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\begin{document}

\title[A nonlinear anisotropic problem with no-flux boundary condition]{The variational analysis of a nonlinear anisotropic problem with no-flux boundary condition}

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


\keywords{Degenerate anisotropic Sobolev spaces; Variable exponent; Boundary value condition; Variational methods. \\
$^\ast$ Corresponding author \\
\hspace*{.3cm} {\it 2000 Mathematics Subject Classifications}: 35J62, 35J70, 46E35.}

%\date{\bf 9 November 2013}

\begin{abstract}

We study the nonlinear degenerate anisotropic problem
$$
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)+b(x)|u|^{p_M(x)-2}u
 = \lambda|u|^{q(x)-2}u \quad \text{in } \Omega,\\
u(x)=\text{constant}  \quad \text{on } \partial\Omega\\
\sum_{i=1}^N\int_{\partial\Omega}a_i(x,\partial_{x_i}u)\nu_id\sigma =0,
\end{array}\right.
$$
where $\Omega\subset\R^N$ is a bounded domain with smooth boundary.
 The constant value of the boundary data is not specified, whereas the  zero integral term corresponds to a no-flux boundary condition. In the case when $|u|^{q(x)-2}u$ ``dominates" the left-hand side, we show that a nontrivial solution exists for all positive values of $\lambda$. If the term $|u|^{q(x)-2}u$ is dominated by the left-hand side, we prove that a solution exists either for small or for large values of $\lambda>0$.
 The proofs combine variational arguments with energy estimates.

\end{abstract}

\maketitle

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

In this paper we are concerned with the existence of weak nontrivial solutions for a class of anisotropic equations in bounded domains of
the Euclidean space. Our main results in the present paper continue and extend the work by Boureanu and Udrea \cite{boud}.

Anisotropic operators appear in several places in the literature.
Recent relevant applications include models in physics \cite{rref13,rref17,rref18,rref23,rref24}, biology \cite{rref10,rref11}, and
image processing (see, for instance, the monograph by Weickert \cite{rref50}). By definition,
anisotropic operators involve directional derivatives with distinct weights.
Relevant references on the theory of anisotropic
Sobolev spaces are Besov \cite{rref12}, Kruzhkov and Kolodii \cite{rref28}, Kruzhkov and
Korolev \cite{rref29}, Nikol�skii \cite{rref36}, Rakosnik \cite{rref40,rref41}, Troisi \cite{rref48}, and Ven�-tuan
\cite{rref49}. We also refer to Fragala,  Gazzola,  Kawohl \cite{gazzola} and El Hamidi, V\'etois \cite{hamidi} as basic references in the treatment of nonlinear anisotropic problems and to the book by Antontsev,  Diaz,  Shmarev \cite{anton} as a source of valuable energy methods in the qualitative analysis of nonlinear boundary value problems.

No-flux problems were studied for the first time by Berestycki and Brezis \cite{bere}, in relationship with models arising in plasma physics.  As stated in \cite{bere}, these problems stem ``from a model describing the equilibrium of a plasma confined in a toroidal  cavity (a Tokamak machine)"; see also \cite{ber}.


\section{Statement of the problem}
The purpose of this work is to analyze the existence of weak solutions of the anisotropic problem
\begin{equation}\label{e2.1}
\left\{\begin{array}{ll}
- \sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)+b(x)|u|^{p_M(x)-2}u
 = \lambda|u|^{q(x)-2}u \quad \text{in } \Omega,\\
u(x)=\text{constant}  \quad \text{on } \partial\Omega\\
\displaystyle\sum_{i=1}^N\int_{\partial\Omega}a_i(x,\partial_{x_i}u)\nu_id\sigma =0,
\end{array}\right.
\end{equation}
where $\Omega\subset\R^N~(N\geq 2)$ is a bounded domain with smooth boundary, $b\in L^{\infty}(\Omega)$ and $a_i:\overline{\Omega}\times \R\rightarrow \R$ are Carath\'{e}odory functions fulfilling some adequate hypotheses.

We notice that the constant value of the boundary data in problem \eqref{e2.1} is not specified and corresponds to the one-dimensional case $u(0)=u(1)$, whereas the requirement in one-dimension that $u'(0)=u'(1)$, corresponds to the {\it no-flux} boundary integral term, in the case that $a_i(x,\xi)=\xi$ for all $i=1,\ldots,N$. We also point out that Zou, Li, Liu and Lv \cite{zou1,zou2} studied the existence of solutions for the problem
\begin{equation*}
\left\{\begin{array}{ll}
- \Delta u=f(x,u) \quad \text{in } \{u>0\},\\
-\Delta u=0  \quad \text{in } \{u\leq0\},\\
u=c~\text{(a negative constant)} \quad \text{on } \partial\Omega\\
-\displaystyle\int_{\partial\Omega}\frac{\partial u}{\partial n}d\sigma=I~\text{(a given positive constant)},
\end{array}\right.
\end{equation*}
where $\Omega$  is bounded, open and connected subset of $\R^{2}$ with regular boundary with the outward unit normal $n$. Problems of this type are related to plasma fusion  and plasma confinement in Tokamak devices. The set $\{u>0\}$ represents the region filled by the plasma, the set $\{u<0\}$ represents the vacuum region, and the set $\{u=0\}$ corresponds to to the free boundary that separate the plasma and the vacuum. The case studied in problem \eqref{e2.1} corresponds to {\it nonresonant surfaces}, namely no-flux surfaces on which the wave number of the perturbation parallel to the equilibrium magnetic field is zero.
Unfortunately, we are not able to give some {\it a posteriori} information on the constant value of the
solution on the boundary in problem \eqref{e2.1}.


