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\begin{document}

\title[Anisotropic elliptic equations with variable exponent]{Qualitative analysis of solutions for a class of anisotropic elliptic equations with variable exponent }

\centerline{}
%--------------------------------------------------------------------------------------------------------------
\author[G.A. Afrouzi, M. Mirzapour]{G.A. Afrouzi, M. Mirzapour}
\address{G.A. Afrouzi,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto:
G.A. Afrouzi <afrouzi@umz.ac.ir>}{afrouzi@umz.ac.ir}}
%-------------------------------------------------------------------------------------------------------------
\address{M. Mirzapour,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto: M.
Mirzapour <mirzapour@stu.umz.ac.ir>}{mirzapour@stu.umz.ac.ir}}
%--------------------------------------------------------------------------------------------------------------
\author[V.D. R\u adulescu]{Vicen\c tiu D. R\u adulescu$^\ast$}
\address{Vicen\c tiu D. R\u adulescu,
Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia} \email{\href{mailto: V.D. R\u adulescu
<vicentiu.radulescu@math.cnrs.fr>}{vicentiu.radulescu@math.cnrs.fr}}
%--------------------------------------------------------------------------------------------------------------

\keywords{Degenerate anisotropic Sobolev spaces; variable exponent; Dirichlet boundary value condition; Fountain theorem; Ekeland variational principle. \\
 {\it 2000 Mathematics Subject Classifications}: 35J62, 35J70, 46E35.\\ $^\ast$ Corresponding author.}

%\date{\bf 7 August 2013}

\begin{abstract}
We are concerned with the
 degenerate anisotropic problem
$$
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)+b(x)|u|^{P^+_+-2}u
 = f(x,u) \quad \text{in } \Omega,\\
u=0  \quad \text{on } \partial\Omega.
\end{array}\right.
$$
We first establish the existence of an unbounded sequence of weak solutions.
We also obtain the existence of a nontrivial weak solution if the nonlinear term $f$ has a special form. The proofs rely on the fountain theorem and Ekeland's variational principle.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
%--------------------------------------------------------------------------------------------------------------
\section{Introduction}
Let $\Omega\subset\R^N~(N\geq 3)$ be a bounded domain with smooth boundary.
The purpose of this paper is to analyze the existence of multiple weak solutions to the anisotropic problem
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)+b(x)|u|^{P^+_+-2}u
 = f(x,u) \quad \text{in } \Omega,\\
u=0  \quad \text{on } \partial\Omega,
\end{array}\right.
\end{equation}
where $b\in L^{\infty}(\overline{\Omega})$, $f:\overline{\Omega}\times \R\rightarrow \R$ and $a_i:\overline{\Omega}\times \R\rightarrow \R$ are Carath\'{e}odory functions fulfilling some natural hypotheses.

Problems of type \eqref{e1.1} have a strong motivation and
important research efforts have been made recently with the aim of
understanding anisotropic phenomena described by nonhomogeneous
differential operators. We remember that equations of this type
can be regarded as models for phenomena arising in the study of
electrorheological fluids (see Halsey \cite{hal}, Diening
\cite{D}, Ru\v zi\v cka \cite{R}), elasticity (see Zhikov
\cite{Z1}) or image processing and restoration (see Chen, Levine
and Rao \cite{CLR}, Esedoglu and Osher \cite{esed}). A survey of the
history of this research field with a comprehensive bibliography
is provided by Diening,  H\"ast\"o and  Nekvinda \cite{dhn}.

The anisotropic differential operator $\sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)$ is a $\overrightarrow{p}(\cdot)$--Laplace type operator, where $\overrightarrow{p}(x)=(p_1(x),p_2(x),\dots,p_N(x))$, and $P^+_+=\max_{i\in \{1, \dots ,N\}}\{\sup_{x\in \Omega}p_i(x)\}$. For $i=1, \dots ,N$, we assume that $p_i$ is a continuous function on $\overline{\Omega}$.

Let
$a_i(x,\eta)$ denote the continuous derivative with respect to $\eta$ of the mapping $A_i:\overline{\Omega}\times \R^N\rightarrow \R^N$, $A_i=A_i(x,\eta)$, that is, $a_i(x,\eta)=\frac{\partial}{\partial\eta}A_i(x,\eta)$. Throughout this paper we assume  that the following hypotheses are fulfilled:
\begin{enumerate}
\item[$(\mathbf{A_0})$] $A_i(x,0)=0$ for a.e. $x\in \Omega$.
\item[$(\mathbf{A_1})$] There exists a positive constant $\overline{c}_i$ such that $a_i$ satisfies the growth condition
$$
|a_i(x,\eta)|\leq \overline{c}_i(1+|\eta|^{p_i(x)-1}),
$$
for  all $x\in \overline{\Omega}$ and $\eta\in \R^N$.
\item[$(\mathbf{A_2})$] The inequalities
$$
|\eta|^{p_i(x)}\leq a_i(x,\eta)\eta\leq p_i(x)A_i(x,\eta),
$$
hold for all $x\in \overline{\Omega}$ and $\eta\in \R^N$.
\item[$(\mathbf{A_3})$] There exists $k_i>0$ such that
$$
A_i\left(x,\frac{\eta+\xi}{2}\right)\leq \frac{1}{2}A_i(x,\eta)+\frac{1}{2}A_i(x,\xi)-k_i|\eta-\xi|^{p_i(x)},
$$
for all $x\in \overline{\Omega}$ and $\eta,\xi\in \R^N$, with equality if and only if $\eta=\xi$.
\item[$(\mathbf{A_4})$] The mapping $A_i$ is even with respect to its second variable, that is,
$$
A_i(x,-\eta)=A_i(x,\eta),
$$
for all $x\in \overline{\Omega}$ and $\eta\in \R^N$.
\end{enumerate}

