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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 155, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/155\hfil Existence of positive solutions]
{Existence of positive solutions for  $p(x)$-Laplacian
equations with a singular nonlinear term}

\author[J. Liu, Q. Zhang, C. Zhao\hfil EJDE-2014/155\hfilneg]
{Jingjing Liu, Qihu Zhang, Chunshan Zhao}  % in alphabetical order

\address{Jingjing Liu \newline
College of Mathematics and Information Science, Zhengzhou
University of Light Industry,
Zhengzhou, Henan 450002, China}
\email{jingjing830306@163.com}

\address{Qihu Zhang (corresponding author)\newline
College of Mathematics and Information Science, Zhengzhou
University of Light Industry,
Zhengzhou, Henan 450002, China}
\email{zhangqihu@yahoo.com, zhangqh1999@yahoo.com.cn}

\address{Chunshan Zhao\newline
Department of Mathematical Sciences, Georgia Southern
University, Statesboro, GA 30460, USA}
\email{czhao@GeorgiaSouthern.edu}

\thanks{Submitted July 2, 2013. Published July 7, 2014.}
\subjclass[2000]{35J25, 35J65, 35J70}
\keywords{$p(x)$-Laplacian; singular nonlinear term;
sub-supersolution method}

\begin{abstract}
 In this article, we study the existence of positive solutions
 for the $p(x)$-Laplacian Dirichlet problem
 $$
  -\Delta _{p(x)}u=\lambda f(x,u)
 $$
 in a bounded domain $\Omega \subset \mathbb{R}^{N}$.
 The singular nonlinearity term $f$ is allowed to be either
 $f(x,s)\to +\infty $, or $f(x,s)\to +\infty $ as $s\to 0^{+}$
 for each $x\in \Omega $. Our main results  generalize  the results
 in \cite{g1} from constant exponents to variable exponents. In particular,
 we give the asymptotic behavior of solutions of a simpler equation which is
 useful for finding supersolutions of differential equations with variable
 exponents, which is of independent interest.
\end{abstract}

\dedicatory{Dedicated to Professor Xianling Fan on his 70th birthday}

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


\section{Introduction}

Let $\Omega \subset \mathbb{R}^{N}$ be an open bounded domain with
 $C^{2}$ boundary. We consider the existence of positive solutions
for elliptic problems with variable exponent of the form
\begin{equation}
\begin{gathered}
-\Delta _{p(x)}u=\lambda f(x,u), \quad \text{in }\Omega , \\
u(x)>0, \quad \text{in }\Omega , \\
u(x)=0, \quad \text{on }\partial \Omega ,
\end{gathered} \label{1.1}
\end{equation}
where $-\Delta _{p(x)}u=-\operatorname{div}(| \nabla u|^{p(x)-2}\nabla u$)
 with $\nabla u=(\partial _{x_1}u,\partial_{x_2}u,\dots ,\partial _{x_{N}}u)$
 which is so-called $p(x)$-Laplacian, $p(\cdot )$ is a function which satisfies
some conditions specified below, $f:\Omega \times
(0,\infty )\to [ 0,\infty )$ is a continuous function, and
$\lambda >0$ is a real parameter. Throughout this paper, we will
denote $d(x)=d(x,\partial \Omega )$.

In recent years, the study of differential equations and variational
problems with nonstandard $p(x)$-growth condition has been an interesting
topic. The $p(x)$-Laplacian arises from the study of nonlinear elasticity,
electrorheological fluids and image restoration etc. For example,
electrorheological fluids have an extensive applications in robotics,
aircraft and aerospace. We refer readers to
\cite{a1,c3,h1,r1,r2,z4} for more detailed background of
applications. There are many reference papers related to the study of
differential equations and variational problems with variable exponent. Far
from being complete, we refer readers to
\cite{a1,b1,c1,d1,f1,f2,f3,f4,f5,f6,f8,h2,h3,h4,h5,h6,h7,k1,l4,m1,m2,m3,m4,m5,m6,p2,r2,z1,z2,z3,z4}
 and references cited therein. For example, the regularity of
weak solutions for differential equations with variable exponent was studied
in \cite{a1,f1}, and existence of solutions for variable
exponent problems was studied in a series of papers
\cite{c1,f3,f6,h2,h6,l4,m3,m6,p2,z2,z3}.
 Recently, the applications of variable exponent
analysis in image restoration attracted more and more attention
\cite{g2,g3,h5,l2}.
In this paper, our aim is to study the
existence of positive solution for problem \eqref{1.1} with singular
nonlinear term $f$.

Clearly, if $p(\cdot )\equiv p$, a constant, the operator is the well-known $
p$-Laplacian, and \eqref{1.1} is the usual $p$-Laplacian equation, but for
non-constant $p(\cdot )$, $p(x)$-Laplacian problems are more complicated due
to the non-homogeneity of $p(x)$-Laplacian. For example, if $\Omega $ is a
smooth bounded domain, the Rayleigh quotient
\[
\lambda _{p(\cdot )}=\underset{u\in W_0^{1,p(\cdot )}(\Omega )\backslash
\{0\}}{\inf }\frac{\int_{\Omega }\frac{1}{p(x)}| \nabla
u| ^{p(x)}dx}{\int_{\Omega }\frac{1}{p(x)}| u|
^{p(x)}dx}
\]
is zero in general, and $\lambda _{p(\cdot )}>0$ only under some special
conditions (see \cite{f7}). It is also possible the first eigenvalue and
eigenfunction of $p(x)$-Laplacian do not exist, even though the existence of
the first eigenvalue and eigenfunction is very important in the study of
elliptic problems related to $p$-Laplacian problems. For example, in
\cite{g1}, the author use the first eigenfunction and the first eigenvalue to
construct subsolutions.  Fan \cite{f2} considered the eigenvalue problem
of $p(x)$-Laplacian equation with the Neumann boundary condition, the
existence of infinite many eigenvalues has been established.
Benouhiba \cite{b1} studied the eigenvalue problem
\[
-\Delta _{p(x)}u=\lambda V(x)| u| ^{q(x)-2}u, \quad x\in\mathbb{R}^{N},
\]
where $1<p(\cdot )$; $q(\cdot )\in C(\mathbb{R}^{N})$ and $V(\cdot )$
is an indefinite weight function. The results show
that the spectrum of such problems contains a continuous family of
eigenvalues.

There are many papers deal with the existence of positive solution for a
class $p$-Laplacian equation with singular nonlinearity (see
\cite{g1,g4,m7,p1,q1}
 Mohammed \cite{m7},
Perera and Silva |cite{p1},  Qing and Yang \cite{q1} and
Guo et al \cite{g4} studied the solvability of \eqref{1.1} with
$\lambda =1,p(\cdot )\equiv p\neq 2$ and $f(\cdot ,\cdot )$ satisfies
various conditions. In  \cite{p1}, the authors considered a boundary
condition in a more general sense.

 Mohammed \cite{m7} considered the existence and uniqueness of weak
solutions of the singular boundary value problem with constant exponent as
follows.
\begin{gather*}
-\Delta _{p}u=f(x,u), \quad \text{in }\Omega , \\
u(x)>0, \quad \text{in }\Omega , \\
u(x)=0, \quad  \text{on }\partial \Omega ,
\end{gather*}
where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$ with $C^{1,\omega }$
boundary for some $0<\omega <1$, and singular
nonlinearity term $f(x,t)$ could show up when $t\to 0^{+}$. Mohammed
make the following two assumptions:
\begin{itemize}
\item[(1)]  For each $\theta \in (0,1)$, there is a constant $C_{\theta }\geq
1 $ such that $g(\theta t)\leq C_{\theta }g(t)$ for all $t>0$;

\item[(2)] $f(x,s)\geq a(x)$ for any $(x,s)\in \Omega \times (0,\infty )$.
\end{itemize}

In \cite{g1}, the authors studied the existence of solutions of the
nonlinear elliptic problem with constant exponent,
\begin{gather*}
-\Delta _{p}u=\lambda f(x,u), \quad \text{in }\Omega , \\
u(x)>0, \quad \text{in }\Omega , \\
u(x)=0, \quad \text{on }\partial \Omega ,
\end{gather*}
where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$,
$1<p<N$, $f:\Omega \times (0,\infty )\to [ 0,\infty )$ is
a suitable function and $\lambda >0$ is a real parameter. The nonlinearity
term $f$ is allowed to be either
$f(x,s)\to +\infty $ or $f(x,s)\to +\infty $
as $s\to 0^{+}$ for each $x\in \Omega $, and the assumptions (1) and (2)
are not assumed.

Results on elliptic problems with singular nonlinearity are rare
(see \cite{l3,z2}).
In \cite{z2}, by using the sub-supersolution method, we
studied the existence and the boundary asymptotic behavior of solutions of
the elliptic problem with variable exponent,
\begin{gather*}
-\Delta _{p(x)}u=\frac{\lambda }{u^{\gamma (x)}}, \quad \text{in }\Omega , \\
u(x)>0, \quad \text{in }\Omega , \\
u(x)=0, \quad \text{on }\partial \Omega ,
\end{gather*}
where $\Omega \subset \mathbb{R}^{N}$ is a domain with $C^{2}$ boundary,
$\lambda $ is a positive parameter which is large enough.

 Liu \cite{l3} generalized the results of \cite{m7} to $p(x)$-Laplacian
by making the similar assumptions. The condition (1) implies
that $g(t)\leq Ct^{-a}$ when $t\leq 1$ for some $a>0$, which is invalid for
$g(t)=e^{1/t}$, and the condition (2) is a bit strong in some
sense. Motivated by \cite{g1}, in this rticle we partly generalized the
results to $p(x)$-Laplacian.

Before stating our main results, we make the following assumptions
throughout this paper:
\begin{itemize}
\item[(H0)] $p(\cdot )\in C^{1}( \overline{\Omega })$,
$1<p^{-}:=\inf_{\Omega} p(x)\leq p^{+}:=\sup_{\Omega}
p(x)<\infty$, $p(\cdot )<N$.

\item[(H1)] $f(x,s)\leq b(x)g(s)$ for all
$( x,s) \in \Omega \times ( 0,\infty ) $, where
$g:(0,+\infty )\to ( 0,+\infty ) $ is a continuous function,
$sg(s)$ is decreasing for $s\leq 1$; and
$b:\Omega \to [ 1,\infty )$,
$b(\cdot )\in L^{\alpha (\cdot ) }(\Omega )$,
$1<\alpha ( \cdot ) \in C( \overline{\Omega }) $, and
$\frac{1}{\alpha ( x) }+\frac{1}{p^{\ast}( x) }<1$,
for all $x\in \overline{\Omega }$.

\item[(H2)] $f(x,s)$ satisfies
\begin{equation}
\liminf_{s\to 0^{+}} \frac{f(x,s)}{
s^{p^{-}-1}| \ln s| ^{p^{-}}}=+\infty \quad
\text{uniformly for }x\in \Omega.  \label{7.2}
\end{equation}
\end{itemize}

\begin{theorem} \label{thm1.1}
 Assume that {\rm (H0), (H1), (H2)} hold.
Then problem \eqref{1.1} has a solution when $\lambda $ is small enough.
\end{theorem}

\begin{theorem} \label{thm1.2}.
Assume that {\rm (H0), (H1), (H2)} hold.
Also assume that
\begin{itemize}
\item[(i)] there is a small $\delta >0$ such that $p( x) \equiv p$ (a
constant) for any $x\in \Omega $ with $d(x)\leq \delta $;

\item[(ii)] $\underset{s\to +\infty }{\lim }\frac{g( s) }{
s^{p^{-}-1-\varepsilon }}:=g_{\infty }\in [ 0,+\infty ) $, where $
\varepsilon >0$ is small enough;

\item[(iii)] $\alpha ( \cdot ) >N$ on $\overline{\Omega }$.
\end{itemize}
Then problem \eqref{1.1} has a solution for any positive $\lambda $.
\end{theorem}

\begin{theorem} \label{thm1.3}
Assume that {\rm (H0), (H1), (H2)} hold.
Also assume that
\begin{itemize}
\item[(i)] $\frac{\partial p( \cdot ) }{\partial \nu }<0$ on $\partial
\Omega $, where $\nu $ is the inward unit normal vector of $\partial \Omega $;

\item[(ii)] $\lim_{s\to +\infty } \frac{g( s) }{
s^{p^{-}-1-\varepsilon }}:=g_{\infty }\in [ 0,+\infty ) $, where
$\varepsilon >0$ is small enough;

\item[(iii)] $\alpha ( \cdot ) >N$ on $\overline{\Omega }$.
\end{itemize}
Then problem \eqref{1.1} has a solution for any positive constant $\lambda $.
\end{theorem}

\begin{theorem} \label{thm1.4}
Assume that {\rm (H0), (H1), (H2)} hold.
Also assume that
\begin{itemize}
\item[(i)] Equation \eqref{1.1} is radial;

\item[(ii)] $\lim_{s\to +\infty } \frac{g( s) }{
s^{p^{-}-1-\varepsilon }}:=g_{\infty }\in [ 0,+\infty ) $, where $
\varepsilon >0$ is small enough.
\end{itemize}
Then problem \eqref{1.1} has a solution for any positive $\lambda $.
\end{theorem}

This paper is organized as follows. In section 2, we will recall some basic
facts about the variable exponent Lebesgue and Sobolev spaces which we will
use later, and we will also give a general principle of sub-supersolution
method. Proofs of our results will be presented in section 3.