The differential operator $\sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)$ is a $\overrightarrow{p}(\cdot)$-Laplace type operator, $\overrightarrow{p}(x)=(p_1(x),p_2(x),\dots,p_N(x))$. For $i=1, \dots ,N$, $p_i(x)$ and $q(x)$ are continuous functions on $\overline{\Omega}$, while
$a_i(x,\eta)$ is 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[$(\mathbf{A_4})$] $a_i(x,0)=0$ for all $x\in \partial\Omega$.
\item[$(\mathbf{B})$] $b\in L^{\infty}(\Omega)$ and there exists $b_0>0$ such that $b(x)\geq b_0$ for all $x\in \Omega$.
\end{enumerate}

The operator presented above is the {\it anisotropic $\overrightarrow{p}(x)$-Laplace  operator}  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 $i\in \{1, \dots ,N\}$. Therefore
\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*}

There are many other operators deriving from $\sum_{i=1}^N\partial_{x_i} a_i(x,\partial_{x_i}u)$. Indeed, 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 $i\in \{1, \dots ,N\}$ and we obtain the {\it 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*}

 A feature of the present paper is that we do not assume a zero Dirichlet boundary condition, but we work in the anisotropic variable exponent space of  functions that are constant on the boundary and fulfill a no-flux boundary integral condition.

\section{Abstract setting}
In this section, we recall some definitions and basic properties of the spaces with variable exponent together with some results that are needed in the sequel.

For any $\Omega\subset\R^N$, we set
\begin{align*}
C_+(\overline{\Omega})=\{h\in C(\overline{\Omega});\ 1<\min_{x\in \Omega}h(x)\}.
\end{align*}
Define
$$
h^+=\max\{h(x);~x \in\Omega\},\qquad h^-=\min\{h(x);~x \in\Omega\}.$$
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  $\emph{Luxemburg norm}$
\begin{align*}
\|u\|_{L^{p(x)}(\Omega)}=\inf \Big\{\mu>0;~\int_{\Omega}|\frac{u(x)}{\mu}|^{p(x)}dx\leq 1\Big\}.
\end{align*}
Then $(L^{p(x)}(\Omega),\|\cdot\|_{L^{p(x)}(\Omega)})$ is a separable and reflexive Banach space \cite[Theorem 2.5, Corollary 2.7]{Kov}.
Also, by Theorem 2.8 in \cite{Kov}, the embedding $L^{p_2(x)}({\Omega})\hookrightarrow L^{p_1(x)}({\Omega})$ is continuous, provided that
$\Omega$ is bounded and $p_1$, $p_2\in C_+(\overline\Omega)$ are such that $p_1 \leq p_2$
in ${\Omega}$.



The {\it isotropic Sobolev space} $W^{1,p(x)}(\Omega)$ is defined by
$$
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega);~\partial_{x_i}u\in L^{p(x)}(\Omega),~i\in \{1, \dots ,N\}\}.
$$
If equipped with the norm
$$
\|u\|_{W^{1,p(x)}(\Omega)}=\|u\|_{L^{p(x)}(\Omega)}+\sum_{i=1}^N\|\partial_{x_i}u\|_{L^{p(x)}(\Omega)},
$$
then ($W^{1,p(x)}(\Omega),\|\cdot\|_{W^{1,p(x)}(\Omega)}$) is a separable and reflexive Banach space (see \cite[Theorem 1.3]{Kov}).

The application $\rho_{p(x)}(u)=\int_{\Omega}|u|^{p(x)}dx$ called the {\it $p(x)$-modular} of $L^{p(x)}(\Omega)$ space, is useful in handling the Lebesgue space with variable exponent. Indeed, cf. \cite[Theorems 1.3 and 1.4]{Fan2}, if $u\in L^{p(x)}(\Omega)$  then
\begin{gather}\label{mod1}
\|u\|_{L^{p(x)}({\Omega})}<1\;(=1;\,>1)\;\Leftrightarrow\;
\rho_{p(x)}(u)<1\;(=1;\,>1);
\\
\label{mod2}
\|u\|_{L^{p(x)}({\Omega})}>1\;\Rightarrow\;
\|u\|_{L^{p(x)}({\Omega})}^{p^-}\leq\rho_{p(x)}(u)
\leq\|u\|_{L^{p(x)}({\Omega})}^{p^+};
\\
\label{mod3}
\|u\|_{L^{p(x)}({\Omega})}<1\; \Rightarrow\;
\|u\|_{L^{p(x)}({\Omega})}^{p^+}\leq \rho_{p(x)}(u)
\leq\|u\|_{L^{p(x)}({\Omega})}^{p^-};
\\
\label{mod4}
\|u\|_{L^{p(x)}({\Omega})}\to 0\;(\to\infty)\; \Leftrightarrow\;
\rho_{p(x)}(u) \to 0\; (\to\infty).
 \end{gather}
In addition, if $(u_n)\subset L^{p(x)}(\Omega)$, then
\begin{eqnarray*}
\lim_{n\rightarrow\infty}\|u_n-u\|_{L^{p(x)}(\Omega)}=0\Leftrightarrow\lim_{n\rightarrow\infty}\rho_{p(x)}(u_n-u)=0\\
\Leftrightarrow  u_n \text{ converges to }u \text{ in measure and }\lim_{n\rightarrow\infty}\rho_{p(x)}(u_n)=\rho_{p(x)}(u).
\end{eqnarray*}