The differential operator $\sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)$ is the anisotropic $\overrightarrow{p}(x)$--Laplace type operator (where $\overrightarrow{p}(x)=(p_1(x), \dots ,p_N(x)$) because when we take
\begin{align*}
a_i(x,\eta)=|\eta|^{p_i(x)-2}\eta,
\end{align*}
for all $i\in \{1, \dots ,N\}$, we have $A_i(x,\eta)=\frac{1}{p_i(x)}|\eta|^{p_i(x)}$ for all $i\in \{1, \dots ,N\}$, that is,
\begin{align*}
\Delta_{\overrightarrow{p}(x)}(u)=\sum_{i=1}^N\partial_{x_i}\Big{(}|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big{)}.
\end{align*}
Obviously, there are many other operators deriving from $\sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)$. Indeed, to give another interesting example, if we take
\begin{align*}
a_i(x,\eta)=(1+|\eta|^{2})^\frac{(p_i(x)-2)}{2}\eta,
\end{align*}
for all $i\in \{1, \dots ,N\}$, we have $A_i(x,\eta)=\frac{1}{p_i(x)}[(1+|\eta|^{2})^\frac{{p_i(x)}}{2}-1]$ for all $i\in \{1, \dots ,N\}$. Thus, we obtain the anisotropic variable mean curvature operator
\begin{align*}
\sum_{i=1}^N\partial_{x_i}\Big[(1+|\partial_{x_i}u|^{2})^{\frac{(p_i(x)-2)}{2}}\partial_{x_i}u\Big].
\end{align*}
In \cite{Maria2,Mih1}, it is studied the anisotropic quasilinear  elliptic problem
\begin{equation}\label{e1.2}
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} \Big{(}|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big{)}
 = f(x,u) \quad \text{in } \Omega,\\
u=0  \quad \text{on } \partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega\subset \R^N$ $(N\geq 3)$ is a bounded domain with smooth boundary. In \cite{Maria2}, the authors have considered problem \eqref{e1.2} where $f$ is a                                                                          Carath\'{e}odory function verifying some appropriate conditions. Their arguments are based on the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz \cite{AmbRab}. In \cite{Mih1}, the authors  studied eigenvalue problems where $f(x,u)=\lambda |u|^{q(x)-2}u$ and established the multiplicity of the solution combining the mountain-pass theorem of Ambrosetti and Rabinowitz  with the Ekeland variational principle \cite{Ek}.
Kone et al \cite{Kone} established the existence and uniqueness of a weak energy solution to the nonlinear problem
\begin{equation}\label{e1.3}
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)
 = f \quad \text{in } \Omega,\\
u=0  \quad \text{on } \partial\Omega.
\end{array}\right.
\end{equation}
In \cite{Mih2}, the authors considered \eqref{e1.3} where $f=\lambda(x)|u|^{q(x)-2}u$. Combining the mountain-pass theorem with the Ekeland's variational principle, they proved that under suitable conditions, \eqref{e1.3} has two nontrivial weak solutions. Boureanu \cite{Maria3} proved that problem \eqref{e1.3} has a sequence of weak solutions by means of the symmetric mountain-pass theorem. Motivated by above papers and the work \cite{MihRad}, the goal of this paper is to establish the existence of a sequence of high energy solutions of problem \eqref{e1.1}.
In addition, we consider problem \eqref{e1.1} in a case where the function $f$ has a special form.
A central role in our arguments will be played by the fountain theorem, which is due to Bartsch \cite{ref91}. This result is nicely presented in Willem \cite{Will} by using the quantitative deformation lemma. We also point out that the dual version of the fountain theorem is due to Bartsch and Willem, see \cite{Will}. Both the fountain theorem and its dual form are effective tools for studying the existence of infinitely many large or small energy solutions. It should be noted that the Palais-Smale condition plays an important role for these theorems and their applications.

\section{Abstract framework}
We recall in what follows some definitions and basic properties
of the Lebesgue and Sobolev spaces with variable exponent $L^{p(x)}(\Omega)$
and $W_0^{1,p(x)}(\Omega)$, where $\Omega$ is
a bounded domain in $\RR^N$. As pointed out in \cite{Mih1}, anisotropic Lebesgue and
Sobolev spaces are functional spaces of Lebesgue's and Sobolev's type in which
different space directions have different roles.

Set
$C_+(\overline\Omega)=\{h\in C(\overline\Omega)\,:\,\min_{x\in\overline\Omega}h(x)>1\}.$
For any $h\in C_+(\overline\Omega)$ we define
$$h^+=\sup_{x\in\Omega}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in\Omega}h(x).$$
If $p\in C_+(\overline\Omega)$, we define {\it the variable exponent
Lebesgue space}
$$L^{p(x)}(\Omega)=\{u\,:\, u \mbox{ is a
 measurable real--valued function such that }
\int_\Omega|u(x)|^{p(x)}\;dx<\infty\},$$
endowed with the  {\it Luxemburg norm} defined by
$$|u|_{p(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\},$$
which is a separable and
reflexive Banach space (see \cite{KR}).

If $p\in C_+(\overline\Omega)$, the {variable exponent Sobolev space} $W^{1,p(x)}(\Omega)$
contains all functions $u\in L^{p(x)}(\Omega)$ such that
gradient $\nabla u$ exists almost everywhere and belongs to $[L^{p(x)}(\Omega)]^N$. Then $W^{1,p(x)}(\Omega)$ is a separable and reflexive Banach space with respect to the norm
$$\|u\|=|u|_{p(x)}+|\nabla u|_{p(x)},$$
 As shown by Zhikov \cite{z1,z2} in relationship with the {\it Lavrentiev phenomenon},
the smooth functions are in general not dense in $W^{1,p(x)}(\Omega)$. However, if
 $p\in C_+(\overline\Omega)$ is {\it logarithmic H\"older continuous}, that is,
\begin{equation}\label{H}|p(x)-p(y)|\le-\,\frac M{\log(|x-y|)}\quad\mbox{for all $x$, $y\in\Omega$ such
that }|x-y|\le1/2,\end{equation}
then the smooth functions are dense in $W^{1,p(x)}(\Omega)$. Let $W_0^{1,p(x)}(\Omega)$ denote
the Sobolev space of functions with zero boundary values under the norm $\|\cdot\|$. Furthermore, if  $p\in C_+(\overline\Omega)$
satisfies \eqref{H}, then $C^\infty_0(\Omega)$ is dense in $W_0^{1,p(x)}(\Omega)$,
see \cite[Theorem~3.3]{hh}.
Since $\Omega$ is an open  bounded set and $p\in C_+(\overline\Omega)$ satisfies \eqref{H},
the {\it $p(x)$--Poincar\'e inequality}
$$|u|_{p(x)}\le C |\nabla u|_{p(x)}$$
holds for all $u\in W_0^{1,p(x)}(\Omega)$, where $C$ depends on $p$, $|\Omega|$,
diam$(\Omega)$ and $N$, \cite[Theorem~4.3]{hh}, and so
$$\|u\|_{1,p(x)}=|\nabla u|_{p(x)},$$
is an equivalent norm in $W_0^{1,p(x)}(\Omega)$. Of course also the
norm
$$\|u\|_{p(x)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p(x)}$$
is an equivalent norm in $W_0^{1,p(x)}(\Omega)$.
Hence $W_0^{1,p(x)}(\Omega)$ is a separable and
reflexive Banach space. Note that if $s\in C_+(\overline\Omega)$
and $s(x)<p^\star(x)$ for all $x\in\overline\Omega$,
where $p^\star(x)={Np(x)}/[N-p(x)]$
if $p(x)<N$ and $p^\star(x)=\infty$ if $p(x)\geq N$,
then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{s(\cdot)}(\Omega)$
is compact and continuous.