\section{Preliminaries}

Throughout this paper, the letters $c,c_{i},C,C_{i}$ $(i=1,2,\dots )$,
denote positive constants which may vary from line to line, but they are
independent of the terms which will take part in any limit process.

To deal with the $p(x)$-Laplacian problem, we need introduce some
functional spaces $L^{p(\cdot )}( \Omega ) $,
$W^{1,p(\cdot )}( \Omega )$, $W_0^{1,p(\cdot )}( \Omega ) $ and
properties of the $p(x)$-Laplacian which we will use later.
Denote by $S(\Omega ) $ be the set of all measurable real-valued functions
defined in $\Omega $. Note that two measurable functions are considered as the same
element of $S( \Omega ) $ when they are equal almost everywhere.
Let
\[
L^{p(\cdot )}( \Omega ) =\big\{  u\in S( \Omega
) : \int_{\Omega }| u(x)| ^{p(x)}dx<\infty \big\} ,
\]
with the norm
\[
| u| _{p(\cdot )}=| u| _{L^{p(\cdot
)}(\Omega )}=\inf \big\{ \lambda >0 : \int_{\Omega }|
\frac{u(x)}{\lambda }| ^{p(x)}dx\leq 1 \big\} .
\]
The space $( L^{p(\cdot )}( \Omega ) ,|\cdot | _{p(\cdot )}) $ becomes
a Banach space. We call it
variable exponent Lebesgue space. Moreover, this space
 is a separable, reflexive and uniform convex Banach space; see
\cite[Theorems 1.6, 1.10, 1.14]{f8}.

The variable exponent Sobolev space
\[
W^{1,p(\cdot )}( \Omega ) =\big\{ u\in L^{p(\cdot )}(
\Omega ) : | \nabla u| \in L^{p(\cdot)}(\Omega )\big\} ,
\]
can be equipped with the norm
\[
\| u\| =| u|_{p(\cdot )}+| \nabla u| _{p(\cdot )},\quad \forall
u\in W^{1,p(\cdot )}( \Omega ) .
\]
Note that  $W_0^{1,p(\cdot )}( \Omega ) $ is the closure of
$C_0^{\infty}( \Omega ) $ in $W^{1,p(\cdot )}( \Omega ) $.
The spaces $ W^{1,p(\cdot )}( \Omega ) $ and $W_0^{1,p(\cdot )}( \Omega
) $ are separable, reflexive and uniform convex Banach spaces (see
\cite[Theorem 2.1]{f8}.

For $u,v \in S( \Omega ) $, we write $u\leq v$ if
$u(x)\leq v(x)$ for a.e. $x\in \Omega $.
Let $\rho (x,s)$ be a Carath\'eodory function on
$\Omega \times\mathbb{R}$ with property that for any $s_0>0$
there exists a constant $A$ such that
\begin{equation} \label{e2b}
| \rho (x,s)| \leq A\quad \text{for a.e. $x\in \Omega$
and all }s\in [ -s_0,s_0] .
\end{equation}

\begin{definition} \label{def2.1}\rm
 (i) Let $\underline{u},\overline{u}\in
W_{\rm loc}^{1,p(\cdot )}( \Omega ) \cap C_0(\overline{\Omega })$
satisfy $\underline{u},\overline{u}>0$ in $\Omega $.
We say $\underline{u}$
and $\overline{u}$ are a subsolution and a supersolution of \eqref{1.1}
respectively, if
\begin{gather*}
\int_{\Omega }| \nabla \underline{u}| ^{p(x)-2}\nabla
\underline{u}\nabla \phi\,dx \leq \int_{\Omega }\lambda f(x,\underline{u}
)\phi\,dx,  \\
\int_{\Omega }| \nabla \overline{u}| ^{p(x)-2}\nabla
\overline{u}\nabla \phi\,dx \geq \int_{\Omega }\lambda f(x,\overline{u}
)\phi\,dx,
\end{gather*}
for all $\phi \in C_0^{\infty }( \Omega ) $ with $\phi \geq 0$
and $\operatorname{supp}\phi \subset \subset \Omega $.
 We say $u$ is a solution of \eqref{1.1}, if it is both a subsolution
and a supersolution of \eqref{1.1}.

(ii) A function $u\in W^{1,p(\cdot )}( \Omega ) \cap C(\overline{
\Omega })$ is called a weak solution of the problem
\begin{equation}
\begin{gathered}
-\Delta _{p(x)}u=\rho (x,u), \quad \text{in }\Omega , \\
u(x)=\varphi (x), \quad \text{on }\Omega ,
\end{gathered}\label{1.5}
\end{equation}
where $\varphi (\cdot )\in C(\overline{\Omega })$, if
\[
\int_{\Omega }| \nabla u| ^{p(x)-2}\nabla u\nabla \phi\,dx
=\int_{\Omega }\rho (x,u)\phi\,dx,\forall \phi \in C_0^{\infty }(\Omega );
\]

(iii) $\underline{u},\overline{u}\in W^{1,p(\cdot )}( \Omega )
\cap C(\overline{\Omega })$ are called a weak subsolution and a weak
supersolution of the problem \eqref{1.5} respectively if
$\underline{u}\leq \varphi $ and $\overline{u}\geq \varphi $ on
 $\partial \Omega $ and for all $\phi \in C_0^{\infty }( \Omega )$,
$\phi \geq 0$,
\begin{gather*}
\int_{\Omega }| \nabla \underline{u}| ^{p(x)-2}\nabla
\underline{u}\nabla \phi\,dx
\leq \int_{\Omega }\rho (x,\underline{u})\phi \,dx,  \\
\int_{\Omega }| \nabla \overline{u}| ^{p(x)-2}\nabla
\overline{u}\nabla \phi\,dx \geq \int_{\Omega }\rho (x,\overline{u})\phi\,dx.
\end{gather*}
\end{definition}

\begin{lemma}[{\cite[Proposition 2.1]{f6}}] \label{lem2.2}
 The space $(L^{p(\cdot)}(\Omega ),| \cdot | _{p(\cdot )})$ is a separable,
uniform convex Banach space, and its conjugate space is
$L^{p^{0}(\cdot )}(\Omega )$, where $p^{0}(\cdot )$ is the conjugate
function of $p(\cdot )$
satisfying $\frac{1}{p(\cdot )}+\frac{1}{p^{0}(\cdot )}\equiv 1$.
For any $u\in L^{p(\cdot )}(\Omega )$ and $v\in L^{p^{0}(\cdot )}(\Omega )$,
we have the following H\"{o}lder inequality
\[
| \int_{\Omega }uvdx| \leq \int_{\Omega }|uv|\,dx
\leq (\frac{1}{p^{-}}+\frac{1}{(p^{0})^{-}})|u| _{p(\cdot )}| v| _{p^{0}(\cdot )}
\leq 2| u| _{p(\cdot )}| v| _{p^{0}(\cdot)}.
\]
\end{lemma}

\begin{definition} \label{def2.3}\rm
 Let $u,v$ $\in W^{1,p(\cdot )}( \Omega) \cap L^{\infty }(\Omega )$.
We say that $-\Delta _{p(x)}u+\rho(x,u)\leq -\Delta _{p(x)}v+\rho (x,v)$ in
$\Omega $ if
\[
\int_{\Omega }| \nabla u| ^{p(x)-2}\nabla u\nabla \phi
\,dx+\int_{\Omega }\rho (x,u)\phi\,dx\leq \int_{\Omega }| \nabla
v| ^{p(x)-2}\nabla v\nabla \phi\,dx+\int_{\Omega }\rho (x,v)\phi
\,dx
\]
for all $\phi \in C_0^{\infty }( \Omega )$, $\phi \geq 0$.
\end{definition}
Next we give a comparison principle as follows.


\begin{lemma}[{\cite[Lemma 2.3]{z1}}]  \label{lem2.4}
Let $\rho (x,t)$ be a function satisfying \eqref{e2b} and nondecreasing in $t$.
Let  $u,v\in W^{1,p(\cdot )}(\Omega )$ satisfy
\[
-\Delta _{p(x)}u+\rho (x,u)\leq -\Delta _{p(x)}v+\rho (x,v),\quad
( x\in \Omega ) ,
\]
if $u\leq v$ on $\partial \Omega $, then $u\leq v$ in $\Omega $.
\end{lemma}

\begin{lemma}[{\cite[Theorem 8.3.1]{d1}}]  \label{lem2.5}
 For every $u\in W_0^{1,p(\cdot )}(\Omega )$, the inequality
\[
| u| _{p^{\ast }(\cdot )}\leq C| \nabla u| _{p(\cdot )}
\]
holds with a constant $C$ depending only on the dimension $N$ and $p^{+}$
and independent of $\Omega $.
\end{lemma}

\begin{lemma} \label{lem2.6}
 Suppose the domain $\Omega $ has finite measure, i.e. $| \Omega | <+\infty $,
$p(\cdot ),q(\cdot )\in C(\overline{\Omega })$, and
$1<p(x)<q(x)<N$, for all $x\in \overline{\Omega }$.
Then for every $u\in L^{q(\cdot )}(\Omega )$, the following inequality
holds:
\[
| u| _{p(\cdot )}\leq 2| \Omega | ^{
\frac{1}{p(\xi )}-\frac{1}{q(\xi )}}| u| _{q(\cdot )}, \quad
\text{where }\xi \in \overline{\Omega }.
\]
Moreover, $| u| _{p(\cdot )}\leq 2| \Omega
| ^{1/N}| u| _{p^{\ast }(\cdot)}$ for all $u\in W_0^{1,p(\cdot )}(\Omega )$.
\end{lemma}


The basic principle of sub-supersolution method for \eqref{1.1} can be
stated as follows.

\begin{lemma} \label{lem2.7}
 Suppose that {\rm (H0)} holds and $\underline{u},
\overline{u}\in W_{\rm loc}^{1,p(\cdot )}( \Omega ) \cap C_0(
\overline{\Omega })$. Let $\underline{u}$ and $\overline{u}$ be a
subsolution and a supersolution of \eqref{1.1} respectively satisfying
$\underline{u}\leq \overline{u}$. If
$f\in C(\overline{\Omega }\times\mathbb{R},\mathbb{R})$, then \eqref{1.1}
 has a solution
$u\in W_{\rm loc}^{1,p(\cdot )}( \Omega) \cap C_0(\overline{\Omega })$
satisfy $\underline{u}\leq u\leq \overline{u}$.
\end{lemma}

\begin{proof}
 Denote $\Omega _{n}=\{x\in \Omega : d(x)>1/n\}$.
Let
\[
\widetilde{f}(x,u)=\begin{cases}
f(x,\overline{u}), & u\geq \overline{u}, \\
f(x,u), & \underline{u}<u<\overline{u}, \\
f(x,\underline{u}), & u\leq \underline{u}.
\end{cases}
\]
Consider
\begin{equation} \label{eIn}
\begin{gathered}
-\Delta _{p(x)}u=\lambda \widetilde{f}(x,u), \quad \text{in }\Omega _{n}, \\
u(x)>0, \quad \text{in }\Omega _{n}, \\
u(x)=\underline{u}(x), \quad \text{on }\partial \Omega _{n}.
\end{gathered}
\end{equation}
Since $| \widetilde{f}(x,u)| $ is bounded on $\overline{
\Omega _{n}}$ and $\underline{u}\in C_0(\overline{\Omega })$, it is easy
to see that \eqref{eIn} has a solution $u_{n}$, satisfy
$\underline{u}\leq u_{n}\leq \overline{u}$.
By \cite[Theorem 1.2]{f5}, we can see that $\{u_{n}\}_{n\geq n_0+1}$ has
 uniformly bounded $C^{1,\alpha }$ norm on $\overline{\Omega _{n_0}}$.
 By the diagonal method, we can choose a subsequence
$\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that
\[
u_{n_{k}}(x)\to u(x),\quad
\nabla u_{n_{k}}(x)\to \nabla u(x),\quad \forall x\in \Omega ,
\]
where $u\in C_0( \overline{\Omega }) \cap C^{1}( \Omega) $.
Thus $u$ is a solution of \eqref{1.1} and satisfies
$\underline{u}\leq u\leq \overline{u}$.
\end{proof}