Finally, we introduce a natural generalization of the function space $W^{1,p(x)}(\Omega)$ that will enable us to study with sufficient accuracy problem (\ref{e2.1}). For this purpose, let us denote by $\overrightarrow{p}:\overline{\Omega}\rightarrow \R^N$ the vectorial function $\overrightarrow{p}(x)=(p_1(x),p_2(x),...,p_N(x))$ with $p_i(x)\in C_{+}(\overline{\Omega}),~i\in \{1, \dots ,N\}$ and we put
$$
p_M(x)=\max\{p_1(x),\dots,p_N(x)\},\hspace{0.3cm}p_m(x)=\min\{p_1(x),\dots,p_N(x)\}.
$$
The {\it anisotropic Sobolev space with variable exponent} is
$$
W^{1,\overrightarrow{p}(x)}(\Omega)=\{u\in L^{p_M(x)}(\Omega): \partial_{x_i}u\in L^{p_i(x)}(\Omega)\hspace{0.1cm} \mbox{for all}\hspace{0.1cm} i\in\{1,\dots,N\}\}.
$$
This space is endowed with the norm
$$
\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\|u\|_{L^{p_M(x)}(\Omega)}+\sum_{i=1}^N\|\partial_{x_i}u\|_{L^{p_i(x)}(\Omega)}.
$$
The space ($W^{1,\overrightarrow{p}(x)}(\Omega),\|\cdot\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}$) is a reflexive Banach space (see \cite[Theorems 2.1 and 2.2]{Fancomp}).
Set
$$
X=\left\{u\in W^{1,\overrightarrow{p}(x)}(\Omega): u|_{\partial\Omega}\equiv \mbox{constant}\right\}.
$$
Since $X$ is a closed subset of $W^{1,\overrightarrow{p}(x)}(\Omega)$, it follows that $X$ is a reflexive Banach space.

We also recall (see \cite[Theorem 2.2]{Ji}) that if
 $\Omega\subset \R^N$ is a bounded domain with smooth boundary and $q\in C_+(\overline{\Omega})$ satisfies  $q(x)<\frac{Np_m^-}{N-p_m^-}$ for all $x\in \overline{\Omega}$, then  the embedding
$
 W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)
$
is  compact.

In the sequel, we use $c_i$ and $\widetilde{c_i}$ to denote  general nonnegative or positive constants (the exact value may change from line to line).


\section{The first domination case}

We start by giving the definition of weak solution of problem \eqref{e2.1}.

\begin{definition}\label{def4.1}
A function $u\in X$ that verifies
$$
\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i}u)\partial_{x_i}\varphi+b(x)|u|^{p_M(x)-2}u\varphi-\lambda|u|^{q(x)-2}u\varphi\Big{\}}\,dx=0,
$$
for all $\varphi\in X$ is called a weak solution of problem \eqref{e2.1}.
\end{definition}
We associate to problem \eqref{e2.1} the energy functional $I_\lambda:X\rightarrow \R$ defined by
$$
I_\lambda(u)=\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{p_M(x)}|u|^{p_M(x)}-\frac{\lambda}{q(x)}|u|^{q(x)}\Big{\}}\,dx.
$$
Then $I_\lambda$ is well-defined and $I_\lambda\in C^1(X,\R)$ with
$$
\langle I_\lambda'(u),\varphi\rangle=\int_{\Omega}\Big{\{}\sum_{i=1}^N a_i(x,\partial_{x_i}u) \partial_{x_i}\varphi +b(x)|u|^{p_M(x)-2}u\varphi-\lambda|u|^{q(x)-2}u\varphi\Big{\}} \,dx,
$$
for all $u,\varphi \in X$. Hence any critical point $u\in X$ of $I_\lambda$ is a weak solution of problem \eqref{e2.1}.

The next result shows that if the variable exponent $q(\,\cdot\,)$ ``dominates" $p(\,\cdot\,)$ then the solution exists for all positive values of the parameter $\lambda$.
In other words, the right-hand side of problem \eqref{e2.1} is ``stronger" than the other side, even if $\lambda>0$ is small.

\begin{theorem}\label{theo4.2}
Assume that the function $q\in C_+(\overline{\Omega})$ verifies the hypothesis
$$
p^+_M<q^-\leq q^+<\frac{Np_m^-}{N-p_m^-}.
$$
Then for any $\lambda>0$ problem \eqref{e2.1} possesses a nontrivial weak solution.
\end{theorem}

We first prove two auxiliary results.

\begin{lemma}\label{lem4.3}
There exist $\eta>0$ and $\alpha>0$ such that $I_\lambda(u)\geq \alpha>0$ for any $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\eta$.
\end{lemma}
\begin{proof}
First, we point out that
\begin{eqnarray*}
|u(x)|^{q(x)}\leq |u(x)|^{q^-}+|u(x)|^{q^+},\hspace{0.5cm}\textrm{ for all }\hspace{0.2cm}x\in \overline{\Omega}.
\end{eqnarray*}
By the above inequality, $(\mathbf{A_2})$ and $(\mathbf{B})$, we find
\begin{equation}\label{e4.1}\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_M(x)}|u|^{p_M(x)}-\frac{\lambda}{q(x)}|u|^{q(x)}\Big{\}}\,dx\\
&\di \geq\frac{1}{p^{+}_{M}}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}\,dx+\frac{b_0}{p^+_M}\int_{\Omega}|u|^{p_M(x)}\,dx-\frac{\lambda}{q^-}
\Big(\|u\|_{L^{q^-}(\Omega)}^{q^-}+\|u\|_{L^{q^+}(\Omega)}^{q^+}\Big).\end{array}
\end{equation}

>From the hypotheses of Theorem \ref{theo4.2}, $W^{1,\overrightarrow{p}(x)}(\Omega)$ is continuously embedded in $L^{q^-}(\Omega)$ and $L^{q^+}(\Omega)$. Then, there exist two positive constants $c_1$ and $c_2$ such that
 for all $u\in X$
\begin{eqnarray}\label{e4.2}
|u(x)|_{L^{q^-}(\Omega)}\leq c_1\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)} \hspace{0.3cm}\textrm{ and }\hspace{0.3cm} |u(x)|_{L^{q^+}(\Omega)}\leq c_2\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\,.
\end{eqnarray}
Here, we let $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}<1$, so $\|u\|_{L^{p_M(x)}(\Omega)}<1$ and $\|\partial_{x_i}u\|_{L^{p_i(x)}(\Omega)}<1, ~i\in \{1,
\dots ,N\}$.

