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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 158, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/158\hfil $p(x)$-biharmonic problem]
{Existence and non-existence of solutions for a $p(x)$-biharmonic problem}

\author[G. A. Afrouzi, M. Mirzapour, N. T. Chung \hfil EJDE-2015/158\hfilneg]
{Ghasem A. Afrouzi, Maryam Mirzapour, Nguyen Thanh Chung}

\address{Ghasem A. Afrouzi \newline
Department of Mathematics,
Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}

\address{Maryam Mirzapour \newline 
Department of Mathematics,
Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{mirzapour@stu.umz.ac.ir}

\address{Nguyen Thanh Chung \newline
Department of Mathematics, 
Quang Binh University, 
312 Ly Thuong Kiet, Dong Hoi, 
Quang Binh, Vietnam}
\email{ntchung82@yahoo.com}

\thanks{Submitted July 22, 2014. Published June 15, 2015.}
\subjclass[2010]{35J60, 35B30, 35B40}
\keywords{$p(x)$-Biharmonic; variable exponent; critical points;
\hfill\break\indent minimum principle; fountain theorem; dual fountain theorem}

\begin{abstract}
 In this article, we study the following problem with Navier boundary conditions
 \begin{gather*}
 \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u
 =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\\
 u=\Delta u=0  \quad \text{on } \partial\Omega.
 \end{gather*}
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary
 $\partial \Omega$, $N\geq1$. $p(x),q(x)$ and $\gamma(x)$ are continuous
 functions on $\overline{\Omega}$, $\lambda$ and $\mu$ are parameters.
 Using variational methods, we establish some existence and non-existence
 results of solutions for this problem.
\end{abstract}

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

\section{Introduction}

In recent years, the study of differential equations and variational problems 
with $p(x)$-growth conditions was an interesting topic, which arises from 
nonlinear electrorheological fluids and elastic mechanics. In that context 
we refer the reader to Ruzicka \cite{Ru}, Zhikov \cite{Zhikov1} and the 
reference therein; see also \cite{Hamidi,Fan4,Fan5,Fan6}.

Fourth-order equations appears in many context. Some of theses problems 
come from different areas of applied mathematics and physics such as 
Micro Electro-Mechanical systems, surface diffusion on solids, 
flow in Hele-Shaw cells (see \cite{Fer}). In addition, this type of equations 
can describe the static from change of beam or the sport of rigid body.
El Amrouss et al \cite{Amr} studied a class of $p(x)$-biharmonic of the form
\begin{gather*}
 \Delta (|\Delta u|^{p(x)-2}\Delta u)=\lambda |u|^{p(x)-2}u+f(x,u) \quad 
 \text{in } \Omega,\\
 u=\Delta u=0  \quad \text{on } \partial\Omega,
\end{gather*}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, with smooth boundary
$\partial \Omega$, $N\geq 1$, $\lambda\leq 0$ and some assumptions 
on the Carath\'{e}odory function $f:\Omega\times \mathbb{R}\to \mathbb{R}$.
They obtained the existence and multiplicity of solutions.

In a recent article, Lin Li et al \cite{Lin} considered the above problem 
and using variational methods, by the assumptions on the Carath\'{e}odory 
function $f$, they establish the existence of at least one solution and 
infinitely many solutions of the problem.

Inspired by the above references and the work of Jinghua Yao \cite{Yao}, 
the aim of this article is to study the existence and multiplicity of 
weak solutions of the following fourth-order elliptic equation 
with Navier boundary conditions
\begin{equation}\label{e1.1}
\begin{gathered}
\Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u
=\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\\
u=\Delta u=0  \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary
$\partial \Omega$, $N\geq1$, $p(x),q(x)$ and $\gamma(x)$ are continuous 
functions on $\overline{\Omega}$ with 
$\inf_{x\in \overline{\Omega}}p(x)>1,\inf_{x\in \overline{\Omega}}q(x)>1$, 
$\inf_{x\in \overline{\Omega}}\gamma(x)>1$ and $\lambda$ and $\mu$ are parameters. 
Throughout the paper, we assume that $\lambda^{2}+\mu^{2}\neq 0$.