\section{Proofs of main results}

To study the existence of solutions of \eqref{1.1}, we need to
do some preparation work. Note that by \cite[Theorem 4.2]{f4},
the following problem has a weak solution
$\omega _{b}\in W_0^{1,p(\cdot )}( \Omega ) $,
\begin{equation}
\begin{gathered}
-\Delta _{p(x)}\omega =b( x) , \quad \text{in }\Omega , \\
\omega ( x) =0, \quad \text{on }\partial \Omega .
\end{gathered} \label{1.6}
\end{equation}

Since $b( \cdot ) $ is nonnegative, by the comparison principle it
follows that $\omega _{b}$ is nonnegative (see  \cite[Lemma 2.3]{z1}) and
it is positive in $\Omega $ (see \cite[Theorem 1.1]{z1}]).
From \cite[Theorem 4.1]{f4},
 we see that $\omega _{b}$ is bounded. Then we have
$\omega _{b}\in C^{1,\alpha }( \overline{\Omega }) $ and
$\frac{ \partial \omega _{b}}{\partial \nu }>0$ on $\partial \Omega $ from the
following Lemma.

\begin{lemma} \label{lem3.1}
 (i) \cite[Theorem 1.2]{f1} Let $\omega_{b}$ be a bounded solution of
\eqref{1.6}, then $\omega _{b}\in C^{1,\alpha }( \overline{\Omega }) $;

(ii) \cite[Theorem 1.2]{z1}  Let $\omega _{b}$ be a solution
of \eqref{1.6}, $x_1\in \partial \Omega $,
$\omega _{b}\in C^{1}(\Omega \cup \{x_1\})$, $\omega _{b}(x_1)=0$.
If $\Omega $ satisfies the
inward-ball condition at $x_1$, then
$\frac{\partial \omega _{b}}{\partial\nu }(x_1)>0$, where $\nu $
is the inward unit normal vector of $\partial \Omega $ on $x_1$.
\end{lemma}

We will prove the Theorems \ref{thm1.1}--\ref{thm1.4} stated in section 1 by using
Lemma \ref{lem2.7}.
Next we will construct a supersolution of \eqref{1.1} when $\lambda $ is
small enough.
Before we begin the proof of Theorem \ref{thm1.1}, we need some background.

Define
\[
g_{\#}(s)=\begin{cases}
g(s), & \text{when }s<1, \\
g(s), & \text{when }s\geq 1\text{ and }
\limsup_{s\to +\infty } \frac{g(s)}{s^{p^{-}-1-\varepsilon }}<+\infty , \\
g(1)s^{p^{-}-1-\varepsilon }, & \text{when }s\geq 1\text{ and }
\limsup_{s\to +\infty } \frac{g(s)}{s^{p^{-}-1-\varepsilon }}=+\infty .
\end{cases}
\]
Without loss of generality, we assume that $g_{\#}( s)
=C_{\ast }s^{p^{-}-1-\varepsilon }$ for $s\geq 1$.
There exists $M_0=M_0(\delta )$ large enough such that
\begin{equation}
g_{\#}( s) <\delta s^{p^{-}-1},\quad \forall s\geq M_0.  \label{1.8}
\end{equation}

Now we define a continuous function $\hat{g}:( 0,\infty )
\to ( 0,\infty ) $ by
\[
\hat{g}( s) :=\sup \big\{ \frac{g_{\#}( t) }{
t^{p^{-}-1}},t>s\big\} ,\quad s>0.
\]
It follows from \eqref{1.8} and the definition of $\hat{g}$ that
\begin{itemize}
\item[(i)] $\hat{g}$ is non-increasing;

\item[(ii)] $\hat{g}( s) \geq \frac{g_{\#}( s) }{s^{p^{-}-1}},s>0$;

\item[(iii)] $\hat{g}( s) <\delta $, for all $s\geq M_0$.
\end{itemize}
We also define a $C^{1}$-function
\[
H( s) :=\Big( \frac{2}{s}\int_{\frac{s}{2}}^{s}\hat{g}(
t) dt\Big) ^{\frac{1}{p^{-}-1}},\quad s>0.
\]

\begin{lemma} \label{lem3.2}
The function $H$ satisfies
\begin{itemize}
\item[(i)] $H$ is strictly decreasing, and $-H'( s) \geq \frac{
2^{\varepsilon }-1}{p^{-}-1}\frac{H( s) }{s}$;

\item[(ii)] $\hat{g}( s) \leq [ H( s) ]^{p^{-}-1}\leq
\hat{g}(s/2)$, $s>0$;

\item[(iii)] $H( s) \to +\infty $ as $s\to 0^{+}$,
$H( s) \to 0^{+}$, when $s\to +\infty $.
\end{itemize}
\end{lemma}

\begin{proof}  We only need to prove
$-H'( s) \geq \frac{2^{\varepsilon }-1}{p^{-}-1}\frac{H( s) }{s}$, the rest is
easy to be verified.
By computations
\[
-H'( s) =\frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{\frac{s
}{2}}^{s}\hat{g}( t) dt\Big) ^{\frac{1}{p^{-}-1}-1}
\Big( \frac{2}{s^{2}}\int_{\frac{s}{2}}^{s}\hat{g}( t) dt+\frac{2}{s}(\frac{1
}{2}\hat{g}(\frac{s}{2})-\hat{g}( s) )\Big) .
\]
By condition (H1), when $s\leq 1$, $s^{p^{-}}\hat{g}( s) $
is decreasing, then we have
$\frac{1}{2}\hat{g}(\frac{s}{2})-\hat{g}(s) \geq 0$, and then
\[
-H'( s) \geq \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{
\frac{s}{2}}^{s}\hat{g}( t) dt\Big) ^{\frac{1}{p^{-}-1}-1}\frac{
2}{s^{2}}\int_{\frac{s}{2}}^{s}\hat{g}( t) dt\geq \frac{
2^{\varepsilon }-1}{p^{-}-1}\frac{H( s) }{s}.
\]
Here we note that $\hat{g}(s)=C_{\ast }s^{-\varepsilon }$ for $s\geq 1$.
When $s\geq 2$, we have
\begin{align*}
\frac{2}{s}\Big(\frac{1}{2}\hat{g}(\frac{s}{2})-\hat{g}( s)\Big)
&= C_{\ast }\frac{2}{s}\Big( \frac{1}{2}(\frac{s}{2})^{-\varepsilon}
 -s^{-\varepsilon }\Big) \\
&= C_{\ast }\frac{2}{s}(2^{\varepsilon -1}-1)s^{-\varepsilon } \\
&= \frac{2}{s}(2^{\varepsilon -1}-1)\hat{g}( s) \\
&\geq \frac{2}{s}(2^{\varepsilon -1}-1)\frac{2}{s}\int_{\frac{s}{2}}^{s}
\hat{g}( t) dt \\
&= (2^{\varepsilon }-2)\frac{2}{s^{2}}\int_{\frac{s}{2}}^{s}\hat{g}(
t) dt,
\end{align*}
and
\begin{align*}
-H'( s) &= \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{
\frac{s}{2}}^{s}\hat{g}( t) dt\Big) ^{\frac{1}{p^{-}-1}
-1}\Big( \frac{2}{s^{2}}\int_{\frac{s}{2}}^{s}\hat{g}( t) dt+
\frac{2}{s}(\frac{1}{2}\hat{g}(\frac{s}{2})-\hat{g}( s) )\Big)
\\
&\geq \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{\frac{s}{2}}^{s}\hat{g}
( t) dt\Big) ^{\frac{1}{p^{-}-1}-1}\Big( \frac{2}{s^{2}}\int_{
\frac{s}{2}}^{s}\hat{g}( t) dt+(2^{\varepsilon }-2)\frac{2}{s^{2}}
\int_{\frac{s}{2}}^{s}\hat{g}( t) dt\Big) \\
&= \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{\frac{s}{2}}^{s}\hat{g}(
t) dt\Big) ^{\frac{1}{p^{-}-1}-1}\Big( \frac{2}{s^{2}}
(2^{\varepsilon }-1)\int_{\frac{s}{2}}^{s}\hat{g}( t) dt\Big) \\
&= \frac{2^{\varepsilon }-1}{p^{-}-1}\frac{H( s) }{s}.
\end{align*}

Note that $s^{p^{-}}\hat{g}( s) $ is decreasing for $s\leq 1$.
When $1<s<2$, we have
\begin{align*}
\frac{2}{s}\big( \frac{1}{2}\hat{g}(\frac{s}{2})-\hat{g}( s)\big)
&= \frac{2}{s}\big( \frac{1}{2}(\frac{s}{2})^{-p^{-}}(\frac{s}{2}
)^{p^{-}}\hat{g}(\frac{s}{2})-C_{\ast }s^{-\varepsilon }\big) \\
&\geq \frac{2}{s}\big( \frac{1}{2}(\frac{s}{2})^{-p^{-}}(\frac{2}{2}
)^{p^{-}}\hat{g}(\frac{2}{2})-C_{\ast }s^{-\varepsilon }\big) \\
&= C_{\ast }\frac{2}{s}( \frac{1}{2}(\frac{s}{2})^{-p^{-}}-s^{-
\varepsilon }) \\
&= C_{\ast }\frac{1}{s}s^{-\varepsilon }( 2^{p^{-}}s^{\varepsilon
-p^{-}}-2) \\
&\geq C_{\ast }\frac{1}{s}s^{-\varepsilon }( 2^{\varepsilon }-2)
\\
&\geq ( 2^{\varepsilon }-2) \frac{2}{s^{2}}\int_{\frac{s}{2}}^{s}
\hat{g}( t) dt.
\end{align*}
Then we have
\begin{align*}
-H'( s)
&= \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{
\frac{s}{2}}^{s}\hat{g}( t) dt\Big) ^{\frac{1}{p^{-}-1}
-1}\Big( \frac{2}{s^{2}}\int_{\frac{s}{2}}^{s}\hat{g}( t) dt+
\frac{2}{s}(\frac{1}{2}\hat{g}(\frac{s}{2})-\hat{g}( s) )\Big)
\\
&\geq \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{\frac{s}{2}}^{s}\hat{g}
( t) dt\Big) ^{\frac{1}{p^{-}-1}-1}\Big( \frac{2}{s^{2}}\int_{
\frac{s}{2}}^{s}\hat{g}( t) dt+(2^{\varepsilon }-2)\frac{2}{s^{2}}
\int_{\frac{s}{2}}^{s}\hat{g}( t) dt\Big) \\
&= \frac{1}{p^{-}-1}\Big( \frac{2}{s}\int_{\frac{s}{2}}^{s}\hat{g}(
t) dt\Big) ^{\frac{1}{p^{-}-1}-1}\Big( \frac{2}{s^{2}}
(2^{\varepsilon }-1)\int_{\frac{s}{2}}^{s}\hat{g}( t) dt\Big) \\
&= \frac{2^{\varepsilon }-1}{p^{-}-1}\frac{H( s) }{s}.
\end{align*}
By summarizing the above discussion, we have
\[
-H'( s) \geq \frac{2^{\varepsilon }-1}{p^{-}-1}\frac{
H( s) }{s},\quad \forall s>0.
\]
The proof is complete.
\end{proof}

As a consequence of Lemma \ref{lem3.2}, we can define the function
\begin{equation}
\eta ( s) :=\int_0^{s}\frac{1}{H( t) }dt,s\geq 0,
\label{1.9}
\end{equation}
for it is easy to show that $\eta \in C^{2}(0,\infty )$.


\begin{lemma} \label{lem3.3} The function $\eta $ satisfies
\begin{itemize}
\item[(i)] $\eta :( 0,\infty ) \to ( 0,\infty ) $ is
strictly increasing;