Taking into account relations \eqref{mod3} and \eqref{e4.2}, the inequality \eqref{e4.1} reduces to
$$
\begin{array}{ll}
I_\lambda(u)&\geq\displaystyle \frac{\min\{1,b_0\}}{p_M^+(N+1)^{p_M^+-1}}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^+_M}\\ &-\displaystyle \frac{\lambda}{q^-}\Big[(c_1\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)})^{q^-}+(c_2\|u\|
_{W^{1,\overrightarrow{p}(x)}(\Omega)})^{q^+}\Big]\\
&=\displaystyle\Big(c_3-c_4\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{q^--p^+_M}-c_5\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{q^+-p^+_M}\Big)
\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^+_M},\end{array}
$$
for any $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}<1$. Since the function $g:[0,1]\rightarrow \R$ defined by
$$
g(t)=c_3-c_4t^{q^--p^+_M}-c_5t^{q^+-p^+_M}
$$
is positive in a neighborhood of the origin, the conclusion of the lemma follows.
\end{proof}


\begin{lemma}\label{lem4.4}
There exists $e\in X$ with $\|e\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}>\eta$ (where $\eta$ is given in Lemma \ref{lem4.3}) such that $I_\lambda(e)<0$.
\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_6(|\eta|+\frac{1}{p_i(x)}|\eta|^{p_i(x)}),
\end{eqnarray*}
for all $x\in \Omega$ and $\eta\in \R^N$, where $c_6=\max_{i\in\{1, \dots ,N\}}\overline{c}_i$. Therefore
\begin{eqnarray}\label{e4.3}
\int_{\Omega}\sum_{i=1}^N A_i(x,\partial_{x_i}u)dx\leq c_6\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}
Let $\varphi\in X$, $\varphi\neq0$. For any $t>1$, we find
$$
\begin{array}{ll}
I_\lambda(t\varphi)&=\di\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}(t\varphi))+\frac{b(x)}{p_M(x)}|t\varphi|^{p_M(x)}-\frac{\lambda}{q(x)}|t\varphi|^{q(x)}\Big{\}}\,dx\\
&\di \leq c_6 \sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}(t\varphi)|+\frac{|\partial_{x_i}(t\varphi)|^{p_i(x)}}{p_i(x)}\Big{)}dx+\frac{1}{p^+_M}\int_{\Omega}b(x)|t\varphi|^{p_M(x)}\,dx\\
&\di -\lambda\int_{\Omega}\frac{1}{q(x)}|t\varphi|^{q(x)}\,dx\\
&\di \leq c_6 t^{p^+_M}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\varphi|+\frac{1}{p^-_m}|\partial_{x_i}\varphi|^{p_i(x)}\Big)\,dx+\frac{t^{p^+_M}}{p^+_M}
\int_{\Omega}b(x)|\varphi|^{p_M(x)}\,dx\\
&\di -\frac{\lambda t^{q^-}}{q^+}\int_{\Omega}|\varphi|^{q(x)}\,dx.\end{array}
$$
Since $p^+_M<q^-$, we infer that $\lim_{t\rightarrow\infty}I_\lambda(t\varphi)=-\infty$. Then for $t>1$ large enough, we can take $e=t\varphi$ such that $\|e\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}>\eta$ and $I_\lambda(e)<0$.
\end{proof}

Now, we prove the following useful property.

\begin{lemma}\label{lem4.5}
Let $\Omega\subset \R^N$ ($N\geq 2$) be a bounded domain with smooth boundary. Assume that the sequence $(u_n)$ converges weakly to $u$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$ and
$$
\limsup_{n\rightarrow \infty}\int_{\Omega}\sum_{i=1}^N a_i(x,\partial_{x_i}u_n)(\partial_{x_i}u_n-\partial_{x_i}u)\,dx\leq 0.
$$
Then $(u_n)$ converges strongly to $u$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$.
\end{lemma}
\begin{proof}
 We use the fact that $W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{p_M(x)}(\Omega)$ compactly. Since $u_n\rightharpoonup u$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$, we deduce that
\begin{equation}\label{e4.50}
u_n\rightarrow u\hspace{0.2cm}\textrm{ in }\hspace{0.2cm}L^{p_M(x)}(\Omega).
\end{equation}
Then, combining \eqref{e4.50} and \eqref{mod4} we conclude that $u_n\rightarrow u$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$.
\end{proof}



\subsection*{Proof of Theorem \ref{theo4.2}}
By Lemmas \ref{lem4.3}--\ref{lem4.4} and the mountain pass theorem of Ambrosetti and Rabinowitz \cite{AmbRab}, we deduce the existence of a sequence $(u_n)\subset X$ such that
\begin{eqnarray}\label{e4.5}
I_\lambda(u_n)\rightarrow c_7>0\hspace{0.5cm}\textrm{ and }\hspace{0.5cm}I_\lambda'(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, up to a subsequence still denoted by $(u_n)$, we have $\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\rightarrow\infty$ as $n\rightarrow\infty$.

