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\begin{document}
\title{Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids}
%\title{Solutions for $p(x)$-Laplacian equations without AR-condition via Morse theory}
\author{Vicen\c{t}iu D. R\u{a}dulescu$^{\mathrm{a},\mathrm{b}}$\ and\ Binlin Zhang$^{\mathrm{c},}$\\
\small $^{\mathrm{a}}$ Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania\\
\small $^{\mathrm{b}}$ Department of Mathematics, University of Craiova,  200585 Craiova, Romania\\
\small $^{\mathrm{c}}$ Department of Mathematics, Heilongjiang Institute of Technology,
 150050 Harbin, China} \date{ }
 \maketitle


\begin{abstract}
{Many electrorheological fluids are suspensions of solid particles that are exposed to a strong electric field. This causes a dramatic increase of their effective viscosity. In this paper we are concerned with a mathematical problem that is related with this non-Newtonian behavior. More precisely, we study the nonlinear stationary equation $-\mathrm{div}\,(|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u=f(x,u)$ in $\Omega$, under Dirichlet boundary conditions,  where $\Omega$ is a smooth bounded domain in ${\mathbb R}^n$, $p>1$ is a continuous function, and $f(x,u)$ has a sublinear growth near the origin. Under various natural assumptions,
by using the Morse theory in combination with
 local linking arguments, we obtain the existence of nontrivial weak solutions.}
\\
\emph{\bf Keywords:} $p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; critical groups; Morse theory.\\
\emph{\bf 2010 Mathematics Subject Classification:} 35J20; 35J66; 46E30; 58E05.
\end{abstract}

%{\section*{{\bf 1. Introduction }}} \setcounter{section}{1}
 %\setcounter{equation}{0}
\section{Introduction and main results}
The qualitative analysis of nonlinear partial differential equations involving  differential operators with variable exponent
 is motivated by wide applications to various fields.
Materials requiring such more advanced theory have been studied
experimentally since the middle of last century. The first major
discovery
on electrorheological fluids is due to Willis Winslow in  1949. These
fluids have the
interesting property that their effective viscosity depends on the electric field
in
the fluid. The dramatic increase of the effective viscosity (or shear stress) is due to the existence of special particle structures that appear in the presence of an electric field hindering the flow.
Winslow noticed that in such fluids (for instance lithium
polymetachrylate) the effective viscosity in an electrical field is inversely
proportional to the strength of the field. The field induces
string-like formations in the fluid, which are parallel to the field. They can
raise the effective viscosity by as much as five orders of magnitude. This phenomenon
is known as the {\it Winslow effect}. For a general account of the underlying physics
we refer to Halsey \cite{hal}. For overviews of microscopic models in relationship with applications to electrorheology we refer the reader to Parthasarathy and Klingenberg \cite{partha} and R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka \cite{ruzi}.
Electrorheological fluids have been used in robotics and space technology.
The experimental research has been done mainly in the USA, for instance
in NASA laboratories.
We also point out an interesting recent mathematical model developed by Rajagopal and R$\mathring{\mathrm{u}}$\v{z}i\v{c}ka \cite{RR}.
The model takes into account the delicate interaction between
the electromagnetic fields and the moving fluids. Particularly, in the
context of continuum mechanics, these fluids are seen as non-Newtonian
fluids. Other relevant applications of nonlinear equations involving differential operators with variable exponent include obstacle problems \cite{ob2,zhikov}, porous medium equation \cite{ant1,ant2,ant3,ob1}, and Kirchhoff problems \cite{autuori1, autuori2}.

In the recent paper \cite{refa} it is introduced a method to study the existence and multiplicity of solutions for a class of nonlinear stationary
partial differential equations described by a non-standard differential operator with variable exponent. We continue this analysis in the present paper,
where we  describe how powerful tools in modern nonlinear analysis (Morse theory and linking theory) can be used to establish some qualitative properties of solutions to these equations.

\subsection{Statement of the problem}
The goal of this paper is to investigate the existence of nontrivial solutions
to the following $p(x)$-Laplacian problem:
$$\left\{
  \begin{array}{ll}
    -\triangle_{p(x)}u+|u|^{p(x)-2}u:=-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u=f(x,u) & \quad\mbox{in}\ \Omega, \\
    u=0 & \quad \mbox{on}\ \partial\Omega,
  \end{array}
\right.\eqno(1.1)$$ where $\Omega$ is a bounded domain in
$\mathbb{R}^n$ with smooth boundary $\partial\Omega$. Let $p\in C
(\overline{\Omega})$ and $1 < p_{-} := \min_{x\in\bar{\Omega}}p(x)
\leq p(x) \leq p_{+} := \max_{x\in\bar{\Omega}} p(x) < \infty$ and
$F(x,t)=\int_0^t f(x,s)ds$, $\mathcal {F}(x,t)=f(x,t)t-p_+ F(x,t)$.
For simplicity, we denote by $C$ and $C_k$ various positive
constants whose exact value is irrelevant.

We assume that the reaction term $f(x,u)$ satisfies the following hypotheses:
\\
(H1) $f\in C(\overline\Omega\times \mathbb{R})$ with $f(x,0)=0$ and there
exists $C_1 >0$ such that $$|f(x,t)|\leq C_1 (1+|t|^{q(x)-1}),\ \
\forall(x,t)\in \Omega \times \mathbb{R},$$ where $q(x)\in
C(\overline{\Omega})$, $1<q(x)<p^* (x)$ for all
$x\in\overline{\Omega}$ and $p^* =\frac{Np(x)}{N-p(x)}$ if $p(x)<N$,
$p^* (x)=+\infty$ if $p(x)\geq N$;
\\
(H2) $\lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{p_+}}=+\infty$
uniformly for $x\in \bar{\Omega}$;
\\
(H3) there exists $\theta\geq1$ such that $\theta \mathcal
{F}(x,t)\geq \mathcal {F}(x,st)$ for $(x,t)\in \Omega\times
\mathbb{R}$ and $s\in[0,1]$;
\\
(H4) there exists $\nu>0$ such that $$\frac{f(x,t)}{|t|^{p_{+}-2}t}\
\ \ \mathrm{is}\ \mathrm{increasing}\ \mathrm{in}\ t\geq\nu\
\mathrm{and}\ \mathrm{decreasing}\ \mathrm{in}\ t\leq -\nu;$$
\\
(H5) there are small constants $r$ and $R$ with $0<r<R$ such that
$$C_2 |t|^{\alpha(x)}\leq p(x)F(x,t)\leq C_3|t|^{p(x)},\ \
\mathrm{for}\ t\in\mathbb{R}\  \mathrm{with}\  r\leq |t| \leq R,\
\mathrm{a.e.}\  x\in\Omega, \eqno(1.2)$$ where $C_2$ and $C_3$ are
constants with $0<C_2<C_3<1$, $\alpha(x)\in C(\bar{\Omega})$ and
$1<\alpha(x)<p(x)$. Moreover, there exists $C_4>0$ such that
$$F(x,t)\geq -C_4|t|^{p_{+}}\ \  \mathrm{for}\  \mathrm{all} \  (x,t)\in \Omega\times
\mathbb{R}. \eqno(1.3)$$

The assumption (H2) implies that the problem
(1.1) is superlinear at infinity. A lot of works concerning
superlinear elliptic boundary value problem have been done by using
the usual Ambrosetti-Rabinowitz condition, which is originally due
to Ambrosetti and Rabinowitz for the case $p=2$ in \cite{2}, that is,\\
(AR): there exist $\mu>p_+$ and $M>0$ such that
$$0<p_+ F(x,t)\leq f(x,t)t\ \mathrm{for}\ \mathrm{all} \ x\in \Omega\  \mathrm{and}\ |t|\geq M.                 \eqno(1.4)$$
From (1.4) it follows that for some $a,\ b>0$
$$ F(x,t)\geq a |t|^{\mu}-b\ \ \mathrm{for}\ (x,t)\in \Omega\times  \mathbb{R}. \eqno(1.5)$$
Obviously, (1.5) implies the much weaker condition (H2).