\section{Preliminaries}

To study $p(x)$-Laplacian problems, we need some results on the spaces 
$L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$, and properties of $p(x)$-Laplacian, 
which we use later.
Let $\Omega$ be a bounded domain of $\mathbb{R}^N$, denote
\[
C_+(\overline{\Omega})=\{h(x); h(x)\in C(\overline{\Omega}),
h(x)>1, \forall x\in \overline{\Omega}\}.
\]
For any $h\in C_+(\overline{\Omega})$, we define
\[
h^+=\max\{h(x);~x \in\overline{\Omega}\},\quad
h^-=\min\{h(x);~x \in\overline{\Omega}\};
\]
For any $p\in C_+(\overline{\Omega})$, we define the 
\emph{variable exponent Lebesgue space}
\begin{align*}
L^{p(x)}(\Omega)=\Big\{&u; u\textrm{ is a measurable real-valued function such that}\\
& \int_{\Omega}|u(x)|^{p(x)}dx<\infty\Big\},
\end{align*}
endowed with the so-called \emph{Luxemburg norm}
\[
|u|_{p(x)}=\inf \big\{\mu>0;~\int_{\Omega}|\frac{u(x)}{\mu}|^{p(x)}dx\leq 1\big\}.
\]
Then $(L^{p(x)}(\Omega),|\cdot|_{p(x)})$ becomes a Banach space.

\begin{proposition}[\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 $p(x)$, i.e.,
\[
\frac{1}{p(x)}+\frac{1}{q(x)}=1,
\]
for all $x\in \Omega$. For $u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$, 
we have
\[
\big|\int_{\Omega}uvdx\big|
\leq \Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)|u|_{p(x)}|v|_{q(x)}
\leq 2|u|_{p(x)}|v|_{q(x)}.
\]
\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
\[
\|u\|_{k,p(x)}=\sum_{|\alpha|\leq k}|D^{\alpha}u|_{p(x)},
\]
also becomes a separable and reflexive Banach space. For more details, 
we refer the reader to \cite{Fan,Fan2,Mih1,Yao}. Denote
\[
p_{k}^{*}(x)=\begin{cases}
\frac{Np(x)}{N-kp(x)}     & \text{if } kp(x)<N,\\
+\infty       & \text{if } kp(x)\geq N
\end{cases}
\]
for any $x\in \overline{\Omega}$, $k\geq1$.

\begin{proposition}[\cite{Fan2}]\label{prop2.2}
For $p,r\in C_{+}(\overline{\Omega})$ such that $r(x)\leq p^{*}_{k}(x)$ for
 all $x\in \overline{\Omega}$, there is a continuous embedding
\begin{align*}
W^{k,p(x)}(\Omega)\hookrightarrow L^{r(x)}(\Omega).
\end{align*}
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 $W^{k,p(x)}(\Omega)$. Note that the weak solutions of problem \eqref{e1.1} 
are considered in the generalized Sobolev space
\begin{align*}
X=W^{2,p(x)}(\Omega)\cap W_0^{1,p(x)}(\Omega)
\end{align*}
equipped with the norm
\[
\|u\|=\inf \Big\{\mu>0:\int_{\Omega}\Big(\Big|\frac{\Delta u(x)}{\mu}\Big|^{p(x)}
+\lambda\big|\frac{u(x)}{\mu}\big|^{p(x)}\Big)dx\leq 1\Big\}.
\]

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

\begin{proposition}[\cite{Amr}] \label{prop2.4}
If we denote $\rho(u)=\int_{\Omega}(|\Delta u|^{p(x)}+|u|^{p(x)})dx$, 
then for $u,u_n\in X$, we have
\begin{itemize}

\item[(1)] $\|u\|<1$ (respectively=1; $>1$) $\Longleftrightarrow$ 
$\rho(u)<1$ (respectively $=1$; $>1$);

\item[(2)] $\|u\|\leq1\Rightarrow\|u\|^{p^{+}}\leq \rho(u)\leq \|u\|^{p^{-}}$;

\item[(3)] $\|u\|\geq1\Rightarrow\|u\|^{p^{-}}\leq \rho(u)\leq\|u\|^{p^{+}}$;

\item[(4)] $\|u\|\to 0$ (respectively $\to \infty$)
$\Longleftrightarrow$ $\rho(u)\to 0$ (respectively $\to \infty$).
\end{itemize}
\end{proposition}

It is clear that the energy functional associated with \eqref{e1.1} is defined by
$$
I_{\lambda,\mu}(u)=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})dx
-\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx
-\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)}dx.
$$
Let us define the functional
\begin{align*}
J(u)=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})dx.
\end{align*}
It is well known that $J$ is well defined, even and $C^1$ in $X$. 
Moreover, the operator $L=J':X\to X^*$ defined as
\begin{align*}
\langle L(u),v\rangle=\int_{\Omega}(|\Delta u|^{p(x)-2}
\Delta u\Delta v+|u|^{p(x)-2}uv)dx
\end{align*}
for all $u,v\in X$ satisfies the following assertions.

\begin{proposition}[\cite{Amr}] \label{prop2.5}
\begin{itemize}
\item[(1)] $L$ is continuous, bounded and strictly monotone. 

