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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 242, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/242\hfil Stability for noncoercive elliptic equations]
{Stability for noncoercive elliptic equations}

\author[S. Huang, Q. Tian, J. Wang, J. Mu \hfil EJDE-2016/242\hfilneg]
{Shuibo Huang, Qiaoyu Tian, Jie Wang, Jia Mu}

\address{Shuibo Huang (corresponding author)\newline
School of Mathematics and Computer,
Northwest University for Nationalities, \newline
Lanzhou, Gansu 730000, China}
\email{huangshuibo2008@163.com}

\address{Qiaoyu Tian \newline
School of Mathematics and Computer,
Northwest University for Nationalities, \newline
Lanzhou, Gansu 730000, China}
\email{tianqiaoyu2004@163.com}

\address{Jie Wang \newline
Department of Applied Mathematics,
Lanzhou University of Technology, \newline
Lanzhou, Gansu 730050, China}
\email{jiema138@163.com}

\address{Jia Mu \newline
School of Mathematics and Computer,
Northwest University for Nationalities, \newline
Lanzhou, Gansu 730000, China}
\email{mujia88@163.com}

\thanks{Submitted April 29, 2016. Published September 5, 2016.}
\subjclass[2010]{37K45，35J60}
\keywords{Removable singularity; capacity; noncoercive elliptic equation}

\begin{abstract}
 In this article, we consider the stability for elliptic problems
 that have degenerate coercivity in their principal part,
 \begin{gather*}
 -\operatorname{div}\Big(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}}\Big)
 +|u|^{q-1}u=f,\quad x\in\Omega, \\
 u(x)=0,\quad x\in \partial\Omega,
 \end{gather*}
 where $\theta>0$, $\Omega\subseteq \mathbb{R}^N$ is a bounded domain.
 Let $K$ be a compact subset in $\Omega$ with zero $r$-capacity ($p<r\leq N$).
 We prove that if $f_n$ is a sequence of functions which converges strongly
 to $f$ in $ L^1_{\rm loc}(\Omega\backslash K)$ and
 $q>r(p-1)[1+\theta(p-1)]/(r-p)$, and $u_n$ is the sequence of solutions
 of the corresponding problems with datum $f_n$.
 Then $u_n$ converges to the solution $u$.
\end{abstract}

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


\section{Introduction and statement of main results}

Let $\Omega\subseteq \mathbb{R}^N$ be  a bounded smooth domain.
 We are interested in the stability  of  quasilinear elliptic problems 
with principal part having degenerate coercivity,
\begin{equation} \label{1.1}
\begin{gathered}
 -\operatorname{div}a(x,u, \nabla u)+|u|^{q-1}u=f, \quad x\in\Omega, \\
 u(x)=0, \quad x\in \partial\Omega,
 \end{gathered}
\end{equation}
where $\theta>0$,  $1<p<N$ and $f\in L^1(\Omega)$.
The function $a: \Omega\times\mathbb{R}\times\mathbb{R}^N \to\mathbb{R}^N $
is a Carath\'{e}odory function (that is,  $a(\cdot,s,\xi)$  measurable on
$\Omega$ for every $(s,\xi)$ in $\mathbb{R}\times\mathbb{R}^N$,  and
 $a(x,\cdot,\cdot)$ continuous on $\mathbb{R}\times\mathbb{R}^N$
for almost every $x$ in $\Omega$)  satisfying the following assumptions:
\begin{gather}\label{1.2}
a(x,s,\xi)\xi\geq \alpha_1 h^{p-1}(|s|) |\xi|^p, \quad \alpha_1>0, \\
\label{1.3}
|a(x,s,\xi)|\leq \alpha_2|\xi|^{p-1}, \quad \alpha_2>0, \\
\label{1.4}
\langle a(x,s,\xi)-a(x,s,\eta),\xi-\eta\rangle>0,\quad \xi\neq\eta,
\end{gather}
for almost every $x\in \Omega$ and for every
$s\in \mathbb{R},\xi\in \mathbb{R}^N, \eta\in \mathbb{R}^N$, $h(t)$ is  defined as
\begin{equation}\label{1.5}
h(t)=\frac{1}{(1+|t|)^{\theta}}.
\end{equation}