Let us consider the following function (for simplicity we drop the
$x$-dependence):
$$f(x,t)=|t|^{p_{+}-2}t\left(p_{+}{\mathrm{log}(1+|t|)}+\frac{|t|}{1+|t|}\right),\eqno(1.6)$$
then $F(x,t)= |t|^{p_{+}}\mathrm{log}(1+|t|)$. Then $f$ does not satisfy (AR) condition for any $\mu>p_+$,
but it satisfies our conditions (H2) and (H3). Furthermore, we can
show that the function fulfills all hypotheses (H1)--(H5).

In \cite{30}, Tan and Fang studied (1.1) under the
assumptions (H1), (H2), (H3), (1.3) and \\
(F):\ There exists $\delta>0$ such that $F(x,t)\leq 0$ for all $x\in
\Omega$\ and\ $|t|\leq \delta.$\\
Using Morse theory, they obtained the existence of one nontrivial
weak solution in $W^{1,p(x)}_{0}{(\Omega)}$. However, the function
$f(x,t)$ in (1.6) does not satisfy the condition (F). Liu and Su
\cite{26} used Morse theory and local linking to study the existence of
multiple nontrivial solutions for $p$-Laplacian equations under the following assumption:\\
(L):\ There is some $r>0$ small and
$\lambda_{1}<\hat{\lambda}<\bar{\lambda}$ such that
$$\lambda_{1}|t|^{p}\leq pF(x,t)\leq \hat{\lambda}|t|^{p},\ \mathrm{for}\ \mathrm{a.e.} \ x\in \Omega\  \mathrm{and}\ |t|\leq
r,$$ where $\lambda_{1}$ is the first eigenvalues of
$-\triangle_{p}$ on $W^{1,p}_{0}{(\Omega)}$. More attention has been
 focused on the existence of nontrivial solutions for $p$-Laplacian
equations via Morse theory, see for example \cite{10, 24, 25, 26, 27, 28} and
reference therein. In particular, Guo and Liu \cite{20} studied the
existence and multiplicity of nontrivial solutions for a class of
$p$-sublinear $p$-Laplacian equations. Inspired by their works, we
introduce the assumption (1.2). Here the assumption (1.2) means that
$f(x,t)$ is sublinear near zero, that is, the function behaves like
$O(|u|^{\alpha(x)-2}u)$ near zero for some $\alpha(x)\in
C(\bar{\Omega})$ and $1<\alpha(x)<p(x)$. To the best of our
knowledge, our results are new even for the case $p(x)\equiv p$.

\subsection{Main results}
Our first two results in this paper are the following theorems.\\
\\
\textbf{Theorem 1.}\begin{itshape} Assume that (H1), (H2), (H3)
and (H5) hold.
Then the problem (1.1) has at least one nontrivial weak solution in $W^{1,p(x)}_{0}{(\Omega)}$.\end{itshape}\\
\\
\textbf{Theorem 2.} \begin{itshape}Assume that (H1), (H2), (H4)
and (H5) hold. Then the problem (1.1) has at least one nontrivial
weak solution in
$W^{1,p(x)}_{0}{(\Omega)}$.\end{itshape}\\


Before the statement of Theorem 3, we recall some known results
about the eigenvalues of $-\triangle_{p(x)}$ on
$W^{1,p(x)}_{0}{(\Omega)}$. We say that $\lambda$ is an eigenvalue
of $-\triangle_{p(x)}$ with Dirichlet boundary conditions means that
the following boundary value problem:
$$\left\{
  \begin{array}{ll}
    -\triangle_{p(x)}u=\lambda|u|^{p(x)-2}u & x\in\Omega, \\
    u=0 & x\in\partial\Omega,
  \end{array}
\right.\eqno(1.7)$$ has nonzero solution. Fan et al. \cite{19} obtained
the principal eigenvalue $\lambda_{\ast}>0$ by introducing the
following condition: \\
(P): there exists a vector $l\in \mathbb{R}^{N}\backslash\{0\}$ such
that for any $x\in\Omega$, $c(t)=p(x+tl)$ is monotone in $t\in
I_{x}=\{t:x+tl\in\Omega\}.$ \\
\\
\textbf{Theorem 3.}\begin{itshape} Assume that conditions (P), (H1), (H5) are fulfilled and\\
(H6):\ $\limsup_{|t|\rightarrow\infty}\frac{p(x)F(x,t)}{|t|^{p(x)}}<
\frac{p_{-}}{p_{+}}{\lambda_{\ast}}+1$\ uniformly on  $x \in
\bar{\Omega};$ \\
(H7):\ $F(x,t)\geq 0$ for all $x\in
\Omega$\ and\ $|t|\leq r.$\\
 Then the problem (1.1) has at least two nontrivial weak
solutions in $W^{1,p(x)}_{0}{(\Omega)}$.\end{itshape}\\

The following example  illustrates the hypotheses of Theorem 3.\\
\\
\textbf{Example 1.} Let $\Omega=B(x_{0},1):=\{x\in\mathbb{R}^{N}:
|x-x_{0}|<1\}$, where $x_{0}=(x_{01},x_{02},\ldots,x_{0N})$,
$x_{0k}>1, k=1,2,\ldots,N$, $p(x)=|x|$ for all $x\in\Omega$. Then
$p(x)$ satisfies the condition (P). If we take
$f(x,t)=\beta|t|^{p_{-}-2}t,$ $0<
\beta<\frac{p_{-}}{p_{+}}{\lambda_{\ast}}+1$, then it is easy to
verify that $f(x,t)$ satisfies all assumptions of Theorem 3.\\

The paper is organized as follows. In section 2, we present some
necessary preliminary knowledge. In section 3, we verify the $(C)$
condition and $(PS)$ condition. In section 4, we compute the
critical groups at zero
and at infinity. In section 5, we give the proof of Theorems 1-3.



%{\section*{{\bf 2. Preliminaries}}}
\section{Preliminary results}
Most materials can be modeled with sufficient accuracy using classical
Lebesgue and Sobolev spaces, $L^p$ and $W^{1,p}$, where $p$ is a fixed
constant.
With the emergence of nonlinear problems in applied sciences,
Lebesgue and Sobolev spaces $L^p$ and $W^{1,p}$ have
demonstrated their limitations in applications. A class of
nonlinear problems with variable exponent is a new
research field and reflects a new kind of physical phenomena.
For instance, for some materials with inhomogeneities, e.g., the
electrorheological fluids (sometimes referred to as ``smart fluids"),
this
is not adequate, but rather the exponent $p$ should be able to vary.