\item[(2)] $L$ is a mapping of $(S_{+})$ type, namely:
$u_n\rightharpoonup u$,  and 
$\limsup_{n\to+\infty}L(u_n) (u_n-u)\leq 0$  implies $u_n\to u$.

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

\section{Main results and proofs}

In this section, we study the existence and non-existence of weak solutions 
for problem \eqref{e1.1}. We use the letter $c_i$ in order to denote a positive
constant.

\begin{theorem}\label{theo3.2}
Assume that $q(x),\gamma(x)\in C_{+}(\overline{\Omega})$ and 
$p^+<q^-\leq q(x)<p_{2}^{*}(x)$, $\gamma^+<p^-$ for any $x\in \overline{\Omega}$. 
Then we have
\begin{itemize}
\item[(i)] For every $\lambda>0$, $\mu\in \mathbb{R}$, \eqref{e1.1}
 has a sequence of weak solutions $(\pm u_k)$ such that 
  $I_{\lambda,\mu}(\pm u_k)\to+\infty$ as $k\to+\infty$.
  
\item[(ii)] For every $\mu>0$, $\lambda\in \mathbb{R}$, \eqref{e1.1}
 has a sequence of weak solutions $(\pm v_k)$ such that 
  $I_{\lambda,\mu}(\pm v_k)<0$  and $I_{\lambda,\mu}(\pm v_k)\to0$
  as $k\to+\infty$.
\item[(iii)] For every $\lambda<0$, $\mu<0$, \eqref{e1.1} has no 
nontrivial weak solution.
\end{itemize}
\end{theorem}

We will use the following Fountain theorem to prove (i) and the
 Dual of the Fountain theorem to prove (ii).
 
\begin{lemma}[\cite{Zhao}]\label{lem3.3}
Let $X$ be a reflexive and separable Banach space, then there exist 
$\{e_j\}\subset X$ and $\{e^*_j\}\subset X^*$  such that
\[
X=\overline{\operatorname{span}\{e_j:j=1,2,\dots\}},\quad
X^*=\overline{\operatorname{span}\{e_j^*:j=1,2,\dots\}},
\]
and
$$ 
\langle e_i,e_j^*\rangle=\begin{cases}
1    & \text{if } i=j,\\
0    & \text{if } i\neq j,
\end{cases}
$$
\end{lemma}

We define
\begin{equation}\label{e3.2}
X_j=\operatorname{span}\{e_j\},\quad
Y_k=\oplus_{j=1}^{k}X_j,\quad 
Z_k=\overline{\oplus_{j=k}^{\infty}X_j}.
\end{equation}
Then we have the following Lemma.

\begin{lemma}[\cite{Amr}] \label{lem3.4}
If $q(x),\gamma(x)\in C_{+}(\overline{\Omega}),~q(x)<p^{*}_{2}(x)$, and 
$\gamma(x)<p_{2}^{*}(x)$ for all $x\in \overline{\Omega},$ denote
\begin{gather*}
\beta_{k}=\sup\{|u|_{q(x)};\|u\|=1,\, u\in Z_k\}\\
\theta_{k}=\sup\{|u|_{\gamma(x)}; \|u\|=1,\,u\in Z_k\},
\end{gather*}
then $\lim_{k\to\infty}\beta_k=0$, $\lim_{k\to\infty}\theta_k=0$.
\end{lemma}