The interest in  removable singularities for elliptic equations goes 
back to the pioneering work of Brezis\cite{BR1983}. 
 Actually, Brezis shown that if $\{u_n\}$ are the sequence of solutions 
of the  nonlinear elliptic problems
\begin{gather*}
 -\Delta u_n+|u_n|^{q-1}u_n=f_n, \quad x\in\Omega, \\
 u_n(x)=0, \quad x\in \partial\Omega,
 \end{gather*}
where $0\in\Omega$, $q\geq\frac{N}{N-2}$ and $\{f_n\}$ be a sequence 
of $L^1(\Omega)$ functions satisfying
\[
\lim_{n\to\infty}\int_{\Omega\backslash B_\rho(0)}|f_n-f|=0.
\]
Then $u_n$ converges to the unique solution $u$ of the equation
\[
 -\Delta u+|u|^{q-1}u=f.
\]
In particular, surprisingly enough, let $\{f_n\}$ be a sequence in 
$L^1(\Omega)$  such that $f_n\subset B(0,\frac{1}{n})$ and $f_n\to \delta$, 
then $u_n\to 0$. While we would expect $u_n$ converges to the  solution 
$u$ of 
\[
 -\Delta u+|u|^{q-1}u=\delta.
\]
but lt is well known that such a $u$ does not exists if 
$q\geq\frac{N}{N-2}$, see \cite{BB1975}.

The results in \cite{BR1983} were extended by Orsina and  Prignet \cite{OP2000} 
for more general uniformly elliptic, coercive and pseudomonotone operator 
and where $f$ is a measure which is concentrated on a set $E$ of 
zero $r$-capacity. Continuing the studies in \cite{OP2000,BR1983}, 
Orsina and  Prignet \cite{OP2002} obtained stability results of  elliptic equations
\begin{gather*}
 -\operatorname{div}a(x,u, \nabla u)+|u|^{q-1}u=f, \quad x\in\Omega, \\
 u(x)=0, \quad x\in \partial\Omega,
 \end{gather*}
where $a$ is a Carath\'{e}odory function satisfying \eqref{1.3}, \eqref{1.4} and
\begin{align*}
a(x,s,\xi)\xi\geq \alpha_1 |\xi|^p, \quad \alpha_1>0.
\end{align*}

With motivation from the results of the above cited papers, the main purpose 
of this paper is to investigate the stability results of problem \eqref{1.1}. 
The main results show that how the nonlinear term $|u|^{q-1}u$ and the 
singular term $h(u)^{p-1}$ affect the existence of  solutions to \eqref{1.1}.

The main results of this article is the following theorem.

\begin{theorem}\label{thm1.1}
Let $p<r\leq N$,  $f=f^+-f^-$ be a function in $L^1(\Omega)$, $u_n$ 
be a solution to problems
\begin{equation} \label{1.7}
\begin{gathered}
 -\operatorname{div}a(x,u_n, \nabla u_n)+|u_n|^{q-1}u_n=f_n, \quad x\in\Omega, \\
 u_n(x)=0, \quad x\in \partial\Omega,
 \end{gathered}
\end{equation}
where $f_n=f^\oplus_n-f^\ominus _n$, $f^\oplus_n$ and $f^\ominus _n$ be
two sequences of nonnegative $L^\infty(\Omega)$ functions such that
\begin{equation}\label{1.8}
\lim_{n\to\infty}\int_{\Omega\backslash I(K^+)}|f^\oplus_n-f^+|=0,\quad
\lim_{n\to\infty}\int_{\Omega\backslash I(K^-)}|f^\ominus_n-f^-|=0,
\end{equation}
for every neighbourhood $I(K^+)$ of $K^+$ and $I(K^-)$ of $K^-$,
where $K^+$ and $K^-$ be two disjoint compact subsets of $\Omega$ of
zero $r$-capacity. Then, up to subsequences still denoted by $u_n$,
$u_n$ converges to a solution in the sense of distributions of the problems \eqref{1.1} with  datum $f$ provided
\begin{equation}\label{1.9}
q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}.
\end{equation}
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
We emphasize that we do not assume that
$f^\oplus_n$ and $f^\ominus _n$
are the positive and negative part of $f_n$, but only
that they are nonnegative. This is the reason why we use the unconventional
notation $f^\oplus_n$ and $f^\ominus _n$.
\end{remark}