\item[(ii)] let $\psi =\eta ^{-1}$ be the inverse function of $\eta $. Then $\psi
'(s)=H(\psi ( s) ),s>0$.
\end{itemize}
\end{lemma}

Denote
$\Omega _{\sigma }=\{x\in \Omega : \omega _{b}( x) <\sigma \}$,
 where $\sigma >0$ is a small positive constant.

\begin{lemma} \label{lem3.4}
Assume that {\rm (H0)} and {\rm (H1)} hold.  Then there is a
supersolution $v$ of \eqref{1.1} such that
$v\in W_{\rm loc}^{1,p(\cdot)} ( \Omega ) \cap C_0( \overline{\Omega })$ when
$\lambda $ is small enough.
\end{lemma}

\begin{proof} Define
\begin{equation}
v( x) :=\begin{cases}
\psi ( k_1\omega _{b}( x) ) , & x\in \Omega _{\sigma}, \\
\omega _{b}( x) +\psi ( k_1\sigma ) -\sigma , & x\in
\Omega \backslash \Omega _{\sigma },
\end{cases} \label{2.1}
\end{equation}
where $\omega _{b}$ is given by \eqref{1.6}, and $k_1>1$ is a constant.
Obviously,
 $v\in C_0(\overline{\Omega })\cap W_{\rm loc}^{1,p(\cdot )}(\Omega ) $.
From the definition of $g$ and $\hat{g}$, Lemma \ref{lem3.2}, and
Lemma \ref{lem3.3}, it follows that
\begin{gather*}
\psi '( k_1\omega _{b}( x) ) =H( \psi( k_1\omega _{b}( x) ) ) =H( v(
x) ) ,x\in \Omega _{\sigma }, \\
\psi ^{\prime \prime }( s) \leq 0,\text{ for all }s\geq 0.
\end{gather*}
We will prove this Lemma in three steps.
\smallskip

\noindent \textit{Step 1}.
We will prove that $v$ is a super-solution of \eqref{1.1} in $\Omega
_{\sigma }$; i.e.,
\[
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx\geq \int_{\Omega }\lambda bg( v) \phi\,dx
\geq \int_{\Omega }\lambda f( x,v) \phi\,dx,
\]
for any $\phi \in C_0^{\infty }( \Omega _{\sigma }) $ with
$ \phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega _{\sigma }$.
By computation, we have
\begin{align*}
&\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx \\
&= \int_{\Omega }[ k_1\psi '(k_1\omega _{b}) ] ^{p( x) -1}| \nabla
\omega _{b}| ^{p( x) -2}\nabla \omega _{b}\nabla \phi\,dx
\\
&= \int_{\Omega }| \nabla \omega _{b}| ^{p( x)
-2}\nabla \omega _{b}\nabla \left\{ \phi [ k_1\psi '(
k_1\omega _{b}) ]^{p( x) -1}\right\}\,dx \\
&\quad -\int_{\Omega }( k_1) ^{p( x) }| \nabla
\omega _{b}| ^{p( x) }\phi ( p( x)
-1) [ \psi '( k_1\omega _{b}) ]
^{p( x) -2}\psi ^{\prime \prime }( k_1\omega _{b})\,dx
\\
&\quad -\int_{\Omega }( k_1) ^{p( x) -1}| \nabla
\omega _{b}| ^{p( x) -2}\nabla \omega _{b}\nabla p[
\phi \psi '( k_1\omega _{b}) ^{p( x) -1}
] \ln k_1\psi '( k_1\omega _{b})\,dx.
\end{align*}
By Lemma \ref{lem3.2}, we have $-H'( v) \geq \frac{2^{\varepsilon
}-1}{p^{-}-1}\frac{H( v) }{v}$ which implies
\[
-\psi ^{\prime \prime }( k_1\omega _{b}) =-H'(
v) \psi '( k_1\omega _{b}) =-H'(
v) H( v) \geq \frac{2^{\varepsilon }-1}{p^{-}-1}\frac{
H( v) }{v}H( v) .
\]
Note that $0<c_1\leq | \nabla \omega _{b}| \leq c_2$
on $\overline{\Omega _{\sigma }}$. Let $\sigma $ be small enough. We can see
that $v$ is small enough in $\Omega _{\sigma }$, and $H(v)$ is large enough
in $\Omega _{\sigma }$. Then we have
\[
| \nabla \omega _{b}| \frac{2^{\varepsilon }-1}{p^{-}-1}
k_1\frac{H( v) }{v}\geq c_1\frac{2^{\varepsilon }-1}{p^{-}-1}
k_1\frac{H( v) }{v}\geq | \nabla p| \ln
k_1H( v) .
\]

By computations, for any $\phi \in C_0^{\infty }(\Omega )$ with $\phi \geq
0 $, we have
\begin{align*}
&-\int_{\Omega }( k_1) ^{p( x) }| \nabla
\omega _{b}| ^{p( x) }\phi ( p( x)
-1) [ \psi '( k_1\omega _{b}) ]
^{p( x) -2}\psi ^{\prime \prime }( k_1\omega _{b})\,dx
\\
&= -\int_{\Omega }( k_1) ^{p( x) }| \nabla
\omega _{b}| ^{p( x) }\phi ( p( x)
-1) [ H(\psi ( k_1\omega _{b}) )] ^{p(
x) -2}\psi ^{\prime \prime }( k_1\omega _{b})\,dx \\
&= -\int_{\Omega }( k_1) ^{p( x) }| \nabla
\omega _{b}| ^{p( x) }\phi ( p( x)
-1) [ H( v) ] ^{p( x) -2}H'( v) \psi '( k_1\omega _{b})\,dx \\
&\geq \int_{\Omega }( k_1) ^{p( x) }|
\nabla \omega _{b}| ^{p( x) }\phi ( p(
x) -1) [ H( v) ] ^{p( x) -1}
\frac{2^{\varepsilon }-1}{p^{-}-1}\frac{H( v) }{v}dx \\
&\geq \int_{\Omega }( k_1) ^{p( x) -1}|
\nabla \omega _{b}| ^{p( x) -1}| \nabla
p| [ \phi \psi '( k_1\omega _{b})
^{p( x) -1}] | \ln k_1H( v)
|\,dx.
\end{align*}
Here we note that $H(s)\to +\infty $ as $s\to 0^{+}$ and $
v\in C_0(\overline{\Omega })$. Then $v(x)\leq 1$ in $\Omega _{\sigma }$
and $H(v(x))\geq 1$ for any $x\in \Omega _{\sigma }$ when $\sigma $ is small
enough. Thus we have
\begin{align*}
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla v\nabla \phi\,dx
&\geq \int_{\Omega }| \nabla \omega
_{b}| ^{p( x) -2}\nabla \omega _{b}\nabla \{\phi [
k_1\psi '( k_1\omega _{b}) ] ^{p(x) -1}\}dx \\
&= \int_{\Omega }b[ k_1\psi '( k_1\omega _{b})
] ^{p( x) -1}\phi\,dx \\
&\geq \int_{\Omega }b[ k_1H(v)] ^{p^{-}-1}\phi\,dx \\
&\geq \int_{\Omega }bk_1^{p^{-}-1}\hat{g}(v)\phi\,dx \\
&\geq \int_{\Omega }bk_1^{p^{-}-1}\frac{g(v)}{v^{p^{-}-1}}\phi\,dx \\
&\geq \int_{\Omega }bg(v)\phi\,dx,
\end{align*}
for any $\phi \in C_0^{\infty }( \Omega _{\sigma }) $ with
$\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega $.

Then for any $\phi \in C_0^{\infty }( \Omega _{\sigma }) $ with
$\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega _{\sigma }$,
we have
\begin{equation}
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx\geq \int_{\Omega }\lambda b( x) g( v)
\phi\,dx\geq \int_{\Omega }\lambda f( x,v) \phi\,dx.  \label{2.4}
\end{equation}
\smallskip

\noindent\textit{Step 2}.
We will prove that $v$ is a supersolution of \eqref{1.1} in $\Omega
\backslash \overline{\Omega _{\sigma }}$; i.e.,
\[
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx\geq \int_{\Omega }\lambda f( x,v) \phi\,dx,\forall
\phi \in C_0^{\infty }( \Omega \backslash \overline{\Omega _{\sigma }}
) ,\phi \geq 0.
\]
Let $\lambda $ be small enough such that
$\lambda g( v(x)) \leq 1$, for all
$x\in \Omega \backslash \overline{\Omega _{\sigma }}$. For any
$\phi \in C_0^{\infty }( \Omega \backslash \overline{\Omega _{\sigma }}) $
 with $\phi \geq 0$, we have
\begin{equation}
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx=\int_{\Omega }b\phi\,dx\geq \int_{\Omega }\lambda bg(
v) \phi\,dx\geq \int_{\Omega }\lambda f( x,v) \phi\,dx.
\label{2.51}
\end{equation}
\smallskip

\textit{Step 3}.
We will prove that $v$ is a super-solution of \eqref{1.1} in $\Omega $;
i.e.,
\[
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx\geq \int_{\Omega }\lambda f( x,v) \phi\,dx,
\]
for all $\phi \in C_0^{\infty }( \Omega )$  with
$\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega$.
Denote $d_1(x)=d(x,\partial (\Omega \backslash \Omega _{\sigma }))$,
and
\[
\xi _{n}(x)=\begin{cases}
2n(\frac{1}{n}-d_1(x)), & \frac{1}{2n}\leq d_1(x)\leq \frac{1}{n}, \\
1, & 0\leq d_1(x)<\frac{1}{2n},\;n\in\mathbb{N} \\
0, & \text{other wise}.
\end{cases}
\]
For $\phi \in C_0^{\infty }( \Omega _{\sigma }) $, with
$\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega $,
 we have $\phi =\phi_{1,n}+\phi _{2,n}+\phi _{3,n}$, where
$\phi _{1,n}=\phi \chi _{\overline{\Omega _{\sigma }}}(1-\xi _{n})
\in C_0^{\infty }( \Omega ) $
satisfies $\operatorname{supp}\phi _{1,n} \Subset \Omega _{\sigma }$,
$\phi_{2,n}=\phi \xi _{n}\in C_0^{\infty }( \Omega ) $, and
$\phi_{3,n}=\phi \chi _{\Omega \backslash \overline{\Omega _{\sigma }}}(1-\xi
_{n})\in C_0^{\infty }( \Omega \backslash \overline{\Omega _{\sigma }}) $.
Therefore,
\begin{align*}
&\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla v\nabla \phi\,dx \\
&= \lim_{n\to \infty} \Big\{ \int_{\Omega }|
\nabla v| ^{p( x) -2}\nabla v\nabla \phi _{1,n}dx
 +\int_{\Omega }| \nabla v| ^{p( x)-2}\nabla v\nabla \phi _{2,n}dx\\
&\quad +\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla v\nabla \phi _{3,n}dx\Big\} \\
&\geq \lim_{n\to \infty} \Big\{ \int_{\Omega }\lambda
f( x,v) \phi _{1,n}dx+\int_{\Omega }\lambda f( x,v)
\phi _{3,n}dx+\int_{\Omega }| \nabla v| ^{p(
x) -2}\nabla v\nabla \phi _{2,n}dx\Big\} \\
&= \int_{\Omega _{\sigma }}\lambda f( x,v) \phi\,dx+\int_{\Omega
\backslash \Omega _{\sigma }}\lambda f( x,v) \phi\,dx+\underset{
n\to \infty }{\lim }\int_{\Omega }| \nabla v|
^{p( x) -2}\nabla v\nabla \phi _{2,n}dx \\
&= \int_{\Omega }\lambda f( x,v) \phi\,dx+\lim_{n\to \infty} \int_{\Omega }\phi | \nabla v| ^{p(
x) -2}\nabla v\nabla \xi _{n}dx \\
&= \int_{\Omega }\lambda f( x,v) \phi\,dx+\int_{\partial (\Omega
\backslash \Omega _{\sigma })}\phi [ (\psi '( k_1\sigma
) | \nabla k_1\omega _{b}| )^{p( x)
-1}-| \nabla \omega _{b}| ^{p( x) -1}]\,dS.
\end{align*}
Here we note that $\psi '( k_1\sigma ) =H(k_1\sigma )\to +\infty $ as
$\sigma \to 0^{+}$. Thus
\[
\int_{\partial (\Omega \backslash \Omega _{\sigma })}\phi [ (\psi
'( k_1\sigma ) | \nabla k_1\omega
_{b}| )^{p( x) -1}-| \nabla \omega
_{b}| ^{p( x) -1}]\,dS\geq 0,
\]
and then
\[
\int_{\Omega }| \nabla v| ^{p( x) -2}\nabla
v\nabla \phi\,dx\geq \int_{\Omega }\lambda f( x,v) \phi\,dx,
\]
for all $\phi \in C_0^{\infty }( \Omega )$  with
$\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega$.
It means that $v$ is a super-solution of \eqref{1.1}.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
At first, we construct a subsolution for
problem \eqref{1.1}. Since $\partial \Omega $ is $C^{2}$ smooth, there
exists a positive constant $\ell $ such that
$d(\cdot )\in C^{2}(\overline{\partial \Omega _{3\ell }})$, and
$| \nabla d(\cdot )| \equiv 1$, where
$\overline{\partial \Omega _{3\ell }}
=\{x\in \overline{\Omega }: d(x)\leq 3\ell \}$.