Using relation (\ref{e4.5}), for $n$ large enough, we have
\begin{equation}\label{e4.6}\begin{array}{ll}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}&\geq\di I_\lambda(u_n)-\frac{1}{q^-} \langle I_\lambda'(u_n),u_n\rangle\\
& \di \geq \sum_{i=1}^N\int_{\Omega}�\Big{[}A_i(x,\partial_{x_i}u_n)+\frac{1}{p_M(x)}b(x)|u_n|^{p_M(x)}-\frac{\lambda}{q(x)}|u_n|^{q(x)}\Big{]} \,dx\\
& \di-\frac{1}{q^-}\sum_{i=1}^N\int_{\Omega}�\Big{[}a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n+b(x)|u_n|^{p_M(x)}-\lambda|u_n|^{q(x)}\Big{]}\, dx\\
& \di \geq \sum_{i=1}^N\int_{\Omega}�\Big{[} A_i(x,\partial_{x_i}u_n)-\frac{1}{q^-}a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\Big{]}\, dx\\
& \di +
\Big{(}\frac{1}{p_M^+}-\frac{1}{q^-}\Big{)}\int_{\Omega}b(x)|u_n|^{p_M(x)}\,dx+\lambda \int_{\Omega}\Big{(}\frac{1}{q^-}-\frac{1}{q(x)}\Big{)}|u_n|^{q(x)}\,dx.\end{array}
\end{equation}
>From $(\mathbf{A_2})$, for all $x\in \Omega$ and $i\in \{1, \dots ,N\}$ we have
$$
a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\leq p_i(x) A_i(x,\partial_{x_i}u_n)\leq p^+_M A_i(x,\partial_{x_i}u_n),
$$
which implies
$$
-\frac{1}{q^-}a_i(x,\partial_{x_i}u_n)\partial_{x_i}u_n\geq -\frac{p^+_M}{q^-}A_i(x,\partial_{x_i}u_n).
$$
Inserting this inequality into relation \eqref{e4.6} we obtain
$$
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\geq \Big{(}1-\frac{p^+_M}{q^-}\Big{)}\sum_{i=1}^N\int_{\Omega}A_i(x,\partial_{x_i}u_n)\,dx+b_0\Big(\frac{1}{p_M^+}-\frac{1}{q^-}\Big)\int_{\Omega}|u_n|^{p_M(x)}\,dx.
$$
Again from $(\mathbf{A_2})$ we have
$$
A_i(x,\partial_{x_i}u_n)\geq \frac{1}{p_i(x)}|\partial_{x_i}u_n|^{p_i(x)}\geq \frac{1}{p^+_M}|\partial_{x_i}u_n|^{p_i(x)},
$$
for all $x\in \Omega$ and $i\in \{1, \dots ,N\}$, thus
\begin{align}\label{e4.7}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\geq \Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u_n|^{p_i(x)}\,dx+b_0\Big(\frac{1}{p_M^+}-
\frac{1}{q^-}\Big)\int_{\Omega}|u_n|^{p_M(x)}\,dx.
\end{align}
We denote
$$
\ell_1=\{i\in \{1,\dots,N\}:\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}\leq 1\}
$$
and
$$
\ell_2=\{i\in \{1,\dots,N\}:\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}> 1\}.
$$
By \eqref{mod1}, \eqref{mod2}, \eqref{mod3} and Jensen's inequality (applied to the convex function $h:\R^+\rightarrow \R^+$, $h(t)=t^{p^-_m}$, $p^-_m>1$), for $n$ large enough we have
\begin{equation}\label{e4.8}
\begin{array}{ll}
\di \sum_{i=1}^N \int_{\Omega}|\partial_{x_i}u_n|^{p_i(x)}\,dx&=\di \sum_{i\in \ell_1} \int_{\Omega}|\partial_{x_i}u_n|^{p_i(x)}\,dx+\sum_{i\in \ell_2} \int_{\Omega}|\partial_{x_i}u_n|^{p_i(x)}\,dx\\
& \di \geq\sum_{i\in \ell_1}\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}^{p_M^+}+\sum_{i\in \ell_2}\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}^{p_m^-}\\
& \di \geq \sum_{i=1}^N\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}^{p_m^-}-\sum_{i\in \ell_1}\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}^{p_m^-}\\
& \di \geq N\Big{(}\frac{\sum_{i=1}^N\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}}{N}\Big{)}^{p^{-}_{m}}-N.\end{array}
\end{equation}
We analyze now the two cases corresponding to the value of $\|u\|_{L^{p_M(x)}(\Omega)}$.\\
{\it Case} 1: $\|u\|_{L^{p_M(x)}(\Omega)}\geq 1$. By �\eqref{e4.7} and \eqref{e4.8} we have
$$
\begin{array}{ll}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}&\geq \di \Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\Big[ \frac{1}{N^{p_m^--1}}\Big(\sum_{i=1}^N\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}\Big)^{p_m^-} -N \Big]\\
& \di \hspace{0.3cm}+b_0\Big(\frac{1}{p_M^+}-\frac{1}{q^-}\Big)\|u\|_{L^{p_M(x)}(\Omega)}^{p_m^-}.\end{array}
$$
and thus
\begin{equation}\label{e4.9}
\begin{array}{ll}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}&\geq \di \frac{1}{2^{p_m^-}}\Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\min\Big\{\frac{1}{N^{p_m^--1}}, b_0\Big\}  \|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p_m^-}\\
& \di \hspace{0.3cm}-N\Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}.\end{array}
\end{equation}

{\it Case} 2: $\|u\|_{L^{p_M(x)}(\Omega)}< 1$. Then
$$
\begin{array}{ll}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}&\geq \di \Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\Big[ \frac{1}{N^{p_m^--1}}\Big(\sum_{i=1}^N\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}\Big)^{p_m^-} -N \Big]\\
& \di \geq \Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\Big[ \frac{1}{N^{p_m^--1}}\Big(\sum_{i=1}^N\|\partial_{x_i}u_n\|_{L^{p_i(x)}(\Omega)}\Big)^{p_m^-}+\|u_n\|_{L^{p_M(x)}(\Omega)}^{p_m^-} \\
& \di \hspace{0.3cm}-N-1 \Big].\end{array}
$$
We obtain
\begin{align}\label{e4.10}
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}&\geq \frac{1}{2^{p_m^-}}\Big{(}\frac{1}{p^+_M}-\frac{1}{q^-}\Big{)}\min\Big\{\frac{1}{N^{p_m^--1}}, 1\Big\}  \|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p_m^-}-\frac{N+1}{p_M^+}
\end{align}
By \eqref{e4.9} and \eqref{e4.10}, we deduce that there exist $\widetilde{c_1}, \widetilde{c_2}>0$ such that
$$
1+c_7+\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\geq \widetilde{c_1}\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p_m^-}-\widetilde{c_2}.
$$

Dividing the above inequality by $\|u_n\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^-_m}$ and passing to the limit as $n\rightarrow\infty$ we obtain a contradiction.

It follows that $(u_n)$ is bounded in $W^{1,\overrightarrow{p}(x)}(\Omega)$. This information combined with the fact that $W^{1,\overrightarrow{p}(x)}(\Omega)$ is reflexive implies that there exists a subsequence, still denoted by $(u_n)$, and $u_0\in W^{1,\overrightarrow{p}(x)}(\Omega)$ such that $(u_n)$ converges weakly to $u_0$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$.