We now introduce a natural generalization of the function space $W_0^{1,p(x)}(\Omega)$, which
 will play a central role in our statements.
For this purpose,
let us denote by $\overset{\rightarrow} p:\overline\Omega\rightarrow\RR^N$ the vectorial function
$\overset{\rightarrow} p=(p_1,...,p_N).$
We define $W_0^{1,\overset{\rightarrow} p(x)}(\Omega)$, the {\it anisotropic variable exponent Sobolev space},
as the closure of $C_0^\infty(\Omega)$ with respect to the norm
$$\|u\|_{\overset{\rightarrow} p(x)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}.$$
In the case when $p_i\in C_+(\overline\Omega)$ are constant functions  for any $i\in\{1,..,N\}$ the
resulting anisotropic Sobolev space is denoted by  $W_0^{1,\overset{\rightarrow} p}(\Omega)$, where
$\overset{\rightarrow} p$ is the constant vector $(p_1,...,p_N)$. The theory of such spaces was developed in \cite{fraga, rak1,rak2}.
   We point out that $W_0^{1,\overset{\rightarrow} p}(\Omega)$ is a reflexive Banach space for any $\overset{\rightarrow} p\in\RR^N$
with $p_i>1$ for all $i\in\{1,..,N\}$. This result can be easily extended to $W_0^{1,\overset{\rightarrow} p(x)}(\Omega)$.
Indeed, denoting by $X=L^{p_1(\cdot)}(\Omega)\times ... \times L^{p_N(\cdot)}(\Omega)$ and considering the operator
$T:W_0^{1,\overset{\rightarrow} p(x)}(\Omega)\rightarrow X$, defined by $T(u)=\nabla u$, it is clear that
$W_0^{1,\overset{\rightarrow} p(x)}(\Omega)$ and $X$ are isometric by $T$, since $\|Tu\|_X=
\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}=\|u\|_{\overset{\rightarrow} p(x)}$.
Thus, $T(W_0^{1,\overset{\rightarrow} p(x)}(\Omega))$ is a closed subspace of $X$, which is a reflexive Banach space, hence
  $T(W_0^{1,\overset{\rightarrow} p(x)}(\Omega))$ is reflexive and
consequently also $W_0^{1,\overset{\rightarrow} p(x)}(\Omega)$ is a reflexive Banach space.

 We define $X:=W_0^{1,\overrightarrow{p}(x)}(\Omega)$. Since $X$ is reflexive, then
 by \cite{Zhao}, there exist
 $\{e_j\}\subset X$ and $\{e_j^*\}\subset X^*$ such that
$$
X=\overline{\textrm{ span }\{e_j~:j=1,2,...\}},\hspace{0.5cm}X^*=\overline{\textrm{ span }\{e_j^*~:j=1,2,...\}},
$$
and
$$ \langle e_i,e_j^*\rangle=\left\{
\begin{array}{ll}
1        & \mbox { if }\hspace{0.3cm} i=j,\\
0       & \mbox { if }\hspace{0.3cm}i\neq j,
\end{array}
\right.
$$
where $\langle.,.\rangle$ denote the duality product between $X$ and $X^*$. We define
$$
X_j=\textrm{ span }\{e_j\},\hspace{0.5cm}Y_k=\bigoplus_{j=1}^{k}X_j,\hspace{0.5cm}Z_k=\overline{\bigoplus_{j=k}^{\infty}X_j}.
$$



\section{Main results}
An important role in what follows will be played by the vectors
$\overrightarrow{P}_+$, $\overrightarrow{P}_-\in \R^N$ and by the real numbers $P^+_+,~P^+_-,~P^-_+,~P^-_-\in \R^+$ defined as
\begin{gather*}
\overrightarrow P_+ = (p^+_1,p^+_2, \dots , p^+_N), \quad
\overrightarrow P_- = (p^-_1,p^-_2, \dots , p^-_N),\\
P^+_+ = \max\{p^+_1,p^+_2, \dots , p^+_N\}, \quad
P^+_- = \max\{p^-_1,p^-_2, \dots , p^-_N\}, \\
P^-_+ = \min\{p^+_1,p^+_2, \dots , p^+_N\}, \quad
P^-_- = \min\{p^-_1,p^-_2, \dots , {p}^-_N\}.
\end{gather*}

Throughout this paper, we assume that
\begin{align}\label{e1.4}
\sum_{i=1}^N\frac{1}{p_i^-}>1\,.
\end{align}
This condition ensures that the anisotropic space $W_0^{1,\overrightarrow{p}(x)}(\Omega)$ is embedded into some Lebesgue space $L^r(\Omega)$. If hypothesis \eqref{e1.4} is no longer fulfilled, then one has embeddings into Orlicz or H\"older spaces.

Define $P_-^*\in \R^+$ and $P_{-,\infty}\in \R^+$ by
$$
P_-^*=\frac{N}{\sum_{i=1}^N\frac{1}{p_i^-}-1},\hspace{1cm}P_{-,\infty}=\max\{P_-^+,P_-^*\}.
$$

  For the Carath\'{e}odory function $f:\Omega\times\R\rightarrow \R$, we consider the antiderivative $F:\Omega\times\R\rightarrow \R$,
$$
F(x,s)=\int_0^sf(x,t)dt.
$$
With the previous notation, the functions $b,\, f$ satisfy the conditions:
\begin{enumerate}
\item[$(\mathbf{B})$] $b\in L^{\infty}(\Omega)$ and there exists $b_0>0$ such that $b(x)\geq b_0$ for all $x\in \Omega$.
\item[$(\mathbf{F_1})$] There exist a positive constant $c_1$ and $\alpha(x)\in C_+(\overline{\Omega})$ with $\alpha(x)<P_{-,\infty}$ such that
\begin{align}\label{e1.5}
|f(x,t)|\leq c_1(1+|t|^{\alpha(x)-1})\hspace{0.5cm}\textrm{ for all
}\hspace{0.3cm}(x,t)\in \Omega\times\R\,.
\end{align}
\item[$(\mathbf{F_2})$] There exist $M>0,~\theta>P_{+}^+$ such that for all $x\in \Omega$ and all $t\in \R$ with $|t|\geq M,$
\begin{align*}
0<\theta F(x,t)\leq tf(x,t).
\end{align*}
\item[$(\mathbf{F_3})$] The function $f$ is odd with respect to its second variable, that is,
\begin{align*}
f(x,-t)=-f(x,t),
\end{align*}
for all $x\in\Omega$ and $t\in \R$.
\end{enumerate}
\begin{definition}\label{def1.3}
A function $u\in W_0^{1,\overrightarrow{p}(x)}(\Omega)$ that verifies
$$
\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i})\partial_{x_i}\varphi+b(x)|u|^{P^+_+-2}u\varphi-f(x,u)\varphi\Big{\}}dx=0,
$$
for all $\varphi\in W_0^{1,\overrightarrow{p}(x)}(\Omega)$ is called a weak solution of problem \eqref{e1.1}.
\end{definition}