\begin{lemma}[Fountain Theorem \cite{Will}] \label{lem3.5}
Let
\begin{itemize}
\item[(A1)] $I\in C^{1}(X,\mathbb{R})$ be an even functional, where $(X,\|\cdot\|)$
is a separable and reflexive Banach space, the subspaces $X_k,~Y_k$ and $Z_k$ 
are defined by \eqref{e3.2}.
If for each $k\in \mathbb{N}$, there exist $\rho_k>r_k>0$ such that
\item[(A2)] $\inf \{I(u)~:~u\in Z_k,~\|u\|=r_k\}\to+\infty$ as $k\to+\infty$.
\item[(A3)] $\max  \{I(u)~:~u\in Y_k,~\|u\|=\rho_k\}\leq0$.
\item[(A4)] $I$ satisfies the (PS) condition for every $c>0.$
\end{itemize}
Then $I$ has an unbounded sequence of critical points.
\end{lemma}

\begin{lemma}[Dual Fountain Theorem \cite{Will}] \label{lem3.6}
Assume {\rm (A1)} is satisfied and there is $k_0>0$ such that,
for each $k\geq k_0$, there exist $\rho_k>r_k>0$ such that
\begin{itemize}
\item[(B1)] $a_k=\inf \{I(u):u\in Z_k,\,\|u\|=\rho_k\}\geq0$.
\item[(B2)] $b_k=\max  \{I(u):u\in Y_k,\,\|u\|=r_k\}<0$.
\item[(B3)] $d_k=\inf \{I(u): u\in Z_k,\,\|u\|\leq\rho_k\} \to0$ as $k\to+\infty$.
\item[(B4)] $I$ satisfies the $(PS)_{c}^{*}$ condition for every $c\in[d_{k_0},0)$.
\end{itemize}
Then $I$ has a sequence of negative critical values converging to $0$.
\end{lemma}

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

\subsection*{Proof of Theorem \ref{theo3.2}}
(i) First we verify $I_{\lambda,\mu}$ satisfies the (PS) condition. 
Suppose that $(u_n)\subset X$ is (PS) sequence, i.e.,
$$
|I_{\lambda,\mu}(u_n)|\leq c_9,\quad
I'_{\lambda,\mu}(u_n)\to 0\quad \text{as } n\to\infty.
$$
By Propositions \ref{prop2.2} and \ref{prop2.1}, we know that if we denote
$$
\phi(u)=-\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)},dx,\quad
\psi(u)=-\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)},dx,
$$
then they are both weakly continuous and their derivative operators are compact. 
By Proposition \ref{prop2.5}, we deduce that $I'_{\lambda,\mu}=L+\phi'+\psi'$ 
is also of type $(S_{+})$. Thus it is sufficient to verify that $(u_n)$ 
is bounded. Assume $\|u_n\|>1$ for convenience. For $n$ large enough, we have
\begin{align}\label{e3.5}
\begin{split}
&c_9+1+\|u_n\|\\
&\geq I_{\lambda,\mu}(u_n)-\frac{1}{q^-}\langle I'_{\lambda,\mu}(u_n),u_n\rangle\\
&=\Big{[}\int_{\Omega}\frac{1}{p(x)}(|\Delta u_n|^{p(x)}
 +|u_n|^{p(x)})dx-\lambda\int_{\Omega}\frac{1}{q(x)}|u_n|^{q(x)}dx
 -\mu\int_{\Omega}\frac{1}{\gamma(x)}|u_n|^{\gamma(x)}dx\Big{]}\\
&\quad -\frac{1}{q^-}\Big{[}\int_{\Omega}(|\Delta u_n|^{p(x)}
 +|u_n|^{p(x)})dx-\lambda\int_{\Omega}|u_n|^{q(x)}dx
  -\mu\int_{\Omega}|u_n|^{\gamma(x)}dx\Big{]}\\
&\geq \Big(\frac{1}{p^+}-\frac{1}{q^-}\Big)||u_n||^{p^-}
 -c_{10}\|u_n\|^{\gamma^+}.
\end{split}
\end{align}
Since $q^->p^+$ and $p^->\gamma^+$, we know that $\{u_n\}$ is bounded in $X$. 
In the following we will prove that if $k$ is large enough, then there 
exist $\rho_k>r_k>0$ such that (A2) and (A3) hold.