To the best of our knowledge, variable
exponent Lebesgue spaces appeared in the literature for the
first time already in a 1931 article by W.~Orlicz \cite{orl}. In
the 1950's this study was carried on by Nakano \cite{nak}, who
made the first systematic study of spaces with variable exponent
(called {\it modular spaces}). Nakano mentioned explicitly variable
exponent Lebesgue spaces as an example of more general spaces he
considered, see Nakano  \cite[p.~284]{nak}.
Later, the Polish mathematicians investigated the modular function
spaces (see e.g. Musielak \cite{M}).
Since Kov\'{a}\v{c}ik and R\'{a}kosn\'{i}k  discussed
$L^{p(x)}$ space and $W^{k,p(x)}$ space in \cite{23}, many results have
been obtained concerning this kind of variable exponent spaces, see
for instance \cite{7, 8, 9, 17, 18} for basic properties of variable
exponent spaces and [11-16] for the applications of variable
exponent spaces on elliptic equations. For the application
backgrounds of the $p(x)$-Laplacian equations we refer to \cite{29}. For
an overview of variable exponent spaces with various applications to
differential equations we refer to \cite{21}.

In this section we first recall
some necessary facts about variable exponent spaces. For the details
we refer to \cite{7, 18, 23} and the reference therein.


Let $\mathcal {P}(\Omega)$ be the set of all Lebesgue measurable
functions $p :\Omega \rightarrow (1,+\infty)$ and
$$||u||_{p(x)}=\inf \left\{ t>0: \int_{\Omega}\Big|\frac{u}{t}\Big|^{p(x)}dx\leq 1 \right\}.\eqno(2.1)$$
The variable exponent Lebesgue space $L^{p(x)}(\Omega)$  is the
class of all functions $u$ such that
$\int_{\Omega}|u(x)|^{p(x)}dx<\infty$. Then the variable exponent
Sobolev space $W^{1,p(x)}(\Omega)$ is the class of all functions
$u\in L^{p(x)}(\Omega)$ such that $|\nabla u|\in L^{p(x)}(\Omega)$
with the norm
$$||u||_{1,p(x)}=||u||_{p(x)}+||\nabla u||_{p(x)}.\eqno(2.2)$$
For $u\in L^{p(x)}(\Omega)$,  we define
$$|||u|||=\inf\Big\{ t>0:\int_{\Omega} \frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{t^{p(x)}}dx\leq1\Big\} \,.                               \eqno(2.3)$$
We deduce that
$$\frac{1}{2}||u||_{1,p(x)}\leq|||u|||\leq2||u||_{1,p(x)}.\eqno(2.4)$$
With these norms, the function spaces $L^{p(x)}(\Omega)$ and
$W^{1,p(x)}(\Omega)$  are separable and reflexive Banach spaces. We
denote by $W^{1,p(x)}_0 (\Omega)$  the closure of $C^{\infty}_0
(\Omega)$  in $W^{1,p(x)}(\Omega)$, then  $W^{1,p(x)}_0 (\Omega)$
is also a separable and reflexive Banach space.\\
\\
\textbf{Proposition 1.} \begin{itshape} (see \cite{18}) For all $u \in L^{p(x)}(\Omega)$, we define $\rho(u)=\int_{\Omega}|u(x)|^{p(x)}dx$.  Then the following properties are fulfilled:
\\
(1) If $||u||_{p(x)} \geq 1$, then $||u||^{p_{-}}_{p(x)} \leq \rho(u) \leq ||u||^{p_{+}}_{p(x)}$.\\
(2) If $||u||_{p(x)} < 1$, then $||u||^{p_{+}}_{p(x)} \leq \rho(u) \leq ||u||^{p_{-}}_{p(x)}$.\end{itshape}\\
\\
A related result is the following.\\
\\
\textbf{Proposition 2.} \begin{itshape} (see \cite{14}) Let $I(u)=\int_{\Omega}\big(|\nabla u|^{p(x)}+|u|^{p(x)}\big)dx$. If $u\in W^{1,p(x)}(\Omega)$, then
the following properties hold:
\\
(1) If $u\neq0$, then $|||u|||=l\Leftrightarrow I(\frac{u}{l})=1$.\\
(2) If $|||u|||\geq1$, then $|||u|||^{p_-}\leq I(u)\leq|||u|||^{p_+}$.\\
(3) If $|||u|||\leq1$, then $|||u|||^{p_-}\leq I(u)\leq|||u|||^{p_+}$.\\
(4) If $|||u|||=1$, then $I(u)=1$.\end{itshape}\\

For $p(x)\in\mathcal {P}(\Omega)$, let
$p':\Omega\rightarrow\mathbb{R}$ be such that\\
$$\frac{1}{p(x)}+\frac{1}{p'(x)}=1,\ \ \mathrm{a.e.}\
x\in\Omega.$$\\
\textbf{Proposition 3.} \begin{itshape} (see \cite{23}) (i) For any
$u\in L^{p(x)}(\Omega)$, $v\in L^{p'(x)}(\Omega)$, we have
$$ \big|\int_{\Omega}uvdx\big|\leq\left(\frac{1}{p_-}+\frac{1}{p'_-}\right)||u||_{p(x)}||v||_{p'(x)}\leq 2 ||u||_{p(x)}||v||_{p'(x)}.
\eqno(2.5)$$\\
(ii) If $p_{1},p_{2}\in C(\bar{\Omega}),1< p_{1}(x)\leq p_{2}(x)$
 for any $x\in\bar{\Omega}$, then the embedding $L^{p_{2}(x)}(\Omega)\rightarrow
 L^{p_{1}(x)}(\Omega)$ is continuous.\end{itshape}\\
 \\
\textbf{Proposition 4.} \begin{itshape} (see \cite{23} )Let
$F:\Omega\times \mathbb{R}\rightarrow\mathbb{R}$ be a
Carath\'eodory function and assume that
 $$|F(x,t)|\leq a(x)+b|t|^{\frac{p_1 (x)}{p_2 (x)}},\ \forall (x,t)\in \Omega\times \mathbb{R}$$
where $a(x)\in L^{\infty}(\Omega)$, $b$ is a positive constant, $p_1
(x),p_2 (x)\in \mathcal {P}(\Omega)\cap L^{\infty}(\Omega)$. We
define the Nemytsky operator as follows:
$$(N_{F}(u))(x)=F(x,u(x))\,.$$
Then the mapping $N_{F}:L^{p_1 (x)}(\Omega)\rightarrow L^{p_2
(x)}(\Omega)$ is
continuous and bounded.\end{itshape}\\
\\
\textbf{Proposition 5.} \begin{itshape} (see \cite{15}) (i) Let $p\in
C(\bar{\Omega})$  and $1\leq q(x)<p^{\ast}(x)$ for each $x\in
\bar{\Omega}$. Then the embedding
$W^{1,p(x)}(\Omega)\rightarrow L^{q(x)}(\Omega)$ is compact.\\
(ii) There is a constant $C>0$ such that
$$||u||_{p(x)}\leq C||\nabla u||_{p(x)},\ \ \forall u\in
W^{1,p(x)}_{0}(\Omega).$$\end{itshape}