The following result establishes the existence of infinitely many solutions of problem \eqref{e1.1}, provided that the right-hand side is odd.

\begin{theorem}\label{theo1.4}
Problem \eqref{e1.1} admits a sequence $(\pm u_n)$ of weak solutions with high energies.
\end{theorem}
In addition, we consider the case when $f(x,u)=\lambda |u|^{q(x)-2}u$, where the parameter $\lambda$ is positive and $q(x)$ is continuous function on $\overline{\Omega}$. Problem \eqref{e1.1} then become
\begin{equation}\label{e1.6}
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)+b(x)|u|^{P^+_+-2}u
 = \lambda|u|^{q(x)-2}u \quad \text{in } \Omega,\\
 u=0  \quad \text{on } \partial\Omega.
\end{array}\right.
\end{equation}
\begin{definition}\label{def1.5}
A function $u\in W_0^{1,\overrightarrow{p}(x)}(\Omega)$ is said to be a weak solution of problem \eqref{e1.6} if and only if
$$
\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i})\partial_{x_i}\varphi+b(x)|u|^{P^+_+-2}u\varphi-\lambda|u|^{q(x)-2}u\varphi\Big{\}}dx=0,
$$
for all $\varphi\in W_0^{1,\overrightarrow{p}(x)}(\Omega)$.
\end{definition}
Our main result concerning problem \eqref{e1.6} can be described as follow.
\begin{theorem}\label{theo1.6}
Assume that $q^-<P^-_-\leq P^+_+<P_{-, \infty}$ for all $x\in \overline{\Omega}$. Then there exists a positive constant $\lambda^*$ such that for any $\lambda\in (0,\lambda^*)$, problem \eqref{e1.6} has at least one nontrivial weak solution.
\end{theorem}
In the sequel, we use $c_i$ to denote the general nonnegative or positive constant (the exact value may change from line to line).


\section{Infinitely many high energy solutions}
In this section we are concerned with the existence of multiple weak solutions of problem \eqref{e1.1}.
We associated to problem \eqref{e1.1} the energy functional $I:X\rightarrow \R$ defined by
$$
I(u)=\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{P^+_+}|u|^{P^+_+}-F(x,u)\Big{\}}dx.
$$
Due to Lemma 3.4 in \cite{Kone}, by a standard calculus we deduce  that $I$ is well-defined and $I\in C^1(X,\R)$ with
$$
\langle I'(u),\varphi\rangle=\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i}u) \partial_{x_i}\varphi +b(x)|u|^{P^+_+-2}u\varphi-f(x,u)\varphi\Big{\}} dx,
$$
for all $u,\varphi \in X$. Hence any critical point $u\in X$ of $I$ is a weak solution of problem \eqref{e1.1}.
The idea of the proof of Theorem \ref{theo1.4} is to show that all the hypotheses of the fountain theorem (\cite{Will}, Theorem 3.6) are fulfilled. To this end we will prove three corresponding auxiliary properties.
\begin{lemma}\label{lem3.1}
For every $k\in \N$, there exists $r_k>0$ such that $\inf_{u\in Z_k,~||u||=r_k}I(u)\rightarrow\infty$, as $k\rightarrow\infty$.
\end{lemma}
\begin{proof}
By $(\mathbf{A2})$ and $(\mathbf{F_1})$ for any $u\in Z_k, ~||u||=r_k>1$ ($r_k$ will be specified below), we have
\begin{align}\label{e3.1}
I(u)&=\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{P^+_+}|u|^{P^+_+}-F(x,u)\Big{\}}dx\nonumber\\
&\geq\frac{1}{P^{+}_{+}}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}dx+\frac{b_0}{P^+_+}\int_{\Omega}|u|^{P^+_+}dx-c_1\int_{\Omega}(1+|u|^{\alpha(x)})dx\nonumber\\
&\geq\frac{1}{P^{+}_{+}}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}dx+\frac{b_0}{P^+_+}|u|_{L^{P^+_+}(\Omega)}^{P^+_+}- c_2\max\{|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^+},|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^-}\}-c_3.
\end{align}
Using $(\mathbf{B})$, we can write
\begin{align}\label{e3.2}
\frac{1}{P^+_+}\int_{\Omega}b(x)|u|^{P^+_+}dx\geq \frac{b_0}{P^+_+}|u|_{L^{P^+_+}(\Omega)}^{P^+_+}\geq 0.
\end{align}
For each $i \in \{1, 2, \dots , N\}$, we define
\[
\alpha_{i} =
\begin{cases}
P^+_+ & \text{ if } |\partial_{x_i}u|_{p_i(x)} < 1, \\
P^-_- & \text{ if } |\partial_{x_i}u|_{p_i(x)} > 1.
\end{cases}
\]
Using Theorem 1.3 in \cite{Fan2} and Jensen inequality (applied to
the convex function $g:\R^+\rightarrow \R^+$, $g(t)=t^{P^-_-}$,
$P^-_->1$), we have
\begin{equation}\begin{array}{ll}\label{e3.3}
\displaystyle\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}dx&\displaystyle
\geq\sum_{i=1}^N|\partial_{x_i}u|_{p_i(x)}^{\alpha_{i}}\\
&\displaystyle\geq\sum_{i=1}^N|\partial_{x_i}u|_{p_i(x)}^{P_{-}^{-}}-
\sum_{\{i;\alpha_{i}=P^{+}_{+}\}}\Big{(}|\partial_{x_i}u|_{p_i(x)}^{P_{-}^{-}}-|\partial_{x_i}u|
_{p_i(x)}^{P_{+}^{+}}\Big{)}\\
&\displaystyle\geq N\big{(}\frac{\sum_{i=1}^N|\partial_{x_i}u|^{p_i(x)}}{N}\Big{)}^{P^{-}_{-}}-N\\
&\displaystyle=\frac{||u||^{P^{-}_{-}}}{N^{P^{-}_{-}-1}}-N.
\end{array}\end{equation}
Taking into account relations (\ref{e3.2}) and (\ref{e3.3}), the inequality (\ref{e3.1}) reduces to
\begin{eqnarray*}
&&\hspace{-4cm}I(u)\geq\frac{||u||^{P^-_-}}{P^+_+N^{P^{-}_{-}-1}}-c_2\max\{|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^+},|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^-}\}-c_4.
\end{eqnarray*}
Denote
$$
\alpha_k=\sup\{|u|_{L^{\alpha(x)}(\Omega)};~||u||=1,~u\in Z_k\}.
$$
By Proposition 3.5 in \cite{Fan6}, we know that $\lim_{k\rightarrow\infty}\alpha_k=0$.\\
If $\max\{|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^{+}},|u|_{L^{\alpha(x)}(\Omega)}^
{\alpha^{-}}\}=|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^{+}}$, we have
\begin{eqnarray}\label{e3.4}
&&I(u)\geq\frac{||u||^{P^-_-}}{P^+_+N^{P^{-}_{-}-1}}-c_2\alpha_k^{\alpha^+}||u||^{\alpha^+}-c_4.
\end{eqnarray}
Depending on the relation between $P^-_-$ and $\alpha^+$ we distinguish the following cases:\\
$\bf Case\ 1.$ $P^-_->\alpha^+$. For sufficiently large $k$, we have $\alpha_k<\frac{1}{2c_2P^+_+N^{P^{-}_{-}-1}}$, so relation (\ref{e3.4}) yields
\begin{eqnarray*}
&&I(u)\geq\frac{1}{2c_2P^+_+N^{P^{-}_{-}-1}}||u||^{P^-_-}.
\end{eqnarray*}
By choosing $r_k$ such that $r_k\rightarrow\infty$ as $k\rightarrow\infty$, for $u\in Z_k$ with $||u||=r_k$, we have that $I(u)\rightarrow\infty$.\\
$\bf Case2.$ $\alpha^->P^+_+$. Choose $r_k=\Big{(}c_2N^{P^{-}_{-}-1}\alpha^+\alpha_k^{\alpha^+}\Big{)}^{\frac{1}{P^{-}_{-}-\alpha^+}}$, we deduce that
\begin{eqnarray*}
&&I(u)\geq\frac{1}{N^{P^{-}_{-}-1}}\Big{(}\frac{1}{P^+_+}-\frac{1}{\alpha^+}\Big{)}r_k^{P^-_-}-c_4.
\end{eqnarray*}
Since $\alpha_k\rightarrow0$ , $r_k\rightarrow\infty$ as $k\rightarrow\infty$, we obtain $I(u)\rightarrow\infty$.