\begin{remark} \label{rmk1.3} \rm
The preceding theorem can be seen as a non-existence result for
problem \eqref{1.1}:
A particular case of Theorem \ref{thm1.1} is when the sequence $f^\oplus_n$ 
is convergent  to $f$ in the tight topology of measures $f$, where $f$ is a bounded
Radon measure concentrated on a set $K$ of zero harmonic capacity 
 and $f^\ominus _n=0$,  In this case, Theorem \ref{thm1.1} states that 
the sequence $u_n$ tends to zero almost everywhere in $\Omega$. 
 This is exactly the result \cite[Theorem 4.1]{BCO2012}.
\end{remark}

\begin{remark} \label{rmk1.4} \rm
The result of preceding theorem can also be seen as a result of
removable singularities for problem \eqref{1.1}. Indeed it states that sets of zero
$r$-capacity are not seen by the equation if $q$ satisfies \eqref{1.9}. 
Some other results about removable singularities of elliptic equations, 
see \cite{AA2003,AF2015, BI2003,BN1997,MI2015,KU2000,PV2008,PV2006}.
\end{remark}

\begin{remark} \label{rmk1.5} \rm
With minor technical modifications in the proof of \cite[Theorem 1.6]{CR2007}, 
we can obtain the existences of  distributional solutions 
$u_n\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$ to problem \eqref{1.7}.
Indeed, lower order term $|u|^{q-1}u$ has a regularizing effect. 
Roughly speaking, large values of $q$ can compensate the
``bad coercivity'' of the principal part and the poor summability of 
the right hand side.
\end{remark}

\begin{remark} \label{rmk1.6} \rm
The principal part  left-hand of  \eqref{1.1} is defined on  
$W^{1,p}_0(\Omega)$,  but it may not be coercive on the same space as $u$ 
becomes large, due to this lack of coercivity, standard existence theorems 
for solutions of nonlinear elliptic equations cannot be applied.
Furthermore,  $\frac{\nabla u}{(1+|u|)^{\theta(p-1)}}$ tends to zero as 
$u$ tends to infinity, which produces a saturation effect. 
Some  other results of elliptic equations with principal part having 
degenerate coerciveness, see \cite{ABL2013,ABO2001,B2015,BO2015,MPP2011}.
\end{remark}

\begin{remark} \label{rmk1.7} \rm
In this article, we only consider $\theta>0$. The case $\theta\equiv0$  
has been considered by  Orsina and Prignet \cite{OP2002},
\end{remark}


The plan of this article is as follows. 
In Section 2, we briefly recall some notations and known results about measures. 
Section 3  contains the proof of Theorem \ref{thm1.1}.

\section{Preliminaries}


In this section, we first recall some notation and definitions. 
In the following, $C$ will be a constant that may change from an inequality
to another, to indicate a dependence of $C$ on the real parameters $\delta$, 
we shall write $C=C(\delta)$.

For each real number $s$, we define $s^+=\max(s,0)$ and 
$s^-=-\max(-s,0)$. Obviously, $s=s^+-s^-$ and $|s|=s^++s^-$.

For $k>0$, denote by $T_k: \mathbb{R}\to\mathbb{R}$ the usual 
truncation at level $k$; that is,
\begin{align*}
T_k(s)=\max\{-k,\min\{k,s\}\}.
\end{align*}
The ``remainder'' of the truncation $T_k(s)$ is defined as $G_k(s)=s-T_k(s)$.

Note that  we will deal with functions $u$ that may not belong to Sobolev spaces, 
we need to give a suitable definition of gradient.
Consider a measurable function $u:\Omega\to \mathbb{R}$  which is finite 
almost everywhere and satisfies
$T_k(u)\in  W^{1,p}_0(\Omega)$ for every $k>0$. According to
\cite[Lemma 2.1]{BBGGPV1995},   there exists  an unique
measurable function $v:\Omega\to \overline{\mathbb{R}}^N$ such that, 
for each $k>0$,
\begin{align*}
\nabla T_k(u)=v\chi_{|u|\leq k}\quad \text{almost everywhere in }\Omega,
\end{align*}
where $\chi_{|u|\leq k}$ is the characteristic function of $\{|u|\leq k\}$.
We define the gradient $\nabla u$ of $u$ as this function $v$, and denote 
$\nabla u=v$.

