$ and $\{w_n\}$ is bounded in $X$ we
have
\begin{eqnarray*}
|\langle d\Psi_{\lambda}(w_n),w_n-w\rangle|&\leq&
|\langle d\Psi_{\lambda}(w_n),w_n\rangle|+|\langle d\Psi_{\lambda}(w_n),w\rangle|\\
&\leq&\|d\Psi_{\lambda}(w_n)\|_a\| w_n\|_a+ \| d\Psi_{\lambda}(w_n)\|_a \|
w\|_a.
\end{eqnarray*}
Moreover, by Propositions \ref{lim} and \ref{lim1}, we have
$$
\lim_{n\to \infty}\langle d\Psi_{\lambda}(w_n),w_n-w\rangle=0.
$$
Therefore
\begin{eqnarray*}
\displaystyle\lim_{n\to \infty}M\Big(\int_{\Omega}\frac{1}{p(x)}(|\Delta w_n|^{p(x)}+a(x)|w_n|^{p(x)})dx\Big)\ \int_{\Omega}\Big(|\Delta w_n|^{p(x)-2}\Delta w_n (\Delta w_n-\Delta w)+\\a(x)|w_n|^{p(x)-2}w_n(w_n-w)\Big)dx=0.
\end{eqnarray*}
Combining hypothesis $(H_0)$ and Proposition \ref{111},  we deduce that
$\{w_n\}$ converges strongly to $w$ in $X$.
Since $\Psi_{\lambda}\in C^1(X,\mathbb{R})$, we conclude that
\begin{equation}\label{19}
 d\Psi_{\lambda}(w_n)\rightarrow d\Psi_{\lambda}(w),\ \mbox{as}\ n\rightarrow \infty.
\end{equation}

Now, relations \eqref{L1770} and \eqref{19} yield
\begin{equation}\label{critical}
\Psi_{\lambda}(w)=\underline{c}<0\ \mbox{ and}\ \ d\Psi_{\lambda}(w)=0.
\end{equation}

In order to show that $w$ is a
solution of problem $(\ref{1})$, it remains to show that $\Delta w=0$ on $\partial\Omega$.
Due to relation \eqref{critical}, $w\in X\setminus\{0\}$ is a critical point of $\Psi_\lambda$, so
\begin{equation}\label{star}M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega m(x)vdx\quad\mbox{for all}\ v\in X,
\end{equation}
where
$$m(x)=\lambda (V_1(x)|w|^{q(x)-2}w-V_2(x)|w|^{\alpha(x)-2}w)-M(t)a(x)|w|^{p(x)-2}w.$$
Relation (\ref{star}) implies that
\begin{equation}\label{1star}M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega m(x)vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).\end{equation}

Let $\zeta$ be the unique solution of the problem
$$\left\{\begin{array}{lll}
&\Delta\zeta =m(x)\quad&\mbox{in}\ \Omega\\
&\zeta =0\quad&\mbox{on}\ \partial\Omega.\end{array}\right.
$$
Relation (\ref{1star}) yields
$$M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega (\Delta\zeta)vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).$$Using the Green formula we have
$$\int_\Omega (\Delta\zeta)vdx=\int_\Omega \zeta\Delta vdx.$$
Therefore
\begin{equation}\label{2star}
M(t)\int_\Omega |\Delta w|^{p(x)-2}\Delta w\Delta vdx=\int_\Omega\zeta \Delta vdx\quad\mbox{for all}\ v\in C^\infty_0(\Omega).\end{equation}

On the other hand, for all $\tilde{w}\in C^\infty_0(\Omega)$ there exists a unique $v\in C^\infty_0(\Omega)$ such that $\Delta v=\tilde{w}$ in $\Omega$. Thus, relation (\ref{2star}) can be rewritten as
$$\int_\Omega (M(t)|\Delta w|^{p(x)-2}\Delta w-\zeta)\tilde{w}dx=0\quad\mbox{for all}\ \tilde{w}\in C^\infty_0(\Omega).$$
Applying the fundamental lemma of the calculus of variations, we deduce that
$$M(t)|\Delta w|^{p(x)-2}\Delta w-\zeta=0\quad\mbox{in}\ \Omega.$$
According to assertion $(H_0)$ and since $\zeta=0$ on $\partial\Omega$, we conclude that $\Delta w=0$ on $\partial\Omega$.
Thus, $w$ is a  nontrivial weak solution of problem
\eqref{1} such that $\Delta w=0$. Since $\Psi_{\lambda}(|w|)=\Psi_{\lambda}(w)$
then problem \eqref{1} has a non-negative solution. The proof is now complete.\quad$\square$