Using (\ref{e4.5}), we infer that
$$
\lim_{n\rightarrow\infty}\langle I_\lambda'(u_n),u_n-u_0\rangle=0,
$$
more precisely,
\begin{equation}\label{e4.11}\begin{array}{ll}
\di\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)
&\di +b(x)|u_n|^{p_M(x)-2}u_n(u_n-u_0)\\
& \di -\lambda|u_n|^{q(x)-2}u_n(u_n-u_0)\Big{]}dx=0.\end{array}
\end{equation}
Since the space $W^{1,\overrightarrow{p}(x)}(\Omega)$ is compactly embedded in $L^{p_M(x)}(\Omega)$ and $L^{q(x)}(\Omega)$, it follows that $(u_n)$ converges strongly to $u_0$ in $L^{p_M(x)}(\Omega)$ and also in $L^{q(x)}(\Omega)$. Therefore
\begin{equation}\label{e4.12}
\Big|\int_{\Omega}b(x)|u_n|^{p_M(x)-2}u_n(u_n-u_0)\,dx\Big|\leq 2\|b\|_{L^{\infty}(\Omega)}\Big\||u_n|^{p_M(x)-1}\Big\|_{L^{\frac{p_M(x)}{p_M(x)-1}}(\Omega)}\|u_n-u_0\|_{L^{p_M(x)}(\Omega)},
\end{equation}
and
\begin{equation}\label{e4.13}
\Big|\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_0)\,dx\Big|\leq2\Big\||u_n|^{q(x)-1}\Big\|_{L^{\frac{q(x)}{q(x)-1}}(\Omega)}\|u_n-u_0\|_{L^{q(x)}(\Omega)}.
\end{equation}
By \eqref{e4.12}, \eqref{e4.13} and \eqref{mod4}, using the strong convergence of $(u_n)$ to $u_0$ in $L^{p_M(x)}(\Omega)$ and $L^{q(x)}(\Omega)$ we deduce
$$
\lim_{n\rightarrow\infty}\int_{\Omega}b(x)|u_n|^{p_M(x)-2}u_n(u_n-u_0)\,dx=0,
$$
and
$$
\lim_{n\rightarrow\infty}\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_0)\,dx=0.
$$
By the above relations, \eqref{e4.11} reduces to
$$
\lim_{n\rightarrow\infty}\int_{\Omega}\sum_{i=1}^N a_i(x,\partial_{x_i}u_n)(\partial_{x_i}u_n-\partial_{x_i}u_0)\,dx=0.
$$
Using Lemma \ref{lem4.5}, we deduce that $(u_n)$ converges strongly to $u_0$ in $W^{1,\overrightarrow{p}(x)}(\Omega)$. Since $X$ is a closed subspace of $W^{1,\overrightarrow{p}(x)}(\Omega)$ and $(u_n)\subset X$ we obtain that $u_0\in X$. Then by relation \eqref{e4.5}
$$
I_\lambda(u_0)=c_7>0 \hspace{0.3cm} \text{ and }\hspace{0.3cm} I_\lambda'(u_0)= 0,
$$
that is, $u_0$ is a nontrivial weak solution for problem \eqref{e2.1}.\qed

\section{The second domination case}
The next result establishes an interesting concentration property in neighborhoods of the origin and of the infinity. More precisely, under an additional assumption, we show that there are positive numbers
$\lambda^*$ and $\lambda^{**}$ such that problem \eqref{e2.1} has a solution provided that either $\lambda\in (0,\lambda^*)$ or $\lambda\in (\lambda^{**},\infty)$. The existence of a ``gap" between $\lambda^*$ and $\lambda^{**}$ still remains an interesting open problem.

\begin{theorem}\label{theo5.1}
In addition,  we assume that $q\in C_+(\overline{\Omega})$ satisfies the hypothesis
$$
1<q^-\leq q^+<p^-_m.
$$
Then the following properties hold.\\
$\bf(i)$ There exists $\lambda^*>0$ such that for any $\lambda\in (0,\lambda^*)$ problem \eqref{e2.1} possesses a nontrivial weak solution.\\
$\bf(ii)$ There exists $\lambda^{**}>0$ such that for any $\lambda>\lambda^{**}$ problem \eqref{e2.1} possesses a nontrivial weak solution.
\end{theorem}

The assumptions in Theorem \ref{theo5.1} show that the weight $p(\,\cdot\,)$ is dominating with respect to the variable exponent $q(\,\cdot\,)$ that controls
the right-hand side. The above results asserts that, in such a case, a solution exists either if $\lambda>0$ is sufficiently small or for large values of $\lambda$.

First, applying Ekeland's variational principle \cite{Ek}, we show that there exists $\lambda^*>0$ such that for any $\lambda\in (0,\lambda^*)$ the functional $I_\lambda$ has a nontrivial critical point. We start with two auxiliary results.