\begin{remark} \label{rmk2.1} \rm
It is worth pointing out that the gradient defined in this way is not, 
in general, the gradient used in the definition of Sobolev spaces, However, 
$v$ is the distributional gradient of $u$ provided  $v$ belongs to 
$(L^1_{\rm loc} (\Omega))^N$, which also implies that $u$ belongs to 
$W^{1,1}_{\rm loc} (\Omega)$.
\end{remark}

\begin{remark} \label{rmk2.2} \rm
As point out in \cite{BBGGPV1995}, the set of functions $u$ such that 
$T_k(u)$ belongs to $W^{1,p}_0(\Omega)$ for every $k>0$ is not a linear space. 
That is, if $u$ and $v$ are such that both $T_k(u)$ and $T_k(v)$ belong to 
$W^{1,p}_0(\Omega)$ for every $k > 0$, while  $\nabla (u+v)$ may not be defined.
\end{remark}

Denote by $|\Omega|$ the $N$-dimensional Lebesgue measure of a measurable 
set $\Omega$. Let $f(x), g(x)$ are functions defined in $\mathbb{R}^N$ 
and $a,b$ are constants, we set
\[
\{ f(x)>a \}:=\{ x\in \mathbb{R}^N: f(x)>a\},\quad
\{g(x)\leq b\}:=\{x\in \mathbb{R}^N: g(x)\leq b\}.
\]

The $r$-capacity $\operatorname{cap}_{1,p}(K, \Omega)$ of a compact
set $K \subset \Omega$ with respect to $\Omega$ is
defined by
\begin{align*}
\operatorname{cap}_{1,p} (K, \Omega)=\inf\Big\{\int_\Omega|\nabla \phi|^pdx
: \phi\in C_0^\infty(\Omega), \phi\geq \chi_E \Big\}.
\end{align*}

The following technical propositions will be  be useful  throughout the 
paper\cite{MMOP1999}.

\begin{proposition}\label{pro2.3}
Let $K^+$ and $K^-$ be two disjoint compact subsets of $\Omega$ of zero
$r$-capacity and $p<r\leq N$. Then, for every $\delta> 0$ there exist $A^+_\delta$
and $A^-_\delta$, two
disjoint open subsets of $\Omega$, and  $\psi^+_\delta$
and $\psi^-_\delta$
in $C^\infty_c(\Omega)$ such that
\begin{gather}
0\leq  \psi^+_\delta(x) \leq1,\quad 0\leq  \psi^-_\delta(x) \leq1, 
\quad x\in\Omega,\label{2.2}\\
  \psi^+_\delta(x) \equiv 1, \quad x\in K^+,  \quad \psi^-_\delta(x) \equiv 1, 
\quad x\in K^-, \label{2.3}\\
  \operatorname{supp}(\psi^+_\delta(x))=A^+_\delta,  \quad 
\operatorname{supp}(\psi^-_\delta(x))=A^-_\delta, \label{2.4} \\
  \int_\Omega |\nabla \psi^+_\delta(x) |^rdx\leq\delta,\quad
 \int_\Omega |\nabla \psi^-_\delta(x) |^rdx\leq\delta,\label{2.5}\\
  \operatorname{meas}(A^+_\delta)\leq\delta,\quad 
\operatorname{meas}(A^-_\delta)\leq\delta.\label{2.6}
\end{gather}
\end{proposition}

\section{Proof of Theorem \ref{thm1.1}}

The following  arguments are similar to these in \cite{OP2002}, 
and the proof will be done with the aid of the following two lemmas.

\begin{lemma}\label{lem3.1}
There exists $0<C<\infty$ such that for any $k>0$,
\begin{equation}\label{3.3}
\int_\Omega |\nabla T_k (u_n)|^pdx< C k^{q+1+\theta(p-1)}.
\end{equation}
\end{lemma}

\begin{proof}
Choose $T_k(u_n)(1-\psi_\delta)^s$ as a test function in \eqref{1.7}, 
here and elsewhere in the paper
\[
\psi_\delta=\psi_\delta^++\psi_\delta^-,\quad
s=\frac{\beta}{\beta-p+1}.
\]
where $\beta$ appears in \eqref{3.10}.
Thus
\begin{equation}\label{3.4}
\begin{split}
 &\int_\Omega a(x, u_n \nabla u_n)\cdot \nabla T_k(u_n)(1-\psi_\delta)^s dx
 +\int_\Omega|u_n|^{q-1}u_nT_k(u_n)(1-\psi_\delta)^sdx\\
 &=s\int_\Omega a(x, u_n \nabla u_n)\nabla \psi_\delta T_k(u_n)
 (1-\psi_\delta)^{s-1}dx \\
 &\quad +\int_\Omega f^\oplus_nT_k(u_n)(1-\psi_\delta)^sdx
 -\int_\Omega f^\ominus _nH(T_k(u_n)(1-\psi_\delta)^sdx.
 \end{split}
\end{equation}
By \eqref{1.2}, we have
\begin{equation}\label{3.5}
\int_\Omega a(x, u_n, \nabla u_n)\cdot \nabla T_k(u_n)d\mu_\delta
 \geq \alpha_1\int_\Omega\frac{|\nabla T_k(u_n)|^p}{(1+|T_k(u_n)|)^{\theta (p-1)}}
 d\mu_\delta,
\end{equation}
here and the rest of this paper we use the note $d\mu_\delta=(1-\psi_\delta)^s dx$.