(A2) For any $u\in Z_k$, $\|u\|=r_k>1$ ($r_k$ will be specified below), we have
\begin{align*}
I_{\lambda,\mu}(u)
&=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})dx
 -\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx
 -\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)}dx\\
&\geq \frac{1}{p^+}\|u\|^{p^-}
 -\frac{\lambda}{q^-}\int_{\Omega}|u|^{q(x)}dx
 -\frac{c_{11}|\mu|}{\gamma^-}\|u\|^{\gamma^+}.
\end{align*}
Since $p^->\gamma^+$, there exists $r_0>0$ large enough such that 
$\frac{c_{11}|\mu|}{\gamma^-}\|u\|^{\gamma^+}\leq\frac{1}{2p^+}\|u\|^{p^-}$ 
as $r=\|u\|\geq r_0$. If $|u|_{q(x)}\leq 1$ then 
$\int_{\Omega}|u|^{q(x)}dx\leq |u|_{q(x)}^{q^-}\leq 1$. However, if 
$|u|_{q(x)}> 1$ then $\int_{\Omega}|u|^{q(x)}dx\leq |u|_{q(x)}^{q^+}
\leq(\beta_k\|u\|)^{q^+}$. So, we conclude that
\begin{align*}
I_{\lambda,\mu}(u) 
&\geq \begin{cases}
\frac{1}{2p^+}\|u\|^{p^-}-\frac{\lambda c_{12}}{q^-} 
&\text{ if } |u|_{q(x)}\leq 1, \\
 \frac{1}{2p^+}\|u\|^{p^-}-\frac{\lambda }{q^-}(\beta_k\|u\|)^{q^+}
 & \text{if } |u|_{q(x)}> 1.
\end{cases}
\\
&\geq \frac{1}{p^+}\|u\|^{p^-}-\frac{\lambda }{q^-}(\beta_k\|u\|)^{q^+}
-c_{13},
\end{align*}
choose $r_k=\Big(\frac{2\lambda }{q^-}q^+\beta_k^{q^+}\Big)^{\frac{1}{p^{-}-q^+}}$, 
we have
\[
I_{\lambda,\mu}(u)=\frac{1}{2}\Big(\frac{1}{p^+}-\frac{1}{q^+}
\Big)r_k^{p^-}-c_{13}\to \infty\quad \text{ as} k\to\infty,
\]
because of $p^+<q^-\leq q^+$ and $\beta_k\to 0$.

(A3) Let $u\in Y_k$ such that $\|u\|=\rho_k>r_k>1$. Then
\begin{align*}
I_{\lambda,\mu}(u)
&=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})dx
 -\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx
 -\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)}dx\\
&\leq\frac{1}{p^-}\|u\|^{p^+}-\frac{\lambda}{q^+}\int_{\Omega}|u|^{q(x)}dx
 +\frac{|\mu|}{\gamma^-}\int_{\Omega}|u|^{\gamma(x)}dx.
\end{align*}
Since $\operatorname{dim}Y_k<\infty$, all norms are equivalent in $Y_k$, 
we obtain
\begin{align*}
I_{\lambda,\mu}(u)\leq\frac{1}{p^-}\|u\|^{p^+}-\frac{\lambda}{q^+}\|u\|^{q^-}
+\frac{|\mu|}{\gamma^-}\|u\|^{\gamma^+}.
\end{align*}
We get that: $I_{\lambda,\mu}(u)\to -\infty$ as $\|u\|\to +\infty$ 
since $q^->p^+$ and $\gamma^+<p^-$. So (A2) holds. 
From the proof of (A2) and (A3), we can choose $\rho_k>r_k>0$. 
Obviously $I_{\lambda,\mu}$ is even and the proof of (i) is complete.
\smallskip

(ii) We use the Dual Fountain theorem to prove conclusion (ii). 
Now we prove that there exist $\rho_k>r_k>0$ such that if $k$ is large 
enough (B1), (B2) and (B3) are satisfied.