From Proposition 5 (ii) we know that $||\nabla u||_{p(x)}$ and
$||u||_{1,p(x)}$ are equivalent norms on $W^{1,p(x)}_{0}(\Omega)$.
In the following, we will use $||\nabla u||_{p(x)}$ to replace
$||u||_{1,p(x)}$.
\\

%{\section*{{\bf 3. Verification of the $(C)$ condition and $(PS)$
%condition}}}
\section{Verification of the compactness conditions}

From the assumption (H1) we deduce that the energy functional
$\Phi:W^{1,p(x)}_{0}(\Omega) \rightarrow \mathbb{R}$ given by
$$\Phi(u)=\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})dx - \int_{\Omega}F(x,u)dx$$
is well defined and of class $C^{1}$. The derivative of $\Phi$ at
$u$ is given by
$$\langle \Phi'(u), v \rangle = \int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u \cdot \nabla v + |u|^{p(x)-2}u\cdot v)dx - \int_{\Omega}f(x,u)vdx \eqno(3.1)$$
for $v \in W^{1,p(x)}_{0}(\Omega).$\\
Set\\
$$ I(u)=\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)} +
|u|^{p(x)})dx,\ \ J(u)=\int_{\Omega}F(x,u)dx.$$ Then
$\Phi(u)=I(u)-J(u).$\\
\\
\textbf{Lemma 1.} \begin{itshape}(i)(see \cite{14}) We have
$I\in C^{1}(W^{1,p(x)}_{0}(\Omega),\mathbb{R})$ and $$\langle I'(u),
\varphi \rangle = \int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u \nabla
\varphi + |u|^{p(x)-2}u \varphi)dx,\ \ \forall u,\varphi \in
W^{1,p(x)}_{0}(\Omega).$$ Moreover, the mapping
$I':W^{1,p(x)}_{0}(\Omega)\rightarrow
(W^{1,p(x)}_{0}(\Omega))^{\ast}$ is a bounded homeomorphism and is of
type $(S^+)$, namely\\
$$u_{n}\rightharpoonup u\  \mathrm{and}\
\limsup_{n\rightarrow\infty}\langle I'(u_{n})-I'(u), u_{n}-u \rangle
\leq 0 \ \mathrm{imply}\ u_{n} \rightarrow u.$$\\
(ii) (see \cite{15}) Under the assumption (H1), we have $J\in
C^{1}(W^{1,p(x)}_{0}(\Omega),\mathbb{R})$ and $$\langle J'(u),
\varphi \rangle=\int_{\Omega}f(x,u)\varphi dx\ \ \forall u,\varphi
\in W^{1,p(x)}_{0}(\Omega).$$ Moreover, the mapping
$I':W^{1,p(x)}_{0}(\Omega)\rightarrow
(W^{1,p(x)}_{0}(\Omega))^{\ast}$ is weakly-strongly continuous,
namely, $$u_{n}\rightharpoonup u\  \mathrm{implies}\ u_{n}
\rightarrow u,$$ where $\rightharpoonup$ and $\rightarrow$ denote
the weak and strong convergence in $W^{1,p(x)}_{0}(\Omega)$,
respectively.\end{itshape}\\

For simplicity, from now on we denote $X = W^{1,p(x)}_{0}(\Omega)$ and $X^{\ast}= (W^{1,p(x)}_{0}(\Omega))^{\ast}$.\\
\\
\textbf{Definition 1.} The function $u\in X$ is called a weak solution of problem (1.1)
if for any $\phi \in X$
$$\int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u \cdot \nabla \phi + |u|^{p(x)-2}u\cdot \phi)dx - \int_{\Omega}f(x,u)\phi dx =0. \eqno(3.2)$$
From (3.1) and (3.2) we know that the critical points of $\Phi$ are
weak solutions of (1.1).
From (3.2) we know that 0 is a critical point of $\Phi$. Now we are interested in seeking nontrivial critical points of $\Phi$.\\
\\
\textbf{Definition 2.} The functional $\Phi$ satisfies the $(C)$
condition if for $c\in \mathbb{R}$, any sequence $\{u_{n}\}\subset
X$ such that $\Phi(u)\rightarrow c,
(1+||u_{n}||)||\Phi'(u_{n})||_{X^{\ast}} \rightarrow 0$ has a
convergent subsequence. The functional $\Phi$ satisfies the $(PS)$
condition if any sequence $\{u_{n}\}\subset X$ such that
$\Phi(u_{n})$ is bounded and $\Phi'(u_{n}) \rightarrow 0$ has a convergent subsequence.\\
\\
\textbf{Theorem 4.} \begin{itshape}Suppose $\Phi$ satisfies (H1), (H2) and (H3). Then $\Phi$ satisfies the $(C)$ condition.\end{itshape}\\
\\
\textbf{Proof.} Similar to the proof of Lemma 3.1 in \cite{1}, and for
the reader's convenience, we will give the key steps of the proof.
We first  verify the boundedness of $(C)$ sequences. Assume
$\Phi$ has an unbounded $(C)$ sequence $\{u_{n}\}$. Up to a
subsequence we may assume that
$$\Phi(u)\rightarrow c,\ \ |||u_{n}||| \rightarrow \infty,\ \ \langle \Phi'(u_{n}), u_{n} \rangle \rightarrow 0\  \mathrm{as}\ n \rightarrow \infty.$$
Let $v_{n} = |||u_{n}|||^{-1}u_{n}$, then $|||v_{n}||| = 1$. By
Proposition 5, up to a subsequence we have
$$v_{n} \rightharpoonup v \ \mathrm{in} \ X,\  v_{n} \rightarrow v \ \mathrm{in} \ L^{q(x)}(\Omega),\  v_{n} \rightarrow v\ \ \mathrm{a.e.}\ x \in \Omega \eqno(3.3)$$
If $v = 0$, as in \cite{22}, we choose a sequence $\{t_{n}\} \subset
[0,1]$ such that $\Phi(t_{n}u_{n}) = \max \limits_{t\in [0,1]}
\Phi(tu_{n})$. For any positive integer $m$, we can choose $r =
(2mp_{+})^{\frac{1}{p_{-}}}$ such that $r|||u_{n}|||^{-1} \in (0,1)$
as $n$ large enough. Since $|||v_{n}||| = 1$, by Proposition 2,
we see that
$$\int_{\Omega}(|\nabla v_{n}|^{p(x)} + |v_{n}|^{p(x)})dx = 1.\eqno(3.4)$$
Since $v_{n} \rightarrow 0$ in $L^{q(x)}(\Omega)$ and (H1), from
Proposition 4 we know that $F(\cdot , rv_{n}) \rightarrow 0$ in
$L^{1}(\Omega)$. Thus
$$\lim \limits_{n \rightarrow \infty}\int_{\Omega}F(x, rv_{n})dx = 0.\eqno(3.5)$$
Hence, for $n$ large enough, relations (3.4) and (3.5) yield
$$
\Phi(t_{n}u_{n}) \geq \Phi(r|||u_{n}|||^{-1}u_{n}) = \Phi(rv_{n})
\geq \frac{r^{p_{-}}}{p^{+}} - \int_{\Omega}F(x,rv_{n})dx \geq m.
$$
We deduce that $\Phi(t_{n}u_{n}) \rightarrow +\infty$. But $\Phi(0) = 0, \Phi(u_{n}) \rightarrow c$, so $t_{n} \in (0,1)$ and
$$\langle \Phi'(t_{n}u_{n}), t_{n}u_{n}\rangle = t_{n}\frac{d}{dt}\Big|_{t=t_{n}}\Phi(tu_{n}) = 0.$$
Now using (H3), we get
\begin{align*}
\frac{1}{\theta}\Phi(t_{n}u_{n}) &= \frac{1}{\theta} \big(\Phi(t_{n}u_{n}) - \frac{1}{p_{+}}\langle \Phi'(t_{n}u_{n}), t_{n}u_{n}\rangle \big)\\
&\leq \int_{\Omega} t^{p(x)} \big(\frac{1}{p(x)} - \frac{1}{p_{+}}\big)(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx + \frac{1}{p_{+}} \int_{\Omega}\frac{\mathcal {F}(x,t_{n}u_{n})}{\theta}dx\\
&\leq  \int_{\Omega}  \big(\frac{1}{p(x)} - \frac{1}{p_{+}}\big)(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx + \frac{1}{p_{+}} \int_{\Omega}\mathcal {F}(x,u_{n})dx\\
&= \Phi(u_{n}) - \frac{1}{p_{+}} \langle \Phi'(u_{n}), u_{n}\rangle
\rightarrow c.\tag{3.6}
\end{align*}
This contradicts the fact that $\Phi(t_{n}u_{n}) \rightarrow +\infty$.\\
If $v \neq 0$, for the set $\Omega':= \{x\in \Omega :v(x) \neq 0\}$
we have $|u_{n}(x)| \rightarrow +\infty, \forall x \in \Omega'$.
Since the set $\Omega'$ has positive Lebesgue measure and
$|||u_{n}||| > 1$ for $n$ large, using (1.3) and (H2), we have
$$\aligned
\frac{1}{p_{-}} &\geq \frac{1}{p_{-}} \int_{\Omega}\frac{1}{|||u_{n}|||^{p(x)}}(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx\\
&\geq  \int_{\Omega}\frac{1}{p(x)|||u_{n}|||^{p(x)}}(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx\\
&\geq\frac{\Phi(u_{n})}{|||u_{n}|||^{p_{+}}} + \int_{\Omega'}\frac
{F(x,u_{n})}{|||u_{n}|||^{p_{+}}}dx+
\int_{\Omega\backslash\Omega'}\frac {F(x,u_{n})}{|||u_{n}|||^{p_{+}}}dx\\
&\geq \frac{\Phi(u_{n})}{|||u_{n}|||^{p_{+}}} +
\int_{\Omega'}\frac{F(x,u_{n})}{|u_{n}|^{p_{+}}}|v_{n}|^{p_{+}}dx-C_{4}\int_{\Omega\backslash\Omega'}|v_{n}|^{p_{+}}dx
\rightarrow +\infty.
\endaligned$$
This is impossible. Therefore we have proved that $\{u_{n}\}$ is bounded in $X$.