Recall that $u_n T_k(u_n)\geq 0$, which  leads to
\begin{equation}\label{3.6}
\begin{split}
\int_\Omega|u_n|^{q-1}u_nT_k(u_n)(1-\psi_\delta)^sdx
&\geq \int_{ \{|u_n|\geq k\}} |u_n|^{q-1}u_nT_k(u_n) d\mu_\delta\\
&\geq  k^{q+1}\mu_\delta(\{|u_n|\geq k\}).
 \end{split}
\end{equation}

Using \eqref{1.3} and Young's inequality, we find
\begin{equation}\label{3.7}
\begin{split}
&\int_\Omega| a(x, u_n, \nabla u_n)\nabla \psi_\delta T_k(u_n)
 (1-\psi_\delta)^{s-1}|dx\\
&\leq \alpha_2 k \int_\Omega|\nabla u_n|^{p-1}(|\nabla \psi_\delta^+|
 +|\nabla \psi_\delta^-|)(1-\psi_\delta)^{s-1}dx\\
&\leq C k \int_\Omega|\nabla u_n|^{(p-1)r'}(1-\psi_\delta)^{(s-1)r'}dx
 +C k \int_\Omega(|\nabla \psi_\delta^+|^r+|\nabla \psi_\delta^-|^r)dx.
 \end{split}
\end{equation}

Combining \eqref{2.5} and \eqref{3.4}-\eqref{3.7}, we obtain
\begin{equation}\label{3.8}
\begin{split}
 &\int_\Omega\frac{|\nabla T_k(u_n)|^p}{(1+|T_k(u_n)|)^{\theta (p-1)}}
 d\mu_\delta+ k^{q+1}\mu_\delta(\{|u_n|\geq k\})\\
&\leq Ck(\delta+I_1(n,\delta)+I_2(n,\delta)),
 \end{split}
\end{equation}
where
\begin{align*}
I_1(n,\delta)=\int_\Omega(f^\oplus_n+f^\ominus _n)d\mu_\delta,\quad
I_2(n,\delta)=\int_\Omega|\nabla u_n|^{(p-1)r'}(1-\psi_\delta)^{(s-1)r'}dx.
\end{align*}