Let $\sigma \in (0,\ell )$ be small enough. Denote
\[
\phi (x)=\begin{cases}
e^{kd(x)}-1, & d(x)<\sigma , \\
e^{k\sigma }-1+\int_{\sigma }^{d(x)}ke^{k\sigma }(\frac{2\ell -t}{2\ell
-\sigma })^{\frac{2}{p^{-}-1}}dt, & \sigma \leq d(x)<2\ell , \\
e^{k\sigma }-1+\int_{\sigma }^{2\ell }ke^{k\sigma }(\frac{2\ell -t}{2\ell
-\sigma })^{\frac{2}{p^{-}-1}}dt, & 2\ell \leq d(x),
\end{cases}
\]
where $k>0$ is a parameter.
It is easy to see that $\phi \in C_0^{1}(\overline{\Omega })$. By
computations it follows that
\[
-\Delta _{p(x)}\mu \phi 
=\begin{cases}
-k(k\mu e^{kd(x)})^{p(x)-1}\big[(p(x)-1)+(d(x)
+\frac{\ln k\mu }{k})\nabla p(x)\nabla d(x)\\
+\frac{\Delta d(x)}{k}\big], 
\quad \text{if }d(x)<\sigma ,
\\[4pt]
\Big\{\frac{1}{2\ell -\sigma }\frac{2(p(x)-1)}{p^{-}-1}-(\frac{2\ell -d(x)}{
2\ell -\sigma })[(\ln k\mu e^{k\sigma }(\frac{2\ell -d(x)}{2\ell -\sigma })^{
\frac{2}{p^{-}-1}})\nabla p(x)\nabla d(x)\\
+\Delta d(x)]\Big\}
 (k\mu e^{k\sigma })^{p(x)-1}(\frac{2\ell -d(x)}{2\ell -\sigma })^{
\frac{2(p(x)-1)}{p^{-}-1}-1},\\
\quad \text{if }\sigma <d(x)<2\ell , 
 \\[4pt]
0 \quad \text{if } 2\ell <d(x).
\end{cases}
\]

Denote $\sigma =\frac{1}{k}\ln 2^{\frac{1}{p^{+}}}$. Then
\begin{equation}
e^{k\sigma }=2^{\frac{1}{p^{+}}}.  \label{7.5}
\end{equation}
Let
\begin{equation}
\mu =e^{-ak},\quad \text{where }
a=\frac{p^{-}-1}{\max_{x\in \overline{\Omega }} | \nabla p| +1}.  \label{1.2}
\end{equation}
Then $k\mu \leq 1$ when $k$ is large enough. If $k$ is sufficiently large,
it is easy to see that
\begin{equation}
-\Delta _{p(x)}\phi \leq 0\leq \lambda f(x,\phi ), d(x)<\sigma.
 \label{8.1}
\end{equation}

Since $d(\cdot )\in C^{2}(\overline{\Omega _{3\ell }})$, there exists a
positive constant $C_3$ such that
$| \Delta d| \leq C_3$ on $\overline{\Omega _{3\ell }}$.
Note that the definition of $\sigma$ means \eqref{7.5}; i.e.,
$e^{k\sigma }=2^{\frac{1}{p^{+}}}$. When $k$ is large enough, we have
\begin{equation}
\begin{aligned}
&-\Delta _{p(x)}\mu \phi \\
&\leq  (k\mu e^{k\sigma })^{p(x)-1}(\frac{2\ell
-d(x)}{2\ell -\sigma })^{\frac{2(p(x)-1)}{p^{-}-1}-1} 
\Big| \frac{2(p(x)-1)}{(2\ell -\sigma )(p^{-}-1)}\\
&-(\frac{ 2\ell -d(x)}{2\ell -\sigma })
\big[(\ln k\mu e^{k\sigma }(\frac{2\ell -d(x)}{
2\ell -\sigma })^{\frac{2}{p^{-}-1}})\nabla p(x)\nabla d(x)+\Delta
d(x)\big]\Big|   \\
&\leq C_1(k\mu )^{p(x)-1}| \ln k+\ln \mu | , \quad
\sigma <d(x)<2\ell .
\end{aligned}  \label{7.1}
\end{equation}
When $k$ is large enough, we can see that $\mu $ is small enough and
moreover, $k\mu \leq 1$. Combining \eqref{7.2}, \eqref{1.2} and \eqref{7.1}
together, we have
\begin{equation}
\begin{aligned}
-\Delta _{p(x)}\mu \phi
&\leq C_1(k\mu )^{p(x)-1}| \ln \mu |   \\
&\leq C_1(k\mu )^{p^{-}-1}| \ln \mu |   \\
&\leq C_1a^{1-p^{-}}\mu ^{p^{-}-1}| \ln \mu | ^{p^{-}} \\
&\leq C_2(\mu \phi )^{p^{-}-1}| \ln \mu \phi | ^{p^{-}} \\
&\leq \lambda f(x,\mu \phi ), \sigma <d(x)<2\ell .
\end{aligned} \label{7.3}
\end{equation}
Obviously
\begin{equation}
-\Delta _{p(x)}\mu \phi =0\leq \lambda f(x,\mu \phi ), 2\ell <d(x).
\label{7.4}
\end{equation}
Combining \eqref{8.1}, \eqref{7.3} and \eqref{7.4}, we can conclude that
\begin{equation}
-\Delta _{p(x)}\mu \phi \leq \lambda f(x,\mu \phi ),\quad \text{a.e. in
}\Omega .  \label{7.7}
\end{equation}

Here we note that $\mu \phi =v$ on $\partial \Omega $, and
$\mu \nabla \phi \leq \nabla v$ near $\partial \Omega $,
then $\mu \phi \leq v$ on $\Omega $
when $\mu $ is small enough. By Lemma \ref{lem2.7} and Lemma \ref{lem3.4},
\eqref{1.1} has a solution.
The proof is complete.
\end{proof}


To prove Theorem \ref{thm1.2}, we need the following Lemma, which is useful
for finding  supersolutions of \eqref{1.1}.
We denote by $C_0$ the best embedding constant of
$W_0^{1,1}(\Omega )\subset L^{\frac{N}{N-1}}(\Omega )$
(see  \cite[Lemma 5.2]{c2}); i. e.,
\begin{equation}
| u| _{L^{N/(N-1)}(\Omega )}\leq C_0| \nabla
u| _{L^{1}(\Omega )}\quad \text{for }u\in W_0^{1,1}(\Omega ).
\label{2.11}
\end{equation}

\begin{lemma} \label{lem3.5}
 Suppose $0\leq b(\cdot )\in L^{\alpha (\cdot )}(\Omega )$,
 $N<\alpha ( \cdot ) \in C( \overline{\Omega }) $.
Let $M>0$ and $u$ be the unique solution of the problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)=Mb(x), \quad \text{in }\Omega , \\
u=0, \quad \text{on }\partial \Omega .
\end{gathered} \label{2.12}
\end{equation}
Set
\[
\rho _0=\frac{p^{-}}{2| b(\cdot )| _{L^{\alpha
^{-}}(\Omega )}| \Omega | ^{\frac{1}{N}-\frac{1}{\alpha^{-}}}C_0}.
\]
Then $| u| _{\infty }\leq C^{\ast}M^{1/(p^{-}-1)}$ when
$M\geq \rho _0$ and $| u|_{\infty }\leq C_{\ast }M^{1/(p^{+}-1)} $ when
$M<\rho _0$, where $C^{\ast }$ and $C_{\ast }$ are positive constants
depending only on $p^{+}, p^{-}, N, | b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)},| \Omega | $ and $C_0$.
\end{lemma}

\begin{proof}
Let $u$ be the solution of \eqref{2.12}. Then $u\geq 0$. For
$k\geq 0$, set $A_{k}=\{x\in \Omega :u(x)>k\}$. By taking $(u-k)^{+}$ as a
test function of \eqref{2.12}, it follows from \eqref{2.11} and Young
inequality that
\begin{equation}
\begin{aligned}
\int_{A_{k}}| \nabla u| ^{p(x)}dx
&= M\int_{A_{k}}b(u-k)dx \\
&\leq M| b(\cdot )| _{L^{N}(A_{k})}|(u-k)^{+}| _{L^{N/(N-1)}(A_{k})}   \\
&\leq  M| b(\cdot )| _{L^{\alpha ^{-}}(A_{k})}|
A_{k}| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}|
(u-k)^{+}| _{L^{N/(N-1)}(A_{k})}   \\
&\leq M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| A_{k}| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}
}C_0\int_{A_{k}}\varepsilon | \nabla u| \varepsilon
^{-1}dx   \\
&\leq M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| A_{k}| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}
}C_0\int_{A_{k}}\Big( \frac{(\varepsilon | \nabla u|
)^{p(x)}}{p(x)}+\frac{(\varepsilon ^{-1})^{p^{0}(x)}}{p^{0}(x)}\Big)\,dx
 \\
&\leq \frac{M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| \Omega | ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}C_0}{
p^{-}}\int_{A_{k}}\varepsilon ^{p(x)}| \nabla u| ^{p(x)}dx
 \\
&\quad +\frac{M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| A_{k}| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}C_0}{
(p^{+})^{0}}\int_{A_{k}}\varepsilon ^{-p^{0}(x)}dx.
\end{aligned} \label{2.12b}
\end{equation}
When $M\geq \rho _0$ we can take
\begin{equation}
\varepsilon =\Big( \frac{p^{-}}{2M| b(\cdot )|
_{L^{\alpha ^{-}}(\Omega )}| \Omega | ^{\frac{1}{N}-\frac{
1}{\alpha ^{-}}}C_0}\Big) ^{1/p^{-}}=( \frac{\rho _0}{M})
^{1/p^{-}},  \label{2.14}
\end{equation}
then $\varepsilon \leq 1$ and
\begin{align*}
&\frac{M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| \Omega | ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}C_0}{
p^{-}}\int_{A_{k}}\varepsilon ^{p(x)}| \nabla u| ^{p(x)}dx
\\
&\leq \frac{M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}| \Omega | ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}C_0}{
p^{-}}\varepsilon ^{p^{-}}\int_{A_{k}}| \nabla u|
^{p(x)}dx\\
&=\frac{1}{2}\int_{A_{k}}| \nabla u| ^{p(x)}dx.
\end{align*}
Consequently, from the inequality above and \eqref{2.12b} it follows that
\begin{equation}
\begin{aligned}
\int_{A_{k}}| \nabla u| ^{p(x)}dx
&\leq \frac{2M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega )}|
A_{k}| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}}C_0}{(p^{+})^{0}}
\int_{A_{k}}\varepsilon ^{-p^{0}(x)}dx   \\
&\leq \frac{2M| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}C_0\varepsilon ^{-(p^{-})^{0}}}{(p^{+})^{^{0}}}|
A_{k}| ^{1+\frac{1}{N}-\frac{1}{\alpha ^{-}}}.
\end{aligned} \label{2.15}
\end{equation}
Note that $b(\cdot )\geq 1$. From \eqref{2.12} and \eqref{2.15}, we have
\begin{equation}
\int_{A_{k}}(u-k)dx\leq \int_{A_{k}}b(x)(u-k)dx=\frac{1}{M}
\int_{A_{k}}| \nabla u| ^{p(x)}dx\leq \gamma |
A_{k}| ^{1+\frac{1}{N}-\frac{1}{\alpha ^{-}}},  \label{2.16}
\end{equation}
where
\begin{equation}
\gamma =\frac{2| b(\cdot )| _{L^{\alpha ^{-}}(\Omega
)}C_0\varepsilon ^{-(p^{-})^{^{0}}}}{(p^{+})^{0}}.  \label{2.17}
\end{equation}
By the \cite[Lemma 5.1, Chapter 2]{l1}, \eqref{2.16} implies
\begin{equation}
| u| _{\infty }\leq \gamma (\frac{\alpha ^{-}N}{\alpha
^{-}-N}+1)^{2}| \Omega | ^{\frac{1}{N}-\frac{1}{\alpha
^{-}}}.  \label{2.18}
\end{equation}
From \eqref{2.14}, \eqref{2.17} and \eqref{2.18}, we  see that
\begin{equation}
| u| _{\infty }\leq C^{\ast }M^{1/(p^{-}-1)},
\label{2.19}
\end{equation}
where
\begin{equation}
C^{\ast }=\frac{(\frac{\alpha ^{-}N}{\alpha ^{-}-N}+1)^{2}(2C_0|
b(\cdot )| _{L^{\alpha ^{-}}(\Omega )}| \Omega
| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}})^{(p^{-})^{0}}}{
(p^{+})^{0}(p^{-})^{(p^{-})^{0}/p^{-}}}.  \label{2.20}
\end{equation}
When $M<\rho _0$, take
\[
\varepsilon =\Big( \frac{p^{-}}{2M| b(\cdot )|
_{L^{\alpha ^{-}}(\Omega )}| \Omega | ^{\frac{1}{N}-\frac{
1}{\alpha ^{-}}}C_0}\Big) ^{1/p^{+}}=( \frac{\rho _0}{M})
^{1/p^{+}}.
\]
Note that in this case $\varepsilon >1$. Using similar arguments as
above we  obtain
\[
| u| _{\infty }\leq C_{\ast }M^{1/(p^{+}-1)},
\]
where
\[
C_{\ast }=\frac{(\frac{\alpha ^{-}N}{\alpha ^{-}-N}+1)^{2}(2C_0|
b(\cdot )| _{L^{\alpha ^{-}}(\Omega )}| \Omega
| ^{\frac{1}{N}-\frac{1}{\alpha ^{-}}})^{(p^{+})^{0}}}{
(p^{+})^{0}(p^{-})^{(p^{+})^{0}/p^{+}}}.
\]
The proof is complete.
\end{proof}