Next, with the same arguments as in the proof of Lemma 5.1 in \cite{1}, we  prove that any
$(C)$ sequence has a convergent subsequence. Since $\{u_{n}\}$ is
bounded in  $X$, by Proposition 5, we may
assume that, up to a subsequence,
$$u_{n} \rightharpoonup u\ \mathrm{in} \ X,\ \ \  u_{n} \rightarrow u \ \mathrm{in} \ L^{q(x)}(\Omega) \eqno(3.7)$$
By the boundedness of $\{u_{n}\}$ in $L^{p(x)}(\Omega)$, we have
$$\sup \limits_{n} \int_{\Omega}|u_{n}|^{p(x)}dx < \infty. \eqno(3.8)$$
By (H1), (2.5), (3.7) and (3.8), as in the proof of Lemma 3.11 in Willem \cite{31},
we have
$$\int_{\Omega}(f(x,u_{n}) - f(x,u))(u_{n} - u)dx \rightarrow 0\ \mathrm{as}\ n \rightarrow \infty.$$
Note that $\Phi'(u_{n}) \rightarrow 0$, hence
$$\langle I'(u_{n}) - I'(u), u_{n} - u\rangle = \langle \Phi'(u_{n}) - \Phi'(u), u_{n} - u\rangle + \int_{\Omega}(f(x,u_{n}) - f(x,u))(u_{n} - u)dx \rightarrow 0$$
By Lemma 1, we obtain $u_{n} \rightarrow u$ in $X$.
The proof is complete.
\\
\textbf{Theorem 5.} \begin{itshape}Suppose $\Phi$ satisfies (H1),
(H2) and (H4).
Then $\Phi$ satisfies the $(C)$ condition. \end{itshape}\\
\textbf{Proof.} We only need to verify that (3.6) is fulfilled, the other part of the
proof being completely similar to the proof of Theorem 4. From the
proof of Theorem 1.2 in \cite{30} we know that due to (H4) there exists a
constant $C>0$ such that
$$ \mathcal {F}(x,s)\leq \mathcal {F}(x,t)\ \mathrm{for}\ x\in
\Omega\ \mathrm{and}\ 0\leq s \leq t\ \mathrm{or}\ t\leq s\leq 0.$$
Therefore
$$\aligned
\Phi(t_{n}u_{n}) &= \Phi(t_{n}u_{n}) - \frac{1}{p_{+}}\langle \Phi'(t_{n}u_{n}), t_{n}u_{n}\rangle \\
&\leq \int_{\Omega} t^{p(x)} \big(\frac{1}{p(x)} - \frac{1}{p_{+}}\big)(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx +
\frac{1}{p_{+}} \int_{\Omega}\mathcal {F}(x,t_{n}u_{n})dx\\
\endaligned$$
$$\aligned
&\leq  \int_{\Omega}  \big(\frac{1}{p(x)} -
\frac{1}{p_{+}}\big)(|\nabla u_{n}|^{p(x)} + |u_{n}|^{p(x)})dx +
\frac{1}{p_{+}} \int_{\Omega}\mathcal {F}(x,u_{n})dx+\frac{C}{p_{+}}|\Omega|\\
&= \Phi(u_{n}) - \frac{1}{p_{+}} \langle \Phi'(u_{n}),
u_{n}\rangle+\frac{C}{p_{+}}|\Omega| \rightarrow
c+\frac{C}{p_{+}}|\Omega|.
\endaligned$$
as $n\rightarrow \infty,$ where $|\Omega|$ denote the measure of the domain $\Omega$.
\\
\textbf{Theorem 6.}\begin{itshape} Suppose $\Phi$ satisfies (P),
(H1) and (H6).
Then $\Phi$ satisfies the $(PS)$ condition.\end{itshape} \\
\textbf{Proof.} We first prove that $\Phi$ is coercive. It follows
from (H1) and (H6) that for some $\varepsilon >0$ small, there
exists a constant $C_{\varepsilon}>0$ such that
$$|F(x,u)|\leq \frac{1}{p(x)}
\big[\frac{p_{-}}{p_{+}}(\lambda_{\ast}-\varepsilon)+1\big]|u|^{p(x)}+C_{\varepsilon}.\eqno(3.9)$$
Hence by means of (3.9) and (2.4), for $u\in
W^{1,p(x)}_{0}{(\Omega)}$ and $|||u|||\geq2$ we have
$$\aligned
\Phi(u)=&\int_\Omega \frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\int_\Omega F(x,u)dx\\
\geq &\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-
\int_\Omega\frac{1}{p(x)}\big[\frac{p_{-}}{p_{+}}(\lambda_{\ast}-\varepsilon)+1\big]|u|^{p(x)}dx-C_{\varepsilon}|\Omega|\\
=&\int_\Omega\frac{1}{p(x)}|\nabla
u|^{p(x)}dx-\frac{p_{-}}{p_{+}}(\lambda_{\ast}-\varepsilon)\int_\Omega\frac{1}{p(x)}|u|^{p(x)}dx-C_{\varepsilon}|\Omega|\\
\geq&\int_\Omega\frac{1}{p(x)}|\nabla
u|^{p(x)}dx-\frac{\lambda_{\ast}-\varepsilon}{\lambda_{\ast}}\int_\Omega\frac{1}{p(x)}|\nabla
u|^{p(x)}dx-C_{\varepsilon}|\Omega|\\
\geq& C(\varepsilon)\,|||u|||^{p_-}
-C_{\varepsilon}|\Omega|\rightarrow\infty
\endaligned$$
as $|||u|||\rightarrow \infty.$ Next, similar to the idea of the
proof in \cite[Lemma 3.1]{26}, the desired
result immediately follows from Lemma 1.