(B1) For any $u\in Z_k$ we have
\begin{align*}
I_{\lambda,\mu}(u)&=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})d
-\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx
-\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)}dx\\
&\geq \frac{1}{p^+}\|u\|^{p^+}-\frac{c_{14}|\lambda|}{q^-}\|u\|^{q^-}
-\frac{\mu}{\gamma^-}\int_{\Omega}|u|^{\gamma(x)}dx.
\end{align*}
Since $q^->p^+$, there exists $\rho_0>0$ small enough such that
 $\frac{c_{14}|\lambda|}{q^-}\|u\|^{q^-}\leq \frac{1}{2p^+}\|u\|^{p^+}$ 
 as $0<\rho=\|u\|\leq \rho_0$. Then from the proof above, we have
\begin{equation}\label{e3.4}
I_{\lambda,\mu}(u) \geq\begin{cases}
\frac{1}{2p^+}\|u\|^{p^+}-\frac{\mu c_{15}}{\gamma^-} 
 & \text{if } |u|_{\gamma(x)}\leq 1, \\
\frac{1}{2p^+}\|u\|^{p^+}-\frac{\mu}{\gamma^-}(\theta_k\|u\|)^{\gamma^+}
& \text{if } |u|_{\gamma(x)}> 1.
\end{cases}
\end{equation}
Choose $\rho_k=\big(\frac{2p^+\mu\theta_k^{\gamma^+}}{\gamma^-}
\big)^{\frac{1}{p^+-\gamma^+}}$, then
\[
I_{\lambda,\mu}(u)=\frac{1}{2p^+}(\rho_k)^{p^+}
 -\frac{1}{2p^+}(\rho_k)^{p^+}=0.
\]
Since $p^->\gamma^+$, $\theta_k\to 0$, we know $\rho_k\to 0$ as $k\to\infty$.

(B2) For $u\in Y_k$ with $\|u\|\leq 1$, we have
\begin{align*}
I_{\lambda,\mu}(u)
&=\int_{\Omega}\frac{1}{p(x)}(|\Delta u|^{p(x)}+|u|^{p(x)})dx
 -\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx
 -\mu\int_{\Omega}\frac{1}{\gamma(x)}|u|^{\gamma(x)}dx\\
&\leq \frac{1}{p^-}\|u\|^{p^-}+\frac{|\lambda|}{q^-}\int_{\Omega}|u|^{q(x)}dx
 -\frac{\mu}{\gamma^+}\int_{\Omega}|u|^{\gamma(x)}dx.
\end{align*}
Since dim$Y_k=k$, conditions $\gamma^+<p^-$ and $p^+<q^-$ imply that there exists 
a $r_k\in (0,\rho_k)$ such that $I_{\lambda,\mu}(u_n)<0$ when $\|u\|=r_k$. 
So we obtain
\[
\max_{u\in Y_k,\|u\|=r_k}I_{\lambda,\mu}(u)<0,
\]
i.e., (B2) is satisfied.