Similarly, if $\max\{|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^{+}},|u|_{L^{\alpha(x)}(\Omega)}^
{\alpha^{-}}\}=|u|_{L^{\alpha(x)}(\Omega)}^{\alpha^{-}}$, we can deduce for $u\in Z_k$ with $||u||=r_k>1$, $I(u)\rightarrow\infty$ as $k\rightarrow+\infty$ and the proof is complete.
\end{proof}

\begin{lemma}\label{lem3.2}
For every $k\in \N$ there exists $\rho_k>r_k$ ($r_k$ given by Lemma \ref{lem3.1}) such that
\begin{eqnarray*}
&&\max_{u\in Y_k,~||u||=\rho_k}I(u)\leq 0.
\end{eqnarray*}
\end{lemma}

\begin{proof}
From $(\mathbf{A_0})$ and $(\mathbf{A_1})$ we have
\begin{eqnarray*}
&&A_i(x,\eta)=\int_0^1a_i(x,t\eta)\eta dt\leq c_5(|\eta|+\frac{1}{p_i(x)}|\eta|^{p_i(x)}),
\end{eqnarray*}
for all $x\in \overline{\Omega}$ and $\eta\in \R^N$, where $c_5=\max_{i\in\{1, \dots ,N\}}\overline{c}_i$. Therefore
\begin{eqnarray*}
\int_{\Omega}\sum_{i=1}^N A_i(x,\partial_{x_i}u)dx\leq c_5\Big{(}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|dx+\sum_{i=1}^N \int_{\Omega}\frac{|\partial_{x_i}u|^{p_i(x)}}{p_i(x)}dx\Big{)}.
\end{eqnarray*}
Moreover, by rewriting condition $(\mathbf{F_2})$, we can obtain the existence of a positive constant $c_6$ such that
\begin{eqnarray*}
&&F(x,s)\geq c_6|s|^{\theta},
\end{eqnarray*}
for all $x\in \Omega$ and $s\in \R$. Then for any $u\in Y_k\backslash\{0\}$ with $||u||=1$ and $1<\rho_k=t_k$ with $t_k\rightarrow\infty$, we have
\begin{align*}
\begin{split}
I(t_ku)&=\int_{\Omega}\sum_{i=1}^N A_i(x,\partial_{x_i}(t_ku))dx+\frac{1}{P^+_+}\int_{\Omega}b(x)|t_ku|^{P^+_+}dx-\int_{\Omega}F(x,t_ku)dx\\
&\leq c_5 \Big{(}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}(t_ku)|dx+\sum_{i=1}^N \int_{\Omega}\frac{|\partial_{x_i}(t_ku)|^{p_i(x)}}{p_i(x)}dx\Big{)}+\frac{1}{P^+_+}\int_{\Omega}b(x)|t_ku|^{P^+_+}dx\\
&\hspace{0.3cm}-c_6\int_{\Omega}|t_ku|^{\theta}dx+c_7\\
&\leq c_5 t_k^{P^+_+}\sum_{i=1}^N\int_{\Omega}\Big(|\partial_{x_i}u|+\frac{|\partial_{x_i}u|^{p_i(x)}}{P^-_-}\Big) dx+\frac{t_k^{P^+_+}}{P^+_+}\int_{\Omega}b(x)|u|^{P^+_+}dx\\
&\hspace{0.3cm}-c_6 t_k^{\theta}\int_{\Omega}|u|^{\theta}dx+c_7.
\end{split}
\end{align*}
Since dim$\,Y_k<\infty$ and all norms are equivalent in the finite
dimensional space, it is easy to see that
$I(t_ku)\rightarrow-\infty$ as $k\rightarrow\infty$ for $u\in Y_k$,
due to $\theta>P^+_+$. Therefore, we deduce that for $\rho_k$ large
enough ($\rho_k>r_k$),
\begin{eqnarray*}
&&\max_{u\in Y_k,~||u||=\rho_k}I(u)\leq 0.
\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{lemma}\label{lem3.3}
The energy functional $I$ satisfies the Palais-Smale condition.
\end{lemma}
\begin{proof}
Let $(u_n)\subset X$ be a sequence such that
\begin{eqnarray}\label{e3.5}
|I(u_n)|<c_8\hspace{0.5cm}\textrm{ and }\hspace{0.5cm}I'(u_n)\rightarrow0\textrm{ as }\hspace{0.5cm}n\rightarrow\infty.
\end{eqnarray}
We claim that $(u_n)$ is bounded. Arguing by contradiction, we assume that, passing eventually to a subsequence still denote by $(u_n)$, $||u_n||\rightarrow\infty$ as $n\rightarrow\infty$.\\
Using relation (\ref{e3.5}), for $n$ large enough we have
\begin{align}\label{e3.6}
1+c_8+||u_n||&\geq I(u_n)-\frac{1}{\theta} \langle I'(u_n),u_n\rangle\nonumber\\
&\geq \sum_{i=1}^N\int_{\Omega}�\Big{[} A_i(x,\partial_{x_i}u_n)-\frac{1}{\theta}a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\Big{]} dx\nonumber\\
&\hspace{0.3cm}+\Big{(}\frac{1}{P^+_+}-\frac{1}{\theta}\Big{)}\int_{\Omega}b(x)|u|^{P^+_+}dx\nonumber\\
&\hspace{0.5cm}-\Big{[}\int_{\Omega}F(x,u_n)-\frac{1}{\theta}u_nf(x,u_n)\Big{]}dx.
\end{align}
From $(\mathbf{A_2})$, for all $x\in \Omega$ and $i\in \{1, \dots ,N\}$ we have
\begin{eqnarray*}
a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\leq p_i(x) A_i(x,\partial_{x_i}u_n)\leq P^+_+ A_i(x,\partial_{x_i}u_n),
\end{eqnarray*}
which implies
\begin{eqnarray*}
&&-\frac{1}{\theta}a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\geq -\frac{P^+_+}{\theta}A_i(x,\partial_{x_i}u_n).
\end{eqnarray*}
Joining together the previous inequality into relation (\ref{e3.6}) and using $(\mathbf{F_2})$ we obtain
\begin{eqnarray*}
&& 1+c_8+||u_n||\geq \Big{(}1-\frac{P^+_+}{\theta}\Big{)}\sum_{i=1}^N\int_{\Omega}A_i(x,\partial_{x_i}u_n)dx.
\end{eqnarray*}
Again from $(\mathbf{A_2})$ we have
\begin{eqnarray*}
&&A_i(x,\partial_{x_i}u_n)\geq \frac{1}{p_i(x)}|\partial_{x_i}u_n|^{p_i(x)}\geq \frac{1}{P^+_+}|\partial_{x_i}u_n|^{p_i(x)},
\end{eqnarray*}
for all $x\in \Omega$ and $i\in \{1, \dots ,N\}$.\\
Taking into consideration relation (\ref{e3.3}), we obtain
\begin{eqnarray*}
&&1+c_8+||u_n||\geq \Big{(}\frac{1}{P^+_+}-\frac{1}{\theta}\Big{)}\Big{(}\frac{||u_n||^{P^-_-}}{N^{p^{-}_{-}-1}}-N\Big{)}.
\end{eqnarray*}
Dividing the above inequality by $||u_n||^{P^-_-}$ and passing to the limit as $n\rightarrow\infty$ we obtain a contradiction.