\begin{lemma}\label{lem5.2}
There exists $\lambda^*>0$ such that for any $\lambda\in (0,\lambda^*)$ there are $\rho,a>0$ such that $I_\lambda(u)\geq a>0$ for any $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\rho$.
\end{lemma}
\begin{proof}
Under the conditions of Theorem \ref{theo5.1}, $W^{1,\overrightarrow{p}(x)}(\Omega)$ is continuously
embedded in $L^{q(x)}(\Omega)$. Thus, there exists a positive
constant $c_8$ such that
\begin{eqnarray}\label{e5.1}
\|u\|_{L^{q(x)}(\Omega)}\leq c_8\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\hspace{0.5cm}\textrm{ for all }\hspace{0.5cm}u\in X.
\end{eqnarray}
Now, let us assume that $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}<\min \{1,\frac{1}{c_8}\}$, where $c_8$ is the positive constant from above. Then we have $\|u\|_{L^{q(x)}(\Omega)}< 1$. Using \eqref{mod3} we get
\begin{eqnarray}\label{e5.2}
\int_{\Omega}|u|^{q(x)}dx\leq \|u\|_{L^{q(x)}(\Omega)}^{q^-}\hspace{0.5cm}\textrm{ for all }\hspace{0.3cm}u\in X\hspace{0.2cm}\textrm{ with }\hspace{0.2cm}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\rho\in (0,1).
\end{eqnarray}
Relations \eqref{e5.1} and \eqref{e5.2} imply
\begin{eqnarray}\label{e5.3}
\int_{\Omega}|u|^{q(x)}dx\leq c_8^{q^-}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{q^-}\hspace{0.5cm}\textrm{ for all }\hspace{0.3cm}u\in X\hspace{0.2cm}\textrm{ with }\hspace{0.2cm}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\rho.
\end{eqnarray}
Using the hypothesis $(\mathbf{A_2})$, $(\mathbf{B})$ and relation \eqref{e5.3}, we deduce that for any $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\rho$, the
following hold:
\begin{equation}\begin{array}{ll}\label{e5.4}
I_{\lambda}(u)&=\di \int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}u)+\frac{b(x)}{p_M(x)}|u|^{p_M(x)}-\frac{\lambda}{q(x)}|u|^{q(x)}\Big{\}}\,dx\\
&\di \geq\frac{1}{p^{+}_{M}}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}dx+\frac{b_0}{p^+_M}\int_{\Omega}|u|^{p_M(x)}\,dx-\frac{\lambda}{q^-}c_8^{q^-}
\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}
^{q^-}\\
&\di\geq \frac{\min\{1,b_0\}}{p^+_M(N+1)^{p^+_M-1}}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^+_M}-\frac{\lambda}{q^-}c_8^{q^-}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{q^-}\\
&=\di\frac{\min\{1,b_0\}}{p^+_M(N+1)^{p^+_M-1}}\rho^{p^+_M}-\frac{\lambda}{q^-}c_8^{q^-}\rho^{q^-}\\
&\di =\rho^{q^-}\Big(\frac{\min\{1,b_0\}}{p^+_M(N+1)^{p^+_M-1}}\rho^{p^+_M-q^-}-\frac{\lambda}{q^-}c_8^{q^-}\Big).\end{array}
\end{equation}
If we define
\begin{eqnarray}\label{e5.5}
\lambda^*=\frac{\min\{1,b_0\}q^-}{2p^+_M (N+1)^{p^+_M-1}c_8^{q^-}}\rho^{p^+_M-q^-},
\end{eqnarray}
then for any $\lambda\in (0,\lambda^*)$ and $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}=\rho$, there exists $a=\frac{\min\{1,b_0\}\rho^{p^+_M}}{2p^+_M (N+1)^{p^+_M-1}}$ such that $I_\lambda(u)\geq a>0$.
\end{proof}

\begin{lemma}\label{lem5.3}
 Assume that $\lambda\in (0,\lambda^*)$, where $\lambda^*$ is given by \eqref{e5.5}. Then there exists $\psi\in X$ such that $\psi\geq0,~\psi\neq 0$ and $I_\lambda(t\psi)<0$ for all $t>0$ small enough.
\end{lemma}
\begin{proof}
By the conditions of Theorem \ref{theo5.1}, $q^-<p^-_m$. Let $\epsilon_0>0$ be such that $q^-+\epsilon_0<p^-_m$. 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 $q(x)<q^-+\epsilon_0<p^-_m$ for all $x\in \Omega_0$.

Let $\psi \in X$ 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 by \eqref{e4.3} for any $t\in(0,1)$, we have
$$
\begin{array}{ll}
I_\lambda(t\psi)&=\di\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}(t\psi))+\frac{b(x)}{p_M(x)}|t\psi|^{p_M(x)}-\frac{\lambda}{q(x)}|t\psi|^{q(x)}\Big{\}}dx\\
&\di \leq c_6 \sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}(t\psi)|+\frac{|\partial_{x_i}(t\psi)|^{p_i(x)}}{p_i(x)}\Big{)}dx+\frac{1}{p^-_M}\int_{\Omega}b(x)|t\psi|^{p_M(x)}dx\\
&\di -\lambda\int_{\Omega}\frac{1}{q(x)}|t\psi|^{q(x)}dx\\
&\di \leq c_6 t^{p^-_m}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{p^-_m}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{t^{p^-_M}}{p^-_M}
\int_{\Omega}b(x)|\psi|^{p_M(x)}dx\\
&\di -\frac{\lambda}{q^+}\int_{\Omega_0}t^{q(x)}|\psi|^{q(x)}dx\\
&\di \leq c_6 t^{p^-_m}\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{p^-_m}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{t^{p^-_m}}{p^-_M}
\int_{\Omega}b(x)|\psi|^{p_M(x)}dx\\
&\di -\frac{\lambda t^{q^-+\epsilon_0}}{q^+}\int_{\Omega_0}|\psi|^{q(x)}dx.\end{array}
$$
So, $I_\lambda(t\psi)<0$ for $t<\delta^{\frac{1}{P^-_m-q^--\epsilon_0}}$, with
\begin{eqnarray*}
0<\delta<\min\Biggl\{1,\frac{\lambda}{q^+}\frac{\int_{\Omega_0}|\psi|^{q(x)}dx}{c_6
\sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}\psi|+\frac{1}{p^-_m}|\partial_{x_i}\psi|^{p_i(x)}\Big)dx+\frac{1}{p^-_M}\int_{\Omega}b(x)|\psi|^{p_M(x)}dx
}\Biggl\}.
\end{eqnarray*}
\end{proof}