%{\section*{{\bf 4. Computation of critical groups}}}
\section{Computation of critical groups}

Let $X$ be a real Banach space and $\Phi \in C^{1}(X,\mathbb{R}), K
= \{u\in X: \Phi'(u) = 0\}$, then the $q$th critical group of $\Phi$
at an isolated critical point $u \in K$ with $\Phi(u) = c$ is
defined by
$$C_{q}(\Phi,u):= H_{q}(\Phi^{c}\cap U, \Phi^{c}\cap U\setminus \{u\}), \ \ q\in \mathbb{N}:=\{0,1,2,\cdots\},$$
where $\Phi^{c}= \{u \in X: \Phi(u)\leq c\}$, $U$ is any
neighborhood of $u$, containing the unique critical point,
$H_{\ast}$ is the singular relative homology with coefficients in an
Abelian group $G$.

We say that $u\in K$ is an homological nontrivial critical point of
$\Phi$ if at least one of its critical groups is nontrivial.

Now we first present the following critical point theorem which will
be used later.\\
\\
\textbf{Proposition 6.}\begin{itshape}(see \cite[Theorem 2.1]{26}) Let
$X$ be a real Banach space and let $\Phi \in C^{1}(X,\mathbb{R})$
satisfy the $(PS)$ condition and is bounded from below. If $\Phi$
has a critical point that is homological nontrivial and is not a
minimizer of
$\Phi$, then $\Phi$ has at least three critical points.\end{itshape}\\

If $\Phi$ satisfies the condition $(C)$ and the critical values of
$\Phi$ are bounded from below by some $a < \inf \Phi(K)$, then the
critical groups of $\Phi$ at infinity were introduced by Bartsch and
Li \cite{3} as
$$C_{q}(\Phi,\infty):= H_{q}(X, \Phi^{a}),\ \ q\in \mathbb{N}.\eqno(4.1)$$
If $\Phi$ satisfies the condition $(C)$, then $\Phi$ satisfies the deformation condition. By the deformation lemma, the right-hand side of (4.1) does not depend on the choice of $a$.\\
\\
\textbf{Remark 1.} Morse theory \cite{5, 28} tells us that if $K =
\{0\}$ then $C_{q}(\Phi,\infty) = C_{q}(\Phi,0)$ for all $q\in
\mathbb{N}$. It follows that if $C_{q}(\Phi,\infty) \neq
C_{q}(\Phi,0)$ for some $q\in \mathbb{N}$, then $\Phi$ must have a
nontrivial critical point. So, we must compute the critical groups
at zero and at infinity.

For the proofs of our theorems, in what follows we may assume that $\Phi$ has only finitely many critical points. Since $\Phi$ satisfies the condition $(C)$, then the critical groups $C_{q}(\Phi,\infty)$ at infinity make sense.\\
\\
\textbf{Theorem 7.} \begin{itshape}Suppose that $\Phi$ satisfies (H1), (H2) and (H3). Then $C_{q}(\Phi,\infty) = 0$ for all $q\in \mathbb{N}$.\end{itshape}
\\
\textbf{Proof.} We use some ideas from the proof of Theorem 1.1 in \cite{30}. Let $S = \{u\in
X: |||u||| = 1\}$. For $u \in S$, by Fatou lemma and (H2) we have
$$\lim \limits_{t\rightarrow +\infty} \int_{\Omega}\frac{F(x,tu)}{|t|^{p_{+}}}dx \geq \int_{\Omega}\lim \limits_{t\rightarrow +\infty} \frac{F(x,tu)}{|tu|^{p_{+}}}|u|^{p_{+}}dx = +\infty.$$
Therefore
$$
\Phi(tu) \leq \frac{t^{p_{+}}}{p_{-}} - \int_{\Omega}F(x,tu)dx
\leq t^{p_{+}}\big(\frac{1}{p_{-}} -
\int_{\Omega}\frac{F(x,tu)}{|t|^{p_{+}}}dx\big) \rightarrow -\infty\
\mathrm{as}\ t\rightarrow +\infty.
$$
Choose $a< \min \{\inf \limits_{\|u\|\leq 1}\Phi(u), 0\}$, then for
any $u \in S$, there exists $t_{0} > 1$ such that $\Phi(t_{0}u)\leq
a$. By (H3), we have
$$\mathcal {F}(x,m) \geq 0\ \ for\ (x,m)\in \Omega \times \mathbb{R}.\eqno(4.2)$$
Therefore, if
$$\Phi(tu) =\int_{\Omega}\frac{t^{p(x)}}{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})dx - \int_{\Omega}F(x,tu)dx \leq a,$$
then
$$\int_{\Omega}\big(t^{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})\big)dx \leq p_{+}a + \int_{\Omega}p_{+}F(x,tu)dx.$$
Using (4.2), we obtain
$$\aligned
\frac{d}{dt}\Phi(tu) &= \frac{1}{t}\Big[\int_{\Omega}t^{p(x)}(|\nabla u|^{p(x)} +|u|^{p(x)})dx - \int_{\Omega}f(x,tu)tu dx\Big]\\
&\leq \frac{1}{t}\Big[p_{+}a - \int_{\Omega}\mathcal
{F}(x,tu)dx\Big] <0.
\endaligned$$
Then by the implicit function theorem, there exists a unique $T \in
C(S,\mathbb{R})$ such that $\Phi(T(u)u) = a$. As in \cite{24}, using the
function $T$ to construct a strong deformation retract from $X
\setminus \{0\}$ to $\Phi^{a}$. Therefore, we deduce
$$C_{q}(\Phi,\infty)= H_{q}(X, \Phi^{a}) = H_{q}(X, X \setminus \{0\}) = 0, \forall q\in \mathbb{N}.$$
The proof is completed.
\\
\textbf{Theorem 8.}\begin{itshape} Suppose that $\Phi$  satisfies
(H1), (H2) and (H4). Then $C_{q}(\Phi,\infty) = 0$ for all $q\in
\mathbb{N}$.\end{itshape}\\
\\
\textbf{Proof.} The proof is completely similar to
the proof of Theorem 1.2 in \cite{30} and thus it is omitted.