It follows that $(u_n)$ is bounded in $X$. This information combined with the fact that $X$ is reflexive implies that there exist a subsequence, still denote by $(u_n)$, and $u_0\in X$ such that $(u_n)$ converges weakly to $u_0$ in $X$.

Due to Theorem 1 in \cite{Mih1}, the embeddings $X\hookrightarrow L^{\alpha(x)}(\Omega)$ and $X\hookrightarrow L^{P^+_+}(\Omega)$  are compact. Thus $(u_n)$ converges strongly to $u_0$ in $L^{\alpha(x)}(\Omega)$ and also in $L^{P^+_+}(\Omega)$.

Using H\"{o}lder type inequality and $(\mathbf{F_1})$,
we deduce that
\begin{align}
&\lim_{n\rightarrow\infty}\int_{\Omega}f(x,u_n)(u_n-u_0)dx=0,\\
&\lim_{n\rightarrow\infty}\int_{\Omega}b(x)|u_n|^{P^+_+-2}u_n(u_n-u_0)dx=0.
\end{align}
Using (\ref{e3.5}) we infer that
\begin{align*}
\lim_{n\rightarrow\infty}\langle I'(u_n),u_n-u_0\rangle=0,\hspace{7cm}
\end{align*}
more precisely,
\begin{align*}
\lim_{n\rightarrow\infty}\int_{\Omega}\Big{[}\sum_{i=1}^N a_i(x,\partial_{x_i}u_n)(\partial_{x_i}u_n-\partial_{x_i}u_0)+b(x)|u_n|^{P^+_+-2}u_n(u_n-u_0)\\
-f(x,u_n)(u_n-u_0)\Big{]}dx=0.
\end{align*}
Combining the above relation with $(3.7)$ and $(3.8)$ we have
\begin{align*}
\lim_{n\rightarrow\infty}\int_{\Omega}\sum_{i=1}^N a_i(x,\partial_{x_i}u_n)(\partial_{x_i}u_n-\partial_{x_i}u_0)dx=0.\hspace{4cm}
\end{align*}
Using Lemma 1 in \cite{Mih2}, we deduce that $(u_n)$ converges strongly
to $u_0$ in $X$, in other words $I$ satisfies the Palais-Smale
condition.
\end{proof}
\subsection{Proof of Theorem \ref{theo1.4} completed}
The fact that the mapping $A_i$ is even and $f$ is odd with respect to their second variable imply that $I$ is even. The proof follows immediately from Lemma \ref{lem3.1}, Lemma \ref{lem3.2}, Lemma \ref{lem3.3} and the fountain theorem.

\section{Case of small positive parameters}
This section is devoted to the proof of Theorem \ref{theo1.6}, which is essentially based on the the Ekeland variational principle \cite{Ek}.
Let us define the functional $I_\lambda:X\rightarrow \R$ by
$$
I_\lambda(u)=\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{P^+_+}|u|^{P^+_+}-\frac{\lambda}{q(x)}|u|^{q(x)}\Big{\}}dx.
$$
Then, the functional $I_\lambda$ associated with problem \eqref{e1.6} is well defined and of $C^1$ class on $X$. Moreover, we have
$$
\langle I_\lambda'(u),\varphi\rangle=\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i}u) \partial_{x_i}\varphi +b(x)|u|^{P^+_+-2}u\varphi-\lambda|u|^{q(x)-2}u\varphi\Big{\}} dx,
$$
for all $u,\varphi\in X$. Thus, weak solutions of problem \eqref{e1.6} are exactly the critical points of the functional $I_\lambda$. Due to the Lemma 3.5 in \cite{Kone}, we can show that $I_\lambda$ is weakly lower semi-continuous in $X$.