\subsection*{Proof of Theorem \ref{theo5.1}$\bf(i)$}
Let $\lambda^*$ be defined as in \eqref{e5.3} and $\lambda\in (0,\lambda^*)$. By Lemma \ref{lem5.2}, 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{lem5.3}, 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 $u\in B_{\rho}(0)$,
$$
I_\lambda(u)\geq \frac{\min\{1,b_0\}}{p^+_M(N+1)^{p^+_M-1}}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^+_M}-\frac{\lambda}{q^-}c_8^{q^-}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{q^-}.
$$
It follows that
$$
-\infty<c_9=\inf_{\overline{B_{\rho}(0)}}I_\lambda(u)<0.
$$
We let now $0<\varepsilon<\inf_{\partial B_{\rho}(0)}I_\lambda-\inf_{B_{\rho}(0)}I_\lambda$. Applying Ekeland's 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
$$
\begin{array}{ll}
I_\lambda(u_{\varepsilon})& \di<\inf_{\overline{B_{\rho}(0)}}I_\lambda+\varepsilon,\\
I_\lambda(u_{\varepsilon})& \di<I_\lambda(u)+\varepsilon\|u-u_{\varepsilon}\|_{W^{1,\overrightarrow{p}(x)}(\Omega)},\hspace{0.2cm}u\neq u_{\varepsilon}.\end{array}
$$
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 $K_\lambda(u)=I_\lambda(u)+\varepsilon \|u-u_{\varepsilon}\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}$. 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
$$
\frac{I_\lambda(u_{\varepsilon}+tv)-I_\lambda(u_{\varepsilon})}{t}+\varepsilon \|v\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\geq0.
$$
Letting $t\rightarrow0$ it follows that $\langle
I_\lambda'(u_{\varepsilon}),v\rangle+\varepsilon \|v\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}>0$, hence $\|I_\lambda'(u_{\varepsilon})\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\leq \varepsilon$. We
deduce that there exists a sequence $(v_n)\subset B_{1}(0)$ such
that
\begin{equation}\label{e5.6}
I_\lambda(v_n)\rightarrow c_9 \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 $X$. Actually, with similar arguments as those used in the end of Theorem \ref{theo4.2}, we can show that $(v_n)$ converges strongly to $u_1$ in $X$. So, by \eqref{e5.6}
$$
I_\lambda(u_1)=c_9<0 \hspace{0.3cm} \text{ and }\hspace{0.3cm} I_\lambda'(u_1)=0,
$$
that is, $u_1$ is a nontrivial weak solution for the problem \eqref{e2.1}. This completes the proof.\qed

\medskip
Next, we want to construct a global minimizer of the functional $I_\lambda$. We start with the following auxiliary result.
\begin{lemma}\label{lem5.4}
The functional $I_\lambda$ is coercive on $X$.
\end{lemma}
\begin{proof}
Using the hypothesis $(\mathbf{A_2})$, relation \eqref{e4.2} we deduce for all $u\in X$,
\begin{equation}\begin{array}{ll}\label{e5.7}
I_\lambda(u)\geq &\di \frac{1}{p^+_M}\sum_{i=1}^N\int_{\Omega}|\partial_{x_i}u|^{p_i(x)}\,dx+\frac{b_0}{p_M^+}\int_{\Omega}|u|^{p_M(x)}\,dx\\
&\di-\frac{\lambda}{q^-}\Big[(c_1\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)})^{q^-}
+(c_2\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)})^{q^+}\Big].\end{array}
\end{equation}
Now we set $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}>1$. Using the same techniques as in the proof of \eqref{e4.7} combined with \eqref{e5.7} we find that
$$
I_\lambda(u)\geq \widetilde{c_1}\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}^{p^-_m}-\widetilde{c_2}-\frac{\lambda}{q^-}\Big[(c_1\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)})
^{q^-}+(c_2\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)})^{q^+}\Big],
$$
for any $u\in X$ with $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}>1$. Since $q^-\leq q^+<p^-_m$, we infer that $I_\lambda(u)\rightarrow\infty$ as $\|u\|_{W^{1,\overrightarrow{p}(x)}(\Omega)}\rightarrow\infty$, that is, $I_\lambda$ is coercive.
\end{proof}
\subsection*{Proof of Theorem \ref{theo5.1}$\bf(ii)$}
The same arguments as in the proof of Lemma 3 of \cite{BouRad} can be used in order to show that $I_\lambda$ is weakly lower semicontinuous on $X$. By Lemma \ref{lem5.4}, the functional $I_\lambda$ is also coercive on $X$. We know from \cite{Struw} that there exists $u_\lambda \in X$, a global minimizer of $I_\lambda$ and thus weak solution of problem \eqref{e2.1}.\\
We show that $u_\lambda$ is nontrivial if $\lambda$ is large enough. Letting $t_0>1$ be a constant and $\Omega_1$ be an open subset of $\Omega$ with $|\Omega_1|>0$, we assume that $v_0\in X$ is such that $v_0(x)=t_0$ for any $x\in \overline{\Omega_1}$ and $0\leq v_0(x)\leq t_0$ in $\Omega\backslash\Omega_1$. We have
$$
\begin{array}{ll}
I_\lambda(v_0)&=\di\int_{\Omega}\Big{\{}\sum_{i=1}^N A_i(x,\partial_{x_i}(v_0))+\frac{b(x)}{p_M(x)}|v_0|^{p_M(x)}-\frac{\lambda}{q(x)}|v_0|^{q(x)}\Big{\}}\,dx\\
&\di \leq c_6 \sum_{i=1}^N\int_{\Omega}\Big{(}|\partial_{x_i}(v_0)|+\frac{|\partial_{x_i}(v_0)|^{p_i(x)}}{p_i(x)}\Big{)}dx+\frac{1}{p^-_M}\int_{\Omega}b(x)|v_0|^{p_M(x)}\,dx\\
&\di -\lambda\int_{\Omega}\frac{1}{q(x)}|v_0|^{q(x)}dx\leq c_{10}-\frac{\lambda}{q^+}t_0^{q^-}|\Omega_1|.\end{array}
$$
So there exists $\lambda^*>0$ such that $J_\lambda(v_0)<0$ for any $\lambda\in (\lambda^*,+\infty)$. It follows that for any $\lambda\geq \lambda^*$, $u_\lambda$ is a nontrivial weak solution of problem \eqref{e2.1} for $\lambda$ large enough.

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


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\end{document} 