\begin{lemma} \label{lem3.6}
Suppose there is a small $\delta >0$ such that $p(x) \equiv p$ (a constant)
for any $x\in \Omega $ with $d(x)\leq \delta$ and
$N<\alpha ( \cdot ) \in C( \overline{\Omega }) $. Let $M>1$ and $u$
 be the unique solution of the problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)=Mb(x), \quad \text{in }\Omega , \\
u=0, \quad \text{on }\partial \Omega ,
\end{gathered}\label{3.2}
\end{equation}
where $0\leq b(\cdot )\in L^{\alpha (\cdot )}(\Omega )$. Then $|
\nabla u(\cdot )| \leq CM^{\frac{1}{p^{-}-1}}$ on $\partial \Omega$.
\end{lemma}

\begin{proof}
 By Lemma \ref{lem3.5}, we have $u(x)\leq C_{\#}M^{\frac{1}{p^{-}-1}}$ for all
$x\in \Omega $. Let $u_2$ be the solution of the following $p$-Laplacian
equation (with constant exponent)
\begin{gather*}
-\operatorname{div}(|\nabla u_2|^{p-2}\nabla u_2)=\varkappa b(x),
\quad \text{in }\Omega , \\
u_2=0, \quad \text{on }\partial \Omega ,
\end{gather*}
where $\varkappa $ is a positive parameter.

It is easy to see that $u_2\in C^{1,\alpha }(\overline{\Omega })$.
Then $\frac{\partial u_2}{\partial \nu }>0$ on $\partial \Omega $,
where $\nu $ is the inward unit normal vector. We can also see that
$u_2>0$ on $\partial (\Omega \backslash \overline{\Omega _{\varepsilon }})$
 when $\varepsilon \in (0,\delta )$ is small enough. Let $\varkappa $ be large
enough, we have $u_2\geq 2C_{\#}$ on $\partial (\Omega \backslash
\overline{\Omega _{\varepsilon }})$. It means that
$u_2M^{\frac{1}{p^{-}-1}}\geq u$ on $\partial \Omega _{\varepsilon }$.
Define $u_3=u_2M^{\frac{1}{p^{-}-1}}$. Since $p( x) \equiv p$ for any
 $x\in \Omega $ with $d(x)\leq \delta $, we have
\[
-\operatorname{div}(|\nabla u_3|^{p(x)-2}\nabla u_3)=-\operatorname{div}(|\nabla
u_3|^{p-2}\nabla u_3)=M^{\frac{p-1}{p^{-}-1}}\varkappa b(x)\geq Mb(x)
\quad\text{in }\Omega _{\varepsilon }.
\]
Therefore, $u_3=u_2M^{\frac{1}{p^{-}-1}}\geq u$ on
$\overline{\Omega _{\varepsilon }}$ and
$| \nabla u| \leq | \nabla u_3| \leq CM^{\frac{1}{p^{-}-1}}$ on
$\partial \Omega $.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 At first, we  construct a supersolution of \eqref{1.1}.
Denote $k_2=\psi ( k_1\sigma ) $. Let $\omega _2$ be the
solution of the problem
\begin{gather*}
-\Delta _{p(x)}\omega _2=b( x) k_2^{p^{-}-1-\frac{
\varepsilon }{2}}, \quad \text{in }\Omega \backslash
\overline{\Omega _{\sigma }}, \\
\omega _2>0, \quad \text{in }\Omega \backslash \overline{\Omega _{\sigma }},\\
\omega _2=0, \quad \text{on }\partial (\Omega \backslash \overline{\Omega
_{\sigma }}).
\end{gather*}
Define
\begin{equation}
v_2( x) :=\begin{cases}
\psi ( k_1\omega _{b}( x) ) , & x\in \Omega _{\sigma}, \\
\omega _2(x)+\psi ( k_1\sigma ) , & x\in \Omega \backslash
\Omega _{\sigma }.
\end{cases}\label{aa}
\end{equation}
For a large enough constant $k_1$, we will prove that $v_2$ is a
supersolution of \eqref{1.1} in three steps.
\smallskip

\noindent\textit{Step 1}.
When $k_1$ is large enough, we will check that $v_2$ is a supersolution
of \eqref{1.1} in $\Omega _{\sigma }$, namely,
\[
\int_{\Omega }| \nabla v_2| ^{p( x)
-2}\nabla v_2\nabla \phi\,dx\geq \int_{\Omega }\lambda bg(
v_2) \phi\,dx\geq \int_{\Omega }\lambda f( x,v_2) \phi
\,dx,
\]
for any $\phi \in C_0^{\infty }( \Omega _{\sigma }) $ with $
\phi \geq 0$ and $\operatorname{supp}\phi \subset \subset \Omega _{\sigma }$.
As in the proof of Lemma \ref{lem3.4}, we only need to prove that
\begin{gather}
\begin{aligned}
&\int_{\Omega }( k_1) ^{p( x) }| \nabla
\omega _{b}| ^{p( x) }\phi ( p( x)
-1) [ H( v_2) ] ^{p( x) -1}\frac{
2^{\varepsilon }-1}{p^{-}-1}\frac{H( v_2) }{v_2}dx   \\
&\geq \int_{\Omega }( k_1) ^{p( x) -1}|
\nabla \omega _{b}| ^{p( x) -1}| \nabla
p| [ \phi \psi '( k_1\omega _{b})
^{p( x) -1}] | \ln k_1H( v_2)|\,dx,
\end{aligned} \label{5.2}
\\
k_1H(v_2(x))\geq 1,\forall x\in \Omega _{\sigma },  \label{5.3}
\\
\frac{k_1^{p^{-}-1}}{[v_2{}(x)]^{p^{-}-1}}\geq 1,\forall x\in \Omega
_{\sigma }.  \label{5.2b}
\end{gather}
We can see that \eqref{5.2} is valid, provided
\begin{equation}
| \nabla \omega _{b}| \frac{2^{\varepsilon }-1}{p^{-}-1}
k_1\frac{H( v_2) }{v_2}\geq c_1\frac{2^{\varepsilon }-1}{
p^{-}-1}k_1\frac{H( v_2) }{v_2}\geq | \nabla
p\| \ln k_1H( v_2) | \quad \text{in }\Omega _{\sigma }.  \label{3.1}
\end{equation}
According to the assumption on $g$, without loss of generality, we
assume that
\[
g(s)\geq cs^{-1}\text{ for }s\leq 1, \quad
\text{and}\quad g(s)=cs^{\theta }\text{ for }s\geq 1,
\]
 where $\theta =p^{-}-1-\varepsilon$.
Thus
\begin{gather*}
\hat{g}( s) \geq cs^{-p^{-}}\text{ for }s\leq 1, \quad \text{and}\quad
\hat{ g}(s)=cs^{\theta +1-p^{-}}\text{ for }s\geq 1,
\\
H( s) \geq c_1s^{-\frac{p^{-}}{p^{-}-1}}\text{ for }s\leq 1,\quad
\text{and}\quad H(s)=c_2s^{\frac{\theta +1-p^{-}}{p^{-}-1}}\text{ for }s\geq 2,
\\
\eta ( s) \leq c_3s^{1+\frac{p^{-}}{p^{-}-1}}\text{ for }s\leq 1,\quad
\text{and}\quad c_{4}s^{2-\frac{\theta }{p^{-}-1}}\leq \eta (s)\leq c_{5}s^{2-
\frac{\theta }{p^{-}-1}}\text{ for }s\geq 3.
\end{gather*}
Then $\psi (s)$ satisfies
\[
c_{7}s^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\leq \psi (s)\leq c_{8}s^{\frac{1
}{2-\frac{\theta }{p^{-}-1}}}\quad \text{for }s\geq 3.
\]

Let $s_0\geq 3$ such that $\eta (s_0)\geq 3$. Denote
\[
\Omega _{\sigma }^{+}=\{x\in \Omega _{\sigma }: k_1\omega _{b}(x)\geq
\eta (s_0)\},\quad
\Omega _{\sigma }^{-}=\{x\in \Omega_{\sigma }: k_1\omega _{b}(x)<\eta (s_0)\}.
\]
Here we note that $v_2=\psi (k_1\omega _{b})$ on
$\overline{\Omega_{\sigma }}$. Since $\psi $ is strictly increasing,
 we have $k_1\omega _{b}\geq \eta (s_0)$ if and only if
$v_2=\psi (k_1\omega _{b})\geq \psi (\eta (s_0))=s_0\geq 3$.
When $v_2\geq s_0$, we have
\begin{gather*}
c_{7}(k_1\omega _{b})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\leq v_2\leq
c_{8}(k_1\omega _{b})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}} \quad\text{on }
\Omega _{\sigma }^{+},
\\
c_{9}(k_1\omega _{b})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}\frac{\theta
+1-p^{-}}{p^{-}-1}}\leq H( v_2) \leq c_{10}(k_1\omega _{b})^{
\frac{1}{2-\frac{\theta }{p^{-}-1}}\frac{\theta +1-p^{-}}{p^{-}-1}} \quad
\text{on }\Omega _{\sigma }^{+},
\\
| \nabla \omega _{b}| \frac{2^{\varepsilon }-1}{p^{-}-1}
k_1\frac{H( v_2) }{v_2}\geq c_{11}k_1(k_1\omega _{b})^{
\frac{1}{2-\frac{\theta }{p^{-}-1}}(\frac{\theta +1-p^{-}}{p^{-}-1}-1)}=
\frac{c_{11}}{\omega _{b}}\geq \frac{c_{11}}{\sigma } \text{ on }\Omega
_{\sigma }^{+},
\\
\begin{aligned}
| \nabla p\| \ln k_1H( v_2)|
&\leq | \nabla p| (\ln k_1+| \ln H( v_2) | )\\
&\leq | \nabla p| (\ln k_1+c_{12}| \ln k_1\omega _{b}| )
\leq c_{13}\ln k_1
\quad \text{on }\Omega _{\sigma }^{+}.
\end{aligned}
\end{gather*}
Denoting $\sigma =\frac{c_{11}}{c_{13}\ln k_1}$, we obtain
\[
| \nabla \omega _{b}| \frac{2^{\varepsilon }-1}{p^{-}-1}
k_1\frac{H( v_2) }{v_2}\geq c_1\frac{2^{\varepsilon }-1}{
p^{-}-1}k_1\frac{H( v_2) }{v_2}\geq | \nabla
p\| \ln k_1H( v_2) | \quad\text{on }\Omega _{\sigma }^{+}.
\]
Since $\psi $ is strictly increasing, we have
$k_1\omega _{b}\leq \eta(s_0)$ if and only if $v_2=\psi (k_1\omega _{b})\leq s_0$.
 Note that $H( v_2) $ is decreasing. It follows that $H( v_2)
\geq H(s_0)$ on $\Omega _{\sigma }^{-}$. Thus \eqref{3.1} is valid when
$k_1$ is large enough.
Thus \eqref{3.1} is valid, and then \eqref{5.2} is valid.