We establish the following two auxiliary properties.��
\begin{lemma}\label{lem4.1}
There exists $\lambda^*>0$ such that for any $\lambda\in (0,\lambda^*)$ there exist $\rho,a>0$ such that $I_\lambda(u)\geq a>0$ for any $u\in X$ with $||u||=\rho$.
\end{lemma}
\begin{proof}
Under the conditions of Theorem \ref{theo1.6}, $X$ is continuously
embedded in $L^{q(x)}(\Omega)$. Thus, there exists a positive
constant $c_9$ such that
\begin{eqnarray}\label{e4.1}
|u|_{q(x)}\leq c_9||u||\hspace{0.5cm}\textrm{ for all }\hspace{0.5cm}u\in X.
\end{eqnarray}
Now, let us assume that $||u||\leq \min \{1,\frac{1}{c_9}\}$, where $c_9$ is the positive constant from above. Then we have $|u|_{q(x)}<1$. Using Theorem 1.3 in \cite{Fan2} we find
\begin{eqnarray}\label{e4.2}
\int_{\Omega}|u|^{q(x)}dx\leq |u|_{q(x)}^{q^-}\hspace{0.5cm}\textrm{ for all }\hspace{0.3cm}u\in X\hspace{0.2cm}\textrm{ with }\hspace{0.2cm}||u||=\rho\in (0,1).
\end{eqnarray}
Relations \eqref{e4.1} and \eqref{e4.2} yield
\begin{eqnarray}\label{e4.3}
\int_{\Omega}|u|^{q(x)}dx\leq c_9^{q^-}||u||^{q^-}\hspace{0.5cm}\textrm{ for all }\hspace{0.3cm}u\in X\hspace{0.2cm}\textrm{ with }\hspace{0.2cm}||u||=\rho.
\end{eqnarray}
Using the hypothesis $(\mathbf{A_2})$, $(\mathbf{B})$  and
\eqref{e4.2}, we deduce that for any $u\in X$ with $||u||=\rho$, the
following hold:
\begin{equation}\label{e4.4}\begin{array}{ll}
I_{\lambda}(u)&=\di \int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{P^+_+}|u|^{P^+_+}-\frac{\lambda}{q(x)}|u|^{q(x)}\Big{\}}dx\\
&\di \geq\frac{1}{P^{+}_{+}}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}dx+\frac{b_0}{P^+_+}|u|_{L^{P^+_+}(\Omega)}^{P^+_+}-\frac{\lambda}{q^-}c_9^{q^-}||u||
^{q^-}.\end{array}
\end{equation}
Here, we let $||u|<1$, so $|\partial_{x_i}u|_{p_i(x)}<1, ~i\in \{1,
\dots ,N\}$. For such an element $u$, by Theorem 1.3 in \cite{Fan2},
we have
\begin{equation}\label{e4.5}\begin{array}{ll}
\di\sum_{i=1}^N\int_\Omega|\partial_{x_i}u|^{p_i(x)}dx
&\di\geq \sum_{i=1}^N| \partial_{x_i}u|^{p^+_i}_{p_i(x)}
 \geq \sum_{i=1}^N|\partial_{x_i}u|^{P^+_+}_{p_i(x)}\\
&\di\geq N \Big(\frac{\sum_{i=1}^N|\partial_{x_i}u|_{p_i(x)}}{N}\Big)^{P^+_+}
= \frac{||u||^{P^+_+}}{N^{P^+_+-1}}.\end{array}
\end{equation}
Taking into account relations \eqref{e3.2} and \eqref{e4.5}, the inequality \eqref{e4.4} reduces to
$$
\begin{array}{ll}
I_\lambda(u)&\geq\displaystyle \frac{||u||^{P^+_+}}{P^+_+N^{P^+_+-1}}-\frac{\lambda}{q^-}c_9^{q^-}||u||^{q^-}\\
&=\displaystyle\frac{\rho^{P^+_+}}{P^+_+N^{P^+_+-1}}-\frac{\lambda}{q^-}c_9^{q^-}\rho^{q^-}\\
&\displaystyle =\rho^{q^-}\Big(\frac{1}{P^+_+N^{P^+_+-1}}\rho^{P^+_+-q^-}-\frac{\lambda}{q^-}c_9^{q^-}\Big).\end{array}
$$
If we define
\begin{eqnarray}\label{e4.6}
\lambda^*=\frac{q^-}{2P^+_+N^{P^+_+-1}c_9^{q^-}}\rho^{P^+_+-q^-},
\end{eqnarray}
then for any $\lambda\in (0,\lambda^*)$ and $u\in X$ with $||u||=\rho$, there exists $a=\frac{\rho^{P^+_+}}{2P^+_+N^{P^+_+-1}}$ such that $I_\lambda(u)\geq a>0$.
\end{proof}

\begin{lemma}\label{lem4.2}
For any $\lambda\in (0,\lambda^*)$ where $\lambda^*$ is given by \eqref{e4.6}, there exist $\psi\in X$ such that $\psi\geq0,~\psi\neq 0$ and $I_\lambda(t\psi)<0$ for all $t>0$ small enough.
\end{lemma}

\begin{proof}
From $(\mathbf{A_0})$ and $(\mathbf{A_1})$, we have
\begin{eqnarray*}
&&A_i(x,\eta)=\int_0^1a_i(x,t\eta)\eta dt\leq c_{10}(|\eta|+\frac{1}{p_i(x)}|\eta|^{p_i(x)}),
\end{eqnarray*}
for all $x\in \overline{\Omega}$ and $\eta\in \R^N$, where $c_{10}=\max_{i\in\{1, \dots ,N\}}\overline{c}_i$. Therefore
\begin{eqnarray*}
\int_{\Omega}\sum_{i=1}^N A_i(x,\partial_{x_i}u)dx\leq c_{10}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}u|+\frac{|\partial_{x_i}u|^{p_i(x)}}{p_i(x)}\Big{)}dx.
\end{eqnarray*}
By the conditions of Theorem \ref{theo1.6}, $q^-<P^-_-$. Let $\epsilon_0>0$ such that $q^-+\epsilon_0<P^-_-$. Since $q\in C(\overline{\Omega})$, there exists an open set $\Omega_0\subset \Omega$ such that $|q(x)-q^-|<\epsilon_0$ for all $x\in \Omega_0$. It follows that $q(x)<q^-+\epsilon_0<P^-_-$ for all $x\in \Omega_0$.