\subsection{Example}
Let $M(t)=\beta+\gamma t$, where $\beta,\gamma$ are positive constants and $t:=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}dx+a(x)|u|^{p(x)})dx$. We first observe that $$M(t)\geq\beta >0.$$
Taking $\theta=\frac{1}{2}$, we have $$\widehat{M}(t)=\int_{0}^{t}M(s) ds=\beta t+\frac{1}{2}\gamma t^2\geq\frac{1}{2}(\beta+\gamma t)t=(1-\theta)M(t)t.$$
Consider the nonlocal problem
\begin{equation}\label{ex1}
\left\{\begin{array}{lll}M(t)(\Delta^{2}_{p(x)} u+a(x)|u|^{p(x)-2}u)&=\lambda
V_1(x)|u|^{q(x)-2}u,
\quad & x\in\Omega\\
%&u\not\equiv 0,\quad  &\mbox\in\Omega\\
&u=\Delta u=0,\quad   & x\in\partial\Omega,\\
\end{array}\right.
\end{equation}
where $a\in L^{\infty}(\Omega)$ such that ${\rm essinf}_{x\in \Omega}a(x)> 0$, the functions $p, q$ and $V_1$ satisfy hypothesis assertion $(H_2)$. Then, by Ekeland's variational principle, there exists $\lambda^*$ such that any $\lambda\in(0,\lambda^*)$ is eigenvalue for problem (\ref{ex1}). Moreover if we consider the problem
\begin{equation}\label{ex}
\left\{\begin{array}{lll}M(t)(\Delta^{2}_{p(x)} u+a(x)|u|^{p(x)-2}u)&=\lambda(
V_1(x)|u|^{q(x)-2}u-V_2(x)|u|^{\alpha(x)-2}u),
\quad & x\in\Omega\\
%&u\not\equiv 0,\quad  &\mbox\in\Omega\\
&u=\Delta u=0,\quad   & x\in\partial\Omega,\\
\end{array}\right.
\end{equation}
where the functions $p, q$, $\alpha$ and $V_2$ satisfy hypothesis assertion $(H_3)$, then there exists $\lambda_{1}^{*}$ such that any $\lambda\in(0,\lambda_{1}^{*})$ is eigenvalue for problem (\ref{ex}).

\subsection{Final comments}
(i) Problems $\eqref{p1}$ and $\eqref{1}$ correspond to a {\it subcritical} setting, as described by Remark \ref{r1} and Remark \ref{r}. We consider that valuable research directions correspond either to the {\it critical} or to the {\it supercritical} framework (in the sense of Sobolev variable exponents). No results are known even for the {\it almost critical} case with {\it lack of compactness}. More precisely, with the same notations as in  Remark \ref{r1} and \ref{r}, a very interesting open problem is to study the qualitative analysis of solutions of problem $\eqref{r1}$, provided that
there exists $z_1\in\Omega$ such that
 $$\max({r_1(z_1),s_1'(z_1)q(z_1)})=p^*(z_1),$$ but
$$\max({r_1(x),s_1'(x)q(x)})<p^{*}(x)\quad\mbox{for all}\ z\in\Omega\setminus\{z_1\},$$
and those of problem $\eqref{1}$ provided that
 there exists $z_1,\, z_2\in\Omega$ such that
 $$\max({r_1(z_1),s_1'(z_1)q(z_1)})=p^*(z_1)\quad \mbox{and}\quad \max(r_2(z_2)s_2'(z_2)\alpha(z_2))=p^*(z_2)$$ but
 $$\max({r_1(x),s_1'(x)q(x)})<p^{*}(x)\quad\mbox{for all}\ z\in\Omega\setminus\{z_1\}$$
 and $$\max(r_2(x)s_2'(x)\alpha(x))<p^{*}(x)\quad\mbox{for all}\ z\in\Omega\setminus\{z_2\}.$$

(ii)  Another very interesting research direction is to extend the approach developed in this paper to the abstract setting recently studied by Mingione {\it et al.} \cite{mingi1, mingi02, mingi2}, namely{\it double phase} problems with associated energies of the type
$$u\mapsto \int_\Omega [|\Delta u|^{p_1(x)}+V(x)|\Delta u|^{p_2(x)}]dx$$
and
$$u\mapsto \int_\Omega [|\Delta u|^{p_1(x)}+V(x)|\Delta u|^{p_2(x)}\log(e+|x|)]dx,$$
where $p_1(x)\leq p_2(x)$, $p_1\not=p_2$, and $V(x)\geq 0$. Considering two different materials with power hardening exponents $p_1(x)$ and $p_2(x)$ respectively, the coefficient $V(x)$ dictates the geometry of a composite of the two materials. When $V(x)>0$ then $p_2(x)$-material is present, otherwise the $p_1(x)$-material is the only one making the composite.

These problems extend to a bi-harmonic setting with variable exponents the pioneering papers by Marcellini \cite{marce1, marce2} on $(p,q)$-problems, which involve integral functionals of the type
$$u\mapsto\int_\Omega F(x,\nabla u)dx.$$
The integrand $F:\Omega\times{\mathbb R}^N\rightarrow {\mathbb R}$ satisfied unbalanced polynomial growth conditions of the type
$$|\xi|^p \lesssim F(x,\xi)\lesssim |\xi|^q\quad\mbox{with $1<p<q$,}$$
for every $x\in\Omega$ and $\xi\in{\mathbb R}^N$.

(iii) We suggest to extend the methods developed in this paper to the more general framework of Musielak-Orlicz-Sobolev spaces (see \cite[Chaper 4]{radrep} for a collection of stationary problems studied in these function spaces).

\subsection*{Acknowledgements} The authors thank an anonymous referee  for a careful and constructive analysis of this paper and for the remarks and comments, which have considerably improved the
initial version of the present work.
V.D.~R\u adulescu acknowledges the support through a grant  of the Romanian Ministry
of Research and Innovation, CNCS--UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130,
within PNCDI III.



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