As in \cite{14}, since $X$ is a separable and reflexive Banach space,
there exist $\{e_n\}^{\infty}_{n=1}\subset X$ and
$\{f_n\}^{\infty}_{n=1}\subset X^{*}$ such that
$$f_n (e_m)=\delta_{n,m}=\left\{
                           \begin{array}{ll}
                             1, & n\neq m \\
                             0, & n=m
                           \end{array}
                         \right.
$$
$$X=\overline{\mathrm{span}\{e_n:n=1,2,\ldots\}},\ X^*=\overline{\mathrm{span}\{f_n:n=1,2,\ldots\}}.$$
For $k=1,2,\ldots,$  we denote $Y_k
=\mathrm{span}\{e_1,e_2,\ldots,e_k\}$, hence $Y_k$  has a closed
complementing subspace $Z_k$  in $X$. Thus, $X=Y_k\oplus Z_k$  (see \cite{4}).\\
\\
\textbf{Lemma 2.}\begin{itshape}(see \cite[Lemma 3.3]{14}) Assume that
$\varphi:X\rightarrow \mathbb{R}$ is weakly-strongly continuous,
$\varphi(0)=0,\rho>0$ is a given positive number. Set $\eta_k
=\sup_{u\in Z_k,||u||\leq\rho}|\varphi(u)|$. Then
$\eta_k\rightarrow 0 $ as $k\rightarrow\infty.$\end{itshape}\\

In order to compute the critical groups at zero, we need the
following result.\\
\\
\textbf{Proposition 7.}\begin{itshape}(see \cite[Proposition 2.1]{26})
Assume that $\Phi$ has a critical point $u=0$  with  $\Phi(0)=0$. Suppose that
$\Phi$  has a local linking at 0  with respect to $X=V\oplus W$,
$k=\mathrm{dim} V<\infty$, that is, there exists $\rho>0$ small such
that
$$\left\{
    \begin{array}{ll}
      \Phi(u)\leq0, & u\in V,|||u|||\leq \rho; \\
      \Phi(u)>0, & u\in W,0<|||u|||\leq \rho.
    \end{array}
  \right.
$$
Then $C_k(\Phi,0)\neq0$, hence 0 is a homological nontrivial
critical point of $\Phi$.\end{itshape}\\
\\
\textbf{Theorem 9.} \begin{itshape}Supposed that $\Phi$  satisfies
(H1) and (H5). Then there exists $k_0 \in \mathbb{N}$  such that
$C_{k_0}(\Phi,0)\neq 0$.\end{itshape}\\
\\
\textbf{Proof.} Since $f(x,0)=0$, the zero function $0$ is a
critical point of $\Phi$. So we only need to prove that $\Phi$ has a
local linking at 0  with respect to $X=Y_k \oplus Z_k$. We take two
steps:

 \begin{itshape}Step\end{itshape} 1: Take $u\in Y_k$. Since $Y_k$  is finite dimensional, we have
that for given $R>0$, there exists $0<\rho<1$  small such that
$$u\in Y_k, |||u|||<\rho\Rightarrow|u(x)|<R,\ \   \forall x\in\Omega.$$
For $0<r<R$, let
$\Omega_{1}=\{x\in\Omega:|u(x)|<r\},\Omega_{2}=\{x\in\Omega:r\leq|u(x)|\leq
R\},\Omega_{3}=\{x\in\Omega:|u(x)|>R\}.$ Then
$\Omega=\cup^3_{i=1}\Omega_{i}$ and $\Omega_{i}$ are pairwise
disjoint. For the sake of simplicity, let $G(x,u)=F(x,u)-
\frac{C_2}{p(x)}|u|^{\alpha(x)}$. Therefore, from (H5) it follows that

\begin{align*}
\Phi(u)=&\int_\Omega \frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\big(\int_{\Omega_{1}}+\int_{\Omega_{2}}+\int_{\Omega_{3}}\big) F(x,u)dx\\
=&\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\int_\Omega\frac{C_2}{p(x)}|u|^{\alpha(x)}dx-\int_{\Omega_{1}}G(x,u)dx\\
&-\int_{\Omega_{2}}G(x,u)dx-\int_{\Omega_{3}}G(x,u)dx\\
\leq& \int_\Omega(|\nabla u|^{p(x)}+|u|^{p(x)})dx-
\frac{C_2}{p_+}\int_\Omega|u|^{\alpha(x)}dx-\int_{\Omega_{1}}G(x,u)dx.
\end{align*}

In terms of the assumptions on $\alpha(x)$ and $\Omega$ is a bounded
domain in $\mathbb{R}^{N}$, the embedding
$L^{p(x)}(\Omega)\rightarrow L^{\alpha(x)}(\Omega)$ is continuous.
This implies that there exists a constant $C>1$  such that
$$||u||_{\alpha(x)}\leq C||u||_{p(x)}\leq C||u||_{1,p(x)}\leq2 C|||u|||\leq2C\rho .$$
If $\rho\leq\frac{1}{2C}$, then $||u||_{\alpha(x)}\leq1$. Note that
the norms on $Y_k$ are equivalent each other, $||u||_{\alpha(x)}$ is
equivalent to $|||u|||$. As in \cite[Theorem 3.1]{6}, due to
$\alpha(x),p(x)\in C(\bar{\Omega})$ and $\alpha(x)<p(x)$, for each $
x\in \Omega$, there exists an open subset $B_{\delta}(x)$ of
$\bar{\Omega}$ such that
$$\alpha_{x}:=\sup_{x\in B_{\delta}(x)}{\alpha(x)}<\inf_{x\in
B_{\delta}(x)}{p(x)}:=p_{x}.$$ Then
$\{B_{\delta}(x)\}_{x\in\bar{\Omega}}$ is an open covering of
$\bar{\Omega}.$ Since $\bar{\Omega}$ is compact, there is a finite
subcovering $\{B_{\delta}(x_{i})\}_{i=1}^{m}$. We can use all the
hyperplane, for each of which there exists at least one hypersurface
of some $\{B_{\delta}(x_{i})\}_{i=1}^{m}$ lying on it, to divide
$\{B_{\delta}(x_{i})\}_{i=1}^{m}$ into finite open hypercube
$\{\mathcal {Q}\}_{j=1}^{n}$ which mutually have no common points.
It is obvious that
$\overline{\Omega}=\cup_{j=1}^{n}\overline{\mathcal {Q}}_{j}$ and
$$\alpha_{j+}:=\sup_{x\in \mathcal {Q}_{j}}{\alpha(x)}<\inf_{x\in
\mathcal {Q}_{j}}{p(x)}:=p_{j-}.$$ Notice that
$\int_{\Omega_{1}}G(x,u)dx\rightarrow 0$ as $r\rightarrow 0$ and
$|||u|||_{\mathcal {Q}_{j}}\leq|||u|||$. Therefore, there is a
constant $C>0$ such that for $0<\rho<0$ small and $r$ sufficiently
small
$$\aligned
\Phi(u)\leq&\sum_{j=1}^{n}{\bigg[\int_{\mathcal {Q}_{j}}(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-\frac{C_2}{p_-}\int_{\mathcal {Q}_{j}}|u|^{\alpha(x)}dx\bigg]}+\int_{\Omega_{1}}G(x,u)dx\\
\leq& \sum_{j=1}^{n}\big(|||u|||_{\mathcal
{Q}_{j}}^{p_{j-}}-C|||u|||_{\mathcal {Q}_{j}}^{\alpha_{j+}}
\big)-\int_{\Omega_{1}}G(x,u)dx \leq 0.
\endaligned$$