Let $\psi \in C_0^{\infty}(\Omega)$ be such that supp$\,(\psi)\supset \overline{\Omega_0}$, $\psi(x)=1$ for all $x\in \overline{\Omega_0}$ and $0\leq \psi\leq1$ in $\Omega$. Then for any $t\in(0,1)$, we have
$$
\begin{array}{ll}
I_\lambda(t\psi)&=\di\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}(t\psi))+\frac{b(x)}{P^+_+}|t\psi|^{P^+_+}-\frac{\lambda}{q(x)}|t\psi|^{q(x)}\Big{\}}dx\\
&\di \leq c_{10} \sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}(t\psi)|+\frac{|\partial_{x_i}(t\psi)|^{p_i(x)}}{p_i(x)}\Big{)}dx+\frac{1}{P^+_+}\int_{\Omega}b(x)|t\psi|^{P^+_+}dx\\
&\di -\lambda\int_{\Omega}\frac{1}{q(x)}|t\psi|^{q(x)}dx\\
&\di \leq c_{10} t^{P^-_-}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{P^-_-}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{t^{P^+_+}}{P^+_+}
\int_{\Omega}b(x)|\psi|^{P^+_+}dx\\
&\di -\frac{\lambda}{q^+}\int_{\Omega_0}t^{q(x)}|\psi|^{q(x)}dx\\
&\di \leq c_{10} t^{P^-_-}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{P^-_-}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{t^{P^+_+}}{P^+_+}
\int_{\Omega}b(x)|\psi|^{P^+_+}dx\\
&\di -\frac{\lambda t^{q^-+\epsilon_0}}{q^+}\int_{\Omega_0}|\psi|^{q(x)}dx.\end{array}
$$
So, $I_\lambda(t\psi)<0$ for $t<\delta^{\frac{1}{P^-_--q^--\epsilon_0}}$, with
\begin{eqnarray*}
0<\delta<\min\Biggl\{1,\frac{\lambda}{q^+}\frac{\int_{\Omega_0}|\psi|^{q(x)}dx}{c_{10}
\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{P^-_-}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{1}{P^+_+}\int_{\Omega}b(x)|\psi|^{P^+_+}dx
}\Biggl\}.
\end{eqnarray*}
This completes the proof.
\end{proof}

\subsection{Proof of Theorem \ref{theo1.6} completed}
Let $\lambda^*$ be defined as in \eqref{e4.6} and $\lambda\in (0,\lambda^*)$. By Lemma \ref{lem4.1}, it follows that on the boundary of the ball centered at the origin and of radius $\rho$ in $X$, we have
$$
\inf_{\partial B_{\rho}(0)}I_\lambda(u)>0.
$$
On the other hand, by Lemma \ref{lem4.2}, there exists $\psi\in X$ such that
$$
I_\lambda(t\psi)<0 \hspace{0.3cm} \text{ for }\hspace{0.3cm} t>0\hspace{0.3cm} \text{ small enough }.
$$
Moreover, for $u\in B_{\rho}(0)$,
$$
I_\lambda(u)\geq \frac{||u||^{P^+_+}}{P^+_+N^{P^+_+-1}}-\frac{\lambda}{q^-}c_9^{q^-}||u||^{q^-}.
$$
It follows that
$$
-\infty<c_{11}=\inf_{\overline{B_{\rho}(0)}}I_\lambda(u)<0.
$$
We let now $0<\varepsilon<\inf_{\partial B_{\rho}(0)}I_\lambda-\inf_{B_{\rho}(0)}I_\lambda$. Applying Ekeland variational
principle \cite{Ek} to the functional $I_\lambda:\overline{B_{\rho}(0)}\rightarrow \R$, we
find $u_{\varepsilon}\in \overline{B_{\rho}(0)}$ such that
$$
\begin{array}{ll}
I_\lambda(u_{\varepsilon})& \di<\inf_{\overline{B_{\rho}(0)}}I_\lambda+\varepsilon\\
I_\lambda(u_{\varepsilon})& \di<I_\lambda(u)+\varepsilon||u-u_{\varepsilon}||,\hspace{0.2cm}u\neq u_{\varepsilon}.\end{array}
$$
Since
$$
I_\lambda(u_{\varepsilon})\leq\inf_{\overline{B_{\rho}(0)}}I_\lambda+\varepsilon\leq \inf_{B_{\rho}(0)}I_\lambda+\varepsilon<\inf_{\partial B_{\rho}(0)}I_\lambda,
$$
we deduce that $u_{\varepsilon}\in B_{\rho}(0)$. Now, we define $K_\lambda:\overline{B_{\rho}(0)}\rightarrow \R$
by $K_\lambda(u)=I_\lambda(u)+\varepsilon ||u-u_{\varepsilon}||$. It is clear that $u_{\varepsilon}$ is a minimum point of $K_\lambda$ and thus
$$
\frac{K_\lambda(u_{\varepsilon}+tv)-K_\lambda(u_{\varepsilon})}{t}\geq0,
$$
for small $t>0$ and $v\in B_{\rho}(0)$. The above relation yields
$$
\frac{I_\lambda(u_{\varepsilon}+tv)-I_\lambda(u_{\varepsilon})}{t}+\varepsilon ||v||\geq0.
$$
Letting $t\rightarrow0$ it follows that $\langle
I_\lambda'(u_{\varepsilon}),v\rangle+\varepsilon ||v||>0$ and we
infer that $||I_\lambda'(u_{\varepsilon})||\leq \varepsilon$. We
deduce that there exists a sequence $(v_n)\subset B_{1}(0)$ such
that
\begin{equation}\label{e4.7}
I_\lambda(v_n)\rightarrow c_{11} \hspace{0.3cm} \text{ and }\hspace{0.3cm} I_\lambda'(v_n)\rightarrow 0.
\end{equation}
It is clear that $(v_n)$ is bounded in $X$. Thus, there exists $u_1\in X$ such that, up to a subsequence, $(v_n)$
converges weakly to $u_1$ in $X$. Theorem 1 in \cite{Mih1} yields that the embeddings $X\hookrightarrow L^{q(x)}(\Omega)$ and $X\hookrightarrow L^{P^+_+}(\Omega)$ are continuous and compact. Then $(v_n)$ converges strongly to $u_1$ in $L^{q(x)}(\Omega)$ and in $L^{P^+_+}(\Omega)$.\\
Using H\"{o}lder type inequality, we can easily get
that
\begin{align*}
&\lim_{n\rightarrow\infty}\int_{\Omega}|v_n|^{q(x)-2}v_n(v_n-u_1)dx=0,\\
&\lim_{n\rightarrow\infty}\int_{\Omega}b(x)|v_n|^{P^+_+-2}v_n(v_n-u_1)dx=0.\hspace{5cm}
\end{align*}
On the other hand, relation \eqref{e4.7} yields
\begin{align*}
\lim_{n\rightarrow\infty}\langle I'(v_n),v_n-u_1\rangle=0,\hspace{7cm}
\end{align*}
Using the above information, we find
\begin{align*}
\lim_{n\rightarrow\infty}\int_{\Omega}\sum_{i=1}^N a_i(x,\partial_{x_i}v_n)(\partial_{x_i}v_n-\partial_{x_i}u_1)dx=0.\hspace{4cm}
\end{align*}
Using Lemma 1 in \cite{Mih2}, we deduce that $(v_n)$ converges strongly to $u_1$ in $X$. So, by \eqref{e4.7}
$$
I_\lambda(u_1)=c_{11}<0 \hspace{0.3cm} \text{ and }\hspace{0.3cm} I_\lambda'(u_1)=0,
$$
that is, $u_1$ is a nontrivial weak solution for the problem \eqref{e1.6}. This completes the proof.\qed


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