Obviously,
\begin{align*}
k_1H(v_2(x))
&\geq k_1H(\psi (k_1\sigma )) \\
&\geq k_1c_{9}(k_1\frac{c_{11}}{c_{13}\ln k_1})^{\frac{1}{2-\frac{
\theta }{p^{-}-1}}\frac{\theta +1-p^{-}}{p^{-}-1}} \\
&= c_{9}(k_1)^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}(\frac{c_{11}}{
c_{13}\ln k_1})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}\frac{\theta +1-p^{-}}{
p^{-}-1}}\to +\infty,
\end{align*}
for all $x\in \Omega _{\sigma }$  as $k_1\to +\infty$.
Thus \eqref{5.3} is valid.

Note that $\frac{\theta }{p^{-}-1}<1$. Then by the above computation,
\[
v_2\leq c_{8}(k_1\omega _{b})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\leq
c_{8}(k_1\sigma )^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\leq k_1
\]
as $k_1$ is large enough.
Thus \eqref{5.2b} is valid.
\smallskip

\noindent\textit{Step 2.}
We will check that $v_2$ is a supersolution of \eqref{1.1} in
$\Omega \backslash \overline{\Omega _{\sigma }}$ when $k_1$ is large enough;
 i.e.,
\[
\int_{\Omega }| \nabla v_2| ^{p( x)
-2}\nabla v_2\nabla \phi\,dx\geq \int_{\Omega }\lambda bg(
v_2) \phi\,dx\geq \int_{\Omega }\lambda f( x,v_2) \phi
\,dx,
\]
for all $\phi \in C_0^{\infty }( \Omega \backslash \overline{\Omega
_{\sigma }})$, $\phi \geq 0$.
By the definition of $\omega _2$ and Lemmas \ref{lem3.5} and \ref{lem3.6}, we have
\[
\omega _2\leq C_1(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}},\quad
| \nabla \omega _2| \leq C_2(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p-1}}.
\]
Since $v_2=\omega _2+\psi ( k_1\sigma ) $ in $\Omega
\backslash \overline{\Omega _{\sigma }}$, we have
\[
c_{7}(k_1\sigma )^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\leq \psi (
k_1\sigma ) \leq v_2\leq C_1(k_2)^{\frac{p^{-}-1-\frac{
\varepsilon }{2}}{p^{-}-1}}+\psi ( k_1\sigma ) \leq
(C_1+1)\psi ( k_1\sigma ) .
\]
Since $k_1\sigma =\frac{c_{11}k_1}{c_{13}\ln k_1}$ is large enough (as
long as $k_1$ is large enough) and $v_2(\cdot )$ is large enough in
$\Omega \backslash \overline{\Omega _{\sigma }}$,
the assumption (ii) of
Theorem \ref{thm1.2} implies that
 $\frac{g( v_2(x)) }{[v_2(x)]^{p^{-}-1-\frac{
\varepsilon }{2}}}$ is small enough. Therefore, we  see that
\[
\frac{1}{\lambda }>\frac{g( v_2(x)) }{[v_2(x)]^{p^{-}-1-\frac{
\varepsilon }{2}}}.
\] Note that $k_2=\psi ( k_1\sigma ) $. We
have
\[
v_2(x)^{p^{-}-1-\frac{\varepsilon }{2}}\leq [ C_1(k_2)^{\frac{
p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}+\psi ( k_1\sigma )
]^{p^{-}-1-\frac{\varepsilon }{2}}\leq C_3k_2^{p^{-}-1-\frac{\varepsilon
}{2}},\quad \forall x\in \Omega \backslash \overline{\Omega _{\sigma }},
\]
which implies
\[
(k_2)^{p^{-}-1-\frac{\varepsilon }{2}}\geq C_3\lambda (k_2)^{p^{-}-1-
\frac{\varepsilon }{2}}\frac{g( v_2(x)) }{[v_2(x)]^{p^{-}-1-
\frac{\varepsilon }{2}}}\geq \lambda g( v_2(x)) ,\quad \forall x\in
\Omega \backslash \overline{\Omega _{\sigma }}.
\]
We can see that $v_2$ is a supersolution of \eqref{1.1} in
$\Omega \backslash \overline{\Omega _{\sigma }}$; i.e.,
for any $\phi \in C_0^{\infty }( \Omega \backslash \overline{\Omega _{\sigma }}) $
with $\phi \geq 0$, we have
\begin{align*}
\int_{\Omega }| \nabla v_2| ^{p( x)
-2}\nabla v_2\nabla \phi\,dx
&= \int_{\Omega }bk_2^{p^{-}-1-\frac{\varepsilon }{2}}\phi\,dx \\
&\geq \int_{\Omega }\lambda bC_3k_2^{p^{-}-1-\frac{\varepsilon }{2}}
\frac{g( v) }{v_2{}^{p^{-}-1-\frac{\varepsilon }{2}}}\phi \,dx\\
&\geq \int_{\Omega }\lambda f( x,v_2) \phi\,dx.
\end{align*}
\smallskip

\textit{Step 3}.
When $k_1$ is large enough, we will prove that $v_2$ is a supersolution
of \eqref{1.1} in $\Omega $.
When $\omega _{b}( x) =\sigma $, it is easy to check that
\begin{align*}
k_1\psi '( k_1\omega _{b})
&= k_1H(v_2) \\
&\geq k_1c_{9}(k_1\omega _{b})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}
\frac{\theta +1-p^{-}}{p^{-}-1}} \\
&= c_{9}(k_1)^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}\sigma ^{\frac{1}{2-
\frac{\theta }{p^{-}-1}}\frac{\theta +1-p^{-}}{p^{-}-1}} \\
&= c_{9}(k_1)^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}(\frac{c_{11}}{
c_{13}\ln k_1})^{\frac{1}{2-\frac{\theta }{p^{-}-1}}\frac{\theta +1-p^{-}}{
p^{-}-1}}.
\end{align*}
Then
\[
| \nabla \omega _2| \leq C(k_2)^{\frac{p^{-}-1-\frac{
\varepsilon }{2}}{p^{-}-1}}\leq C(\frac{c_{11}k_1}{c_{13}\ln k_1})^{
\frac{1}{2-\frac{\theta }{p^{-}-1}}\frac{p^{-}-1-\frac{\varepsilon }{2}}{
p^{-}-1}}<| \nabla \psi ( k_1\omega _{b}) |
\]
 as $k_1\to +\infty$.
Thus we know that
\[
(k_1\psi '( k_1\omega _{b}( x) )
| \nabla \omega _{b}( x) |
)^{p(x)-1}-| \nabla \omega _2( x) |
^{p(x)-1}>0, \quad\text{when }\omega _{b}( x) =\sigma .
\]
Therefore, when $\sigma =c_{11}/(c_{13}\ln k_1)$ and $k_1$ is
large enough, similar argument as to the step 3 of the proof of Lemma \ref{lem3.4}
implies $v_2$ is a supersolution of \eqref{1.1}.

It is easy to see that $\mu \phi $ defined in the proof of
Theorem \ref{thm1.1} is a subsolution of \eqref{1.1} and $\mu \phi \leq v_2$
 when $\mu $ is small enough. By Lemma \ref{lem2.7}, we can get the
existence of a solution to \eqref{1.1}.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
 At first, similar to the proof of Theorem \ref{thm1.2}, we will prove that
$v_2$ defined by \eqref{aa} is also a
supersolution of \eqref{1.1} for a large enough constant $k_1$.

Similar to the proof of Theorem \ref{thm1.2}, we consider the solution $\omega _2$
of the problem
\begin{equation}
\begin{gathered}
-\Delta _{p(x)}\omega _2=b( x) k_2^{p^{-}-1-\frac{
\varepsilon }{2}}, \quad \text{in }\Omega \backslash \overline{\Omega _{\sigma }}
, \\
\omega _2( x) >0, \quad \text{in }\Omega \backslash \overline{
\Omega _{\sigma }}, \\
\omega _2( x) =0, \quad \text{on }\partial (\Omega \backslash
\overline{\Omega _{\sigma }}),
\end{gathered}
\label{3.11}
\end{equation}
where $k_2=\psi (k_1\sigma )$. We have
\[
\omega _2\leq C_1(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}.
\]
Next we only need to prove that
\[
| \nabla \omega _2| \leq C_2(k_2)^{\frac{p^{-}-1-
\frac{\varepsilon }{2}}{p^{-}-1}}.
\]
Now we consider $(\gamma k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{
p^{-}-1}}\omega _{b}$, where $\gamma \geq 1$ is a constant. Here we note
that $\nabla \omega _{b}\cdot \nu =| \nabla \omega _{b}| $
on $\partial (\Omega \backslash \overline{\Omega _{\sigma }})$ and $\nabla
p\cdot \nu <0$ on $\partial (\Omega \backslash \overline{\Omega _{\sigma }})$,
where $\nu $ is the inward unit normal vector on
$\partial (\Omega \backslash \overline{\Omega _{\sigma }})$.
There exists a small enough
positive constant $\delta >0$ such that $\nabla \omega _{b}\nabla p<0$ in
$(\Omega \backslash \overline{\Omega _{\sigma }})_{\delta }^{\#}:=\{x\in
\Omega \backslash \overline{\Omega _{\sigma }}: d(x,\partial (\Omega
\backslash \overline{\Omega _{\sigma }}))<\delta \}$. By computations it
follows that
\begin{align*}
&-\Delta _{p(x)}(\gamma k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{
p^{-}-1}}\omega _{b} \\
&= (\gamma k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}
}{p^{-}-1}(p(x)-1)}(-\Delta _{p(x)}\omega _{b}-\frac{p^{-}-1-\frac{
\varepsilon }{2}}{p^{-}-1}| \nabla \omega _{b}|
^{p(x)-2}\nabla \omega _{b}\nabla p\ln \gamma k_2) \\
&\geq (\gamma k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}
(p(x)-1)}(-\Delta _{p(x)}\omega _{b}) \\
&\geq b(\gamma k_2)^{p^{-}-1-\frac{\varepsilon }{2}}\text{ in }(\Omega
\backslash \overline{\Omega _{\sigma }})_{\delta }^{\#}.
\end{align*}
Since $\omega _{b}$ is positive and continuous, there exists a large enough
positive $\gamma $ such that
$\gamma ^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{
p^{-}-1}}\omega _{b}>2C_1$ for
$d(x,\partial (\Omega \backslash \overline{
\Omega _{\sigma }}))=\delta $. Therefore
$(\gamma k_2)^{\frac{p^{-}-1- \frac{\varepsilon }{2}}{p^{-}-1}}\omega _{b}$
is a supersolution of \eqref{3.11} in
$(\Omega \backslash \overline{\Omega _{\sigma }})_{\delta }^{\#}$.
By the comparison principle, we have
$(\gamma k_2)^{\frac{p^{-}-1-\frac{
\varepsilon }{2}}{p^{-}-1}}\omega _{b}\geq \omega _2$ in
$(\Omega \backslash \overline{\Omega _{\sigma }})_{\delta }^{\#}$, and then
\[
| \nabla \omega _2| \leq | (\gamma k_2)^{
\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}\nabla \omega _{b}|
\leq C_2(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}
\quad \text{on }\partial (\Omega \backslash \overline{\Omega _{\sigma }}).
\]
Note that
\[
\max_{x\in \overline{\Omega \backslash \Omega _{\sigma }}}
\omega _2(x)\leq C_3(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{
p^{-}-1}}
\]
 and $k_2=\psi (k_1\sigma )$. Since $v_2=\omega _2+\psi
(k_1\sigma )$ in $\Omega \backslash \overline{\Omega _{\sigma }}$, we have
\begin{align*}
c_{7}(k_1\sigma )^{\frac{1}{2-\frac{\theta }{p^{-}-1}}}
&\leq \psi (k_1\sigma ) \leq v_2\\
&\leq \max_{x\in \overline{\Omega \backslash \Omega _{\sigma }}}
 \omega _2(x)+\psi (k_1\sigma )\\
&\leq C_3(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}+\psi
(k_1\sigma )
\leq C_{4}\psi (k_1\sigma ).
\end{align*}
As in the proof of Theorem \ref{thm1.2}, we can see that the $v_2$ defined in
the proof of Theorem \ref{thm1.2} is a supersolution of \eqref{1.1}.