 \begin{itshape}Step\end{itshape} 2: By (H1) and Young's inequality, there exists
$C>0$  such that
$$|F(x,u)|\leq \frac{C_{3}}{p_{+}}|u|^{p(x)}+C|u|^{s(x)},\ \mathrm{for}\ \mathrm{all} \ x\in \Omega\  \mathrm{and}\ |u|\geq R,\eqno(4.3)$$
where $s(x)\in C(\bar{\Omega})$ and $p(x)<s(x)<p^{\ast}(x)$. For the
sake of simplicity, let $H(x,u)=F(x,u)- \frac{C_3}{p(x)}|u|^{p(x)}$.
Therefore, if $u\in Z_k$ and $|||u|||\leq 1$, from (H5) and (4.3) we
deduce
$$\aligned
\Phi(u)=&\int_\Omega \frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\big(\int_{\Omega_{1}}+\int_{\Omega_{2}}+\int_{\Omega_{3}}\big) F(x,u)dx\\
=&\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\int_\Omega\frac{C_3}{p(x)}|u|^{p(x)}dx-\int_{\Omega_{1}}H(x,u)dx\\
&-\int_{\Omega_{2}}H(x,u)dx-\int_{\Omega_{3}}H(x,u)dx\\
\geq& \frac{1-C_{3}}{p_{+}}\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-
C\int_\Omega|u|^{s(x)}dx-\int_{\Omega_{1}}H(x,u)dx.
\endaligned$$

We next consider the functional $\varphi:X\rightarrow \mathbb{R}$,
$\varphi(u)=\int_\Omega|u|^{s(x)}dx$. In view of Proposition 5,
the embedding $X\rightarrow L^{s(x)}(\Omega)$ is compact. Thus it is
easy to show that $\varphi$ is weakly-strongly continuous (see \cite[Theorem 1.4]{18}). Hence by Lemma 2, we have $$\eta_k=\sup_{u\in
Z_k,||u||\leq 1}|\varphi(u)|\rightarrow0\ \  \mathrm{as}\
k\rightarrow\infty. \eqno(4.4)$$ Note that
$\int_{\Omega_{1}}H(x,u)dx\rightarrow 0$ as $r\rightarrow 0.$
Therefore, using the same argument as in
\begin{itshape}Step\end{itshape} 1, we obtain
$$\aligned
\Phi(u)\geq&\sum_{i=1}^{l}{\Big[\frac{1-C_{3}}{p_{+}}\int_{\mathbb{Q}_{i}}(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-C\int_{\mathbb{Q}_{i}}|u|^{s(x)}dx\Big]}+\int_{\Omega_{1}}H(x,u)dx\\
\geq&
\sum_{i=1}^{l}\Big[\frac{1-C_{3}}{p_{+}}|||u|||_{\mathbb{Q}_{i}}^{p_{i+}}-C\eta_k|||u|||_{\mathbb{Q}_{i}}^{s_{i-}}
\Big]-\int_{\Omega_{1}}H(x,u)dx.
\endaligned$$
From (4.4) we know that there exists $k_{0}\in\mathbb{N}$ such that
$\eta_{k_{0}} \leq\frac{1-C_{3}}{2Cp_{+}}$. Then we get
$$\Phi(u)\geq
\sum_{i=1}^{l}\frac{1-C_{3}}{2Cp_{+}}(|||u|||_{\mathbb{Q}_{i}}^{p_{i+}}-|||u|||_{\mathbb{Q}_{i}}^{s_{i-}})-\int_{\Omega_{1}}H(x,u)dx>0$$
as $0<\rho<1$ small and  $r$ sufficiently small. Thus, there
exists $k_0\in\mathbb{N}$ such that $\Phi(u)>0$  as $u\in Z_{k_0}$
and
 $0<|||u|||\leq\rho$.\\

Combining \begin{itshape}Step\end{itshape} 1 with \begin{itshape}Step\end{itshape} 2, we complete the proof of Theorem 9 due to Proposition 7.
\\
\textbf{Remark 2.} From the proof of Theorem 9, we deduce that the conclusion of Theorem 9 still holds under the assumptions (H1), (H5) and (H7).\\

\section{Proof of main results}
%{\section*{{\bf 5. Proof of main results}}}

In this section we
deduce the proofs of our main existence results.\\
\\
\textbf{Proof of Theorem 1.} By Theorem 7 and Theorem 9, we
have $C_{k_0}(\Phi,\infty)\neq C_{k_0}(\Phi,0)$ for some $k_0\in
\mathbb{N}$. Then Theorem 1 follows immediately from Theorem 4
and Remark 1. The proof is complete.
\\
\textbf{Proof of Theorem 2.} By Theorem 8 and Theorem 9, we
have $C_{k_0}(\Phi,\infty)\neq C_{k_0}(\Phi,0)$ for some $k_0\in
\mathbb{N}$. Then Theorem 2 follows immediately from Theorem 4
and Remark 1. This completes the proof.
\\
\textbf{Proof of Theorem 3.} By Theorem 6, $\Phi$ satisfies the
$(PS)$ condition and is bounded from below. By the assumption (H7)
and Remark 2, the trivial solution $u=0$ is homological nontrivial
and is not a minimizer. The conclusion follows from
Proposition 6.
\\
\textbf{Remark 3.} The method used in this paper can also be
applied to study the following more general problem:
$$\left\{
  \begin{array}{ll}
    -\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)+b(x)|u|^{p(x)-2}u=f(x,u), & x\in\Omega \\
    u=0, & x\in\partial\Omega
  \end{array}
\right.$$ where $b(x)\in C(\bar{\Omega})$  and $1<b_- \leq b_+
<\infty$. Through taking the same arguments of this paper, results similar to Theorems 1-3 can be established under the corresponding assumptions.

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

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