(B3) Because $Y_k\cap Z_k\neq\emptyset$ and $r_k<\rho_k$, we have
\[
d_k=\inf_{u\in Z_k,\|u\|\leq\rho_k} I_{\lambda,\mu}(u)
\leq b_k=\max_{u\in Y_k,\|u\|=r_k}I_{\lambda,\mu}(u)<0.
\]
From \eqref{e3.4} , for $u\in Z_k$, $\|u\|\leq \rho_k$ small enough we can write
\[
I_{\lambda,\mu}(u)
\geq \frac{1}{2p^+}\|u\|^{p^+}-\frac{\lambda}{\gamma^-}\theta_k^{\gamma^+}
 \|u\|^{\gamma^+}
\geq -\frac{\lambda}{\gamma^-}\theta_k^{\gamma^+}\|u\|^{\gamma^+},
\]
Since $\theta_k\to 0$ and $\rho_k\to 0$ as $k\to\infty$, (B3) holds.
Finally we verify the $(PS)_{c}^{*}$ condition. Suppose 
$\{u_{n_{j}}\}\subset X$ such that 
$$
n_{j}\to +\infty,\quad  u_{n_{j}}\in Y_{n_{j}},\quad 
I_{\lambda,\mu}(u_{n_{j}})\to c_{16},\quad 
(I_{\lambda,\mu}|_{Y_{n_{j}}})'(u_{n_{j}})\to 0.
$$
If $\lambda\geq 0$, similar to \eqref{e3.5}, we can get the boundedness 
of $\|u_{n_{j}}\|$. Assume $\|u_{n_{j}}\|\geq1$ for convenience. 
If $\lambda<0$, for $n>0$ large enough, we have
\begin{align*}
c_{16}+1+\|u_{n_{j}}\|
&\geq I_{\lambda,\mu}(u_{n_{j}})-\frac{1}{q^+}
 \langle I'_{\lambda,\mu}(u_{n_{j}}),u_{n_{j}}\rangle\\
&=\Big[\int_{\Omega}\frac{1}{p(x)}(|\Delta u_{n_{j}}|^{p(x)}+|u_{n_{j}}|^{p(x)})dx
 -\lambda\int_{\Omega}\frac{1}{q(x)}|u_{n_{j}}|^{q(x)}dx\\
&\quad -\mu\int_{\Omega}\frac{1}{\gamma(x)}|u_{n_{j}}|^{\gamma(x)}dx\Big{]}
 -\frac{1}{q^+}\Big{[}\int_{\Omega}(|\Delta u_{n_{j}}|^{p(x)}
  +|u_{n_{j}}|^{p(x)})dx\\
&\quad -\lambda\int_{\Omega}|u_{n_{j}}|^{q(x)}dx
 -\mu\int_{\Omega}|u_{n_{j}}|^{\gamma(x)}dx\Big]\\
&\geq \Big(\frac{1}{p^+}-\frac{1}{q^+}\Big)\|u_{n_{j}}\|^{p^-}
 -c_{17}\|u_{n_{j}}\|^{\gamma^+}.
\end{align*}
Since $p^->\gamma^+$ and $q^+>p^+$, we know that $\{u_{n_{j}}\}$ is bounded 
in $X$. Hence there exists $u\in X$ such that $u_{n_{j}}\to u$ in $x$. 
Observe now that $X=\overline{\cup_{n_j}Y_{n_{j}}}$, then we can find 
$v_{n_{j}}\in Y_{n_{j}}$ such that $v_{n_{j}}\to u$. We have
\begin{align*}
\langle I'_{\lambda,\mu}(u_{n_{j}}),u_{n_{j}}-u\rangle
=\langle I'_{\lambda,\mu}(u_{n_{j}}),u_{n_{j}}-v_{n_{j}}\rangle
+\langle I'_{\lambda,\mu}(u_{n_{j}}),v_{n_{j}}-u\rangle.
\end{align*}
Having in mind that $(u_{n_{j}}-v_{n_{j}})\in Y_{n_{j}}$, it yields
\begin{equation}\label{e3.6}
\langle I'_{\lambda,\mu}(u_{n_{j}}),u_{n_{j}}-u\rangle
=\langle (I_{\lambda,\mu}|_{Y_{n_{j}}})'(u_{n_{j}}),u_{n_{j}}-v_{n_{j}}\rangle
+\langle I'_{\lambda,\mu}(u_{n_{j}}),v_{n_{j}}-u\rangle
\to 0
\end{equation}
as $n\to\infty$.
By Proposition \ref{prop2.5}, the operator $I'_{\lambda,\mu}$  is obviously 
of $(S_+)$ type. Using this fact with \eqref{e3.6}, we deduce that 
$u_{n_{j}}\to u$ in $X$, furthermore 
$I'_{\lambda,\mu}(u_{n_{j}})\to I'_{\lambda,\mu}(u)$.

We claim now that $u$ is in fact a critical point of $I_{\lambda,\mu}$. 
Taking $\omega_k\in Y_k$, notice that when $n_j\geq k$ we have
\begin{align*}
\langle I'_{\lambda,\mu}(u),\omega_k\rangle
&=\langle I'_{\lambda,\mu}(u)-I'_{\lambda,\mu}(u_{n_{j}}),\omega_k\rangle
 +\langle I'_{\lambda,\mu}(u_{n_{j}}),\omega_k\rangle\\
&=\langle I'_{\lambda,\mu}(u)-I'_{\lambda,\mu}(u_{n_{j}}),\omega_k\rangle
 +\Big{\langle}(I_{\lambda,\mu}|_{Y_{n_{j}}})'(u_{n_{j}}),\omega_k\Big{\rangle}.
\end{align*}
Going to the limit on the right side of the above equation reaches
\[
\langle I'_{\lambda,\mu}(u),\omega_k\rangle=0,\quad \forall \omega_k\in Y_k,
\]
so $I'_{\lambda,\mu}(u)=0$, this show that $I_{\lambda,\mu}$ satisfies the 
$(PS)_{c}^{*}$ condition for every $c\in \mathbb{R}$.

(iii) Assume for the sake of contradiction, $u\in X\backslash\{0\}$ is a 
weak solution of problem \eqref{e1.1}. Then multiplying the equation 
in \eqref{e1.1} by $u$, integrating by parts we obtain
\[
\int_{\Omega}(|\Delta u|^{p(x)}+|u|^{p(x)})dx
=\lambda \int_{\Omega}|u|^{q(x)}+\mu\int_{\Omega}|u|^{\gamma(x)}.
\]
This leads to contradiction and the proof is complete.


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\end{document}