It is easy to see that $\mu \phi $ defined in the proof of Theorem \ref{thm1.1}
is a subsolution of \eqref{1.1}, and $\mu \phi \leq v_2$ when $\mu $ is small
enough. By  Lemma \ref{lem2.7}, we obtain the existence of solution of
\eqref{1.1}. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
The proof is similar to that of Theorem \ref{thm1.2}.
We will prove that $v_2$ defined in \eqref{aa} is a supersolution of
\eqref{1.1} for a large enough constant $k_1$.

Since \eqref{1.1} is radial, we may assume the both solutions
$\omega _{b}( \cdot ) $ and $\omega _2( \cdot ) $ are
radial. We only need to prove that $v_2$ defined in \eqref{aa} is also a
supersolution of \eqref{1.1} for a large enough constant $k_1$.
Since $\omega _2$ is radial, it is easy to see that
\[
\omega _2\leq C_1(k_2)^{\frac{p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}
, | \nabla \omega _2| \leq C_2(k_2)^{\frac{
p^{-}-1-\frac{\varepsilon }{2}}{p^{-}-1}}.
\]
Similar to the proof of Theorem \ref{thm1.2}, we can see that the
 $v_2$ defined in \eqref{aa} is a supersolution of \eqref{1.1}.
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank Professor Julio G. Dix for his
suggestions, and the anonymous referees for their valuable comments.

This research was partially supported by the National Natural Science
Foundation of China (11326161 and 10971087) and the key projects
of Science and Technology Research of the Henan Education Department (14A110011).

\begin{thebibliography}{99}

\bibitem{a1} E. Acerbi, G. Mingione;
 Regularity results for a class of
functionals with nonstandard growth, \textit{Arch. Ration. Mech. Anal.} 156
(2001), 121--140.

\bibitem{b1} N. Benouhiba;
 On the eigenvalues of weighted $p(x)$-Laplacian on $\mathbb{R}^{N}$,
\textit{Nonlinear Anal.,} 74(2011)235-243.

\bibitem{c1} F. Cammaroto, L. Vilasi;
 On a perturbed $p(x)$-Laplacian
problem in bounded and unbounded domains, \textit{J. Math. Anal. Appl.}, 402
(2013), 71-83.

\bibitem{c2} K.C. Chang;
 \textit{Critical point theory and applications},
Shanghai Scientific and Technology press, Shanghai, 1986.

\bibitem{c3} Y. Chen, S. Levine, M. Rao;
 Variable exponent, linear growth
functionals in image restoration, \textit{SIAM J. Appl. Math.}, \textbf{66 }
(2006), No.4, 1383--1406.

\bibitem{d1} L. Diening, P. Harjulehto, P. H\"{a}st\"{o}, M. R\r{u}\v{z}i\v{c}ka;
 \textit{Lebesgue and Sobolev Spaces with Variable Exponents},
Lecture Notes in Mathematics 2017,Springer-Verlag, Berlin, 2011.

\bibitem{f1} X. L. Fan;
Global $C^{1,\alpha }$ regularity for variable
exponent elliptic equations in divergence form, \textit{J. Differ. Equations}
235 (2007) 397-417.

\bibitem{f2} X. L. Fan;
Eigenvalue of the $p(x)$-Laplacian Numann problems,
\textit{Nonlinear Anal.}, 67 (2007), 2982-2992.

\bibitem{f3} X. L. Fan;
On the sub-super solution method for $p(x)$-Laplacian
equations, \textit{J. Math. Anal. Appl.} 330 (2007), 665-682.

\bibitem{f4} X. L. Fan, D. Zhao;
 A class of De Giorgi type and H\"{o}lder
continuity, \textit{Nonlinear Anal., }36 (1999), 295-318.

\bibitem{f5} X. L. Fan, D. Zhao;
The quasi-minimizer of integral functionals
with $m(x)$ growth conditions, \textit{Nonlinear Anal.,} 39 (2000) 807-816.

\bibitem{f6} X. L. Fan, Q. H. Zhang;
Existence of solutions for $p(x)$-Laplacian Dirichlet problem,
 \textit{Nonlinear Anal.,} 52(2003)1843-1852.

\bibitem{f7} X. L. Fan, Q. H. Zhang, D. Zhao;
 Eigenvalues of $p(x)$-Laplacian
Dirichlet problem, \textit{J. Math. Anal. Appl.} 302(2005)306-317.

\bibitem{f8} X. L. Fan, D. Zhao;
On spaces $L^{p(x)}( \Omega ) $ and $W^{m,p(x)}( \Omega ) $,
\textit{J. Math. Anal. Appl.} 263 (2001) 424-446.

\bibitem{g1} J. V. A. Gon\c{c}alves, M. C.Rezende, C. A. Santos;
 Positive solutions for a mixed and singular quasilinear problem, \textit{Nonlinear
Anal.,} 74(2011)132-140.

\bibitem{g2} Z. C. Guo, Q. Liu, J. B. Sun, B. Y. Wu;
Reaction--diffusion systems with $p(x)$-growth for image denoising,
\textit{Nonlinear Anal-Real.,} 12 (2011), 2904--2918.

\bibitem{g3} Z. C. Guo, J. B. Sun, D. Z. Zhang, B. Y. Wu;
Adaptive perona-malik model based on the variable exponent for image denoising,
\textit{IEEE T. Image Process}, 21 (2012), 958-967.

\bibitem{g4} C. M. Guo, C. B. Zhai, R. P. Song;
 An existence and uniqueness result for the singular Lane--Emden--Fowler equation, \textit{Nonlinear
Anal.,} 72 (2010), 1275--1279.

\bibitem{h1} T. C. Halsey;
 \textit{Electrorheological fluids}, Science 258
(5083) (1992), 761-766.

\bibitem{h2} A. El Hamidi;
 Existence results to elliptic systems with
nonstandard growth conditions, \textit{J. Math. Anal. Appl.,} 300 (2004),
30-42.

\bibitem{h3} P. Harjulehto, P. H\"{a}st\"{o}, M. Koskenoja, S. Varonen;
 The Dirichlet energy integral and variable exponent Sobolev spaces with zero
boundary values, \textit{Potential Anal.} 25 (2006), 205--222.

\bibitem{h4} P. Harjulehto, P. H\"{a}t\"{o}, V. Latvala;
 Harnack's inequality for $p(\cdot )$-harmonic functions with unbounded exponent $p$,
\textit{J. Math. Anal. Appl.}, 352 (2009), 345-359.

\bibitem{h5} P. Harjulehto, P. H\"{a}st\"{o}, V. Latvala, O. Toivanen;
Critical variable exponent functionals in image restoration, \textit{Appl.
Math. Lett.}, 26 (2013), 56--60.

\bibitem{h6} P. Harjulehto, P. H\"{a}st\"{o}, \'{U}. V. L\^{e}, M. Nuortio;
Overview of differential equations with non-standard growth,
\textit{Nonlinear Anal.,} 72 (2010), 4551-4574.

\bibitem{h7} P. Harjulehto, P. H\"{a}st\"{o}, V. Latvala;
 Minimizers of the variable exponent, non-uniformly convex Dirichlet energy,
\textit{J. Math. Pure. Appl.}, 89 (2008) (2) , pp. 174-197.

\bibitem{k1} O. Kov\'{a}\v{c}ik, J. R\'{a}kosn\'{\i}k;
 On spaces $L^{p(x)}( \Omega ) $ and $W^{k,p(x)}( \Omega ) $,
\textit{Czech. Math. J.}, 41\textbf{\ }(1991), 592-618.

\bibitem{l1} O. A. Ladyzenskaja, N. N. Ural'tzeva;
 \textit{Linear and Quasilinear Elliptic Equations}, Academic Press, New York, 1968.

\bibitem{l2} F. Li, Z. B. Li, L. Pi;
 Variable exponent functionals in image restoration,
\textit{Appl. Math. Comput.,} 216 (2010), 870--882.

\bibitem{l3} J. J. Liu;
 Positive solutions of the $p(x)$-Laplace equation
with singular nonlinearity, \textit{Nonlinear Anal.,} 72 (2010), 4428-4437.

\bibitem{l4} T. Lukkari;
 Singular solutions of elliptic equations with
nonstandard growth, \textit{Math. Nachr., }282 (2009), 1770--1787.

\bibitem{m1} M. Mih\u{a}ilescu, V. R\u{a}dulescu;
Continuous spectrum for a class of nonhomogeneous differential operators,
\textit{Manuscripta Math}. 125 (2008), 157-167.

\bibitem{m2} M. Mih\u{a}ilescu, V. R\u{a}dulescu;
 On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces
with variable exponent,
\textit{Proc. Amer. Math. Soc.} 135 (2007), 2929-2937.

\bibitem{m3} M. Mih\u{a}ilescu, V. R\u{a}dulescu, D. Stancu-Dumitru;
 A Caffarelli-Kohn-Nirenberg-type inequality with variable exponent and
applications to PDE's, \textit{Complex Var. Elliptic Equa.}, 56 (2011),
659-669.

\bibitem{m4} M. Mih\u{a}ilescu, G. Moro\c{s}anu, D. Stancu-Dumitru;
Equations involving a variable exponent Grushin-type operator, \textit{
Nonlinearity}, 24 (2011), 2663-2680.

\bibitem{m5} M. Mih\u{a}lescu, V. R\u{a}dulescu;
Concentration phenomena in nonlinear eigenvalue problems with variable
exponents and sign-changing potential, \textit{J. Anal. Math.}, 111 (2010), 267-287.

\bibitem{m6} M. Mih\u{a}lescu, V. R\u{a}dulescu, D. Repov\v{s};
 On a non-homogeneous eigenvalue problem involving a potential: An Orlicz-Sobolev
space setting, \textit{J. Math. Pures Appl.}, 93 (2010) (2), 132-148.

\bibitem{m7} A. Mohammed;
Positive solutions of the $p$-Laplace equation with singular nonlin-earity,
\textit{J. Math. Anal. Appl.} 352 (2009), 234-245.

\bibitem{p1} K. Perera, E. A. B. Silva;
On singular $p$-Laplacian problems,
\textit{Differential Integral Equations,} 20 (2007), 105--120.

\bibitem{p2} P. Pucci, Q.H. Zhang;
Existence of entire solutions for a class of variable exponent elliptic equations,
\textit{J. Differ. Equations}, 257 (2014), 1529--1566.

\bibitem{q1} M. Qing, Z. Yang;
Quasilinear elliptic equation involving singular non-linearities,
\textit{Int. J. Comput. Math.} (2009), 1--11.

\bibitem{r1} K. R. Rajagopal, M. R\'{u}\v{z}i\v{c}ka;
 Mathematical modeling of electrorheological fluids,
\textit{Contin. Mech. Thermodyn.} 13 (2001), 59-78.

\bibitem{r2} M. R\r{u}\v{z}i\v{c}ka;
\textit{Electrorheological fluids: modeling and mathematical theory},
Lecture Notes in Math 1748, Springer-Verlag, Berlin, 2000.

\bibitem{z1} Q. H. Zhang;
A strong maximum principle for differential
equations with nonstandard $p(x)$-growth conditions, \textit{J. Math. Anal.
Appl, }\textbf{312}(2005), No.1, 24-32.

\bibitem{z2} Q. H. Zhang;
Existence and asymptotic behavior of positive
solutions to $p(x)$-Laplacian equations with singular nonlinearities,
\textit{J. Ineq. Appl.}, 2007 (2007), Article ID 19349, 9 pages.

\bibitem{z3} Q. H. Zhang;
Existence, nonexistence and asymptotic behavior of
boundary blow-up solutions to $p(x)$-Laplacian problems with singular
coefficient, \textit{Nonlinear Anal.}, 74 (2011), 2045--2061.

\bibitem{z4} V. V. Zhikov;
 Averaging of functionals of the calculus of variations and elasticity theory,
\textit{Math. USSR. Izv}. \textbf{29} (1987), 33-36.

\end{thebibliography}

\end{document}