For a fixed $\rho\geq 0$, thanks to \eqref{3.8}, we have
\begin{align*}
&\mu_\delta(\{|\nabla u_n|\geq \rho\})\\
&=\mu_\delta\big(\{|\nabla u_n|\geq \rho\}\cup \{| u_n|<k\}\big)
+\mu_\delta\big(\{|\nabla u_n|\geq \rho\}\cup \{| u_n|\geq k\}\big)\\
&\leq \frac{1}{\rho^p}\int_\Omega|\nabla T_k(u_n)|^{p}d\mu_\delta+\mu_\delta(\{| u_n|\geq k\})\\
&\leq \frac{(1+k)^{\theta (p-1)}}{\rho^p}\int_\Omega\frac{|\nabla T_k(u_n)|^p}{(1+|T_k(u_n)|)^{\theta (p-1)}} d\mu_\delta+\mu_\delta(\{| u_n|\geq k\})\\
&\leq C (\delta+I_1(n,\delta)+I_2(n,\delta))
\Big(\frac{k^{1+\theta (p-1)}}{\rho^p}+\frac{1}{k^{q}}\Big),
\end{align*}
which implies 
\begin{equation}\label{3.9}
\mu_\delta(\{|\nabla u_n|\geq \rho\})
\leq C \rho^{-\frac{pq}{q+1+\theta(p-1)}}(\delta+I_1(n,\delta)+I_2(n,\delta)).
\end{equation}
Let $\beta$ be such that
\begin{equation}\label{3.10}
(p-1)r'<\beta<\frac{pq}{q+1+\theta(p-1)}.
\end{equation}
It can be easily seen that such a $\beta$ exists by \eqref{1.9}.
In view of \eqref{3.9}, we have
\begin{equation}\label{3.11}
\int_\Omega|\nabla u_n|^{\beta}d\mu_\delta
\leq C(\delta+I_1(n,\delta)+I_2(n,\delta)).
\end{equation}
This fact and  H\"{o}lder's inequality  imply
\begin{align*}
I_2(n,\delta)
\leq C\Big(\int_\Omega|\nabla u_n|^{\beta}d\mu_\delta\Big)^{\frac{(p-1)r'}{\beta}}
\leq C (\delta+I_1(n,\delta)+I_2(n,\delta))^{\frac{(p-1)r'}{\beta}},
\end{align*}
which, combined with the fact that  $X^\gamma\leq C+X$ imply that 
$X$ is bounded provided $\gamma>1$; this yields
\begin{equation}\label{3.12}
I_2(n,\delta)
\leq C (\delta+I_1(n,\delta))\leq C(\delta),
\end{equation}
since $1-\psi_\delta$ is zero both on a neighbourhood of $K^+$ and of 
$K^-$, this fact and \eqref{1.8} show that $I_1(n,\delta)$ is bounded 
with respect to $\delta$.

Using   estimates \eqref{3.7}, \eqref{3.8} and \eqref{3.12}, we conclude that
\begin{gather}\label{3.13}
\int_\Omega|\nabla T_k(u_n)|^p d\mu_\delta \leq C( \delta)k^{1+\theta (p-1)},\\
\label{3.14}
\int_\Omega|u_n|^{q-1}u_nT_k(u_n)d\mu_\delta\leq C( \delta)k, \\
\label{3.15}
 \int_\Omega |\nabla u_n|^{p-1}(|\nabla \psi_\delta^+|+|\nabla \psi_\delta^-|)
(1-\psi_\delta)^{s-1}dx\leq C( \delta).
\end{gather}

Choose $T_k(u_n^+)(1-\psi_\delta^+)^s$  and $-T_k(u_n^-)(1-\psi_\delta^-)^s$  
as a test function in \eqref{1.7}  respectively. Similar arguments show that
\begin{equation}\label{3.16}
\begin{gathered}
\int_\Omega|\nabla T_k(u_n^+)|^p d\mu_\delta^+ \leq C( \delta)k^{1+\theta (p-1)},\\
\int_\Omega|\nabla T_k(u_n^-)|^p d\mu_\delta^- \leq C( \delta)k^{1+\theta (p-1)},
 \end{gathered}
\end{equation}
and
\begin{equation}\label{3.17}
\begin{gathered}
\int_\Omega|u_n^+|^{q-1}u_n^+T_k(u_n^+)d\mu_\delta^+\leq C( \delta)k,\\
\int_\Omega|u_n^-|^{q-1}u_n^-T_k(u_n^-)d\mu_\delta^-\leq C( \delta)k,
 \end{gathered}
\end{equation}
where $d\mu_\delta^+=(1-\psi_\delta^+)^s dx$ and 
$d\mu_\delta^-=(1-\psi_\delta^-)^s dx$.

Now we  choose $(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)$ as a test function 
in \eqref{1.7}. We must emphasize that
\begin{gather*}
(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)=k-T_k(u_n^+), \quad x\in K^+, \\
(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)=0.
\end{gather*}
Apart from the support of $\psi_\delta^+$,
a simple calculation yields
\begin{equation}\label{3.18}
\begin{split}
 &-\int_\Omega a(x, u_n, \nabla u_n)\cdot \nabla T_k(u_n^+)(1-(1-\psi_\delta^+)^s)dx\\
 &+s\int_\Omega a(x, u_n, \nabla u_n)\nabla \psi_\delta^+ (k-T_k(u_n^+))(1-\psi_\delta^+)^{s-1}dx\\
 &+\int_\Omega|u_n|^{q-1}u_n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx\\
 =&\int_\Omega f^\oplus_n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx \\
&\quad -\int_\Omega f^\ominus _n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx.
 \end{split}
\end{equation}
Obviously
\begin{equation}\label{3.19}
\begin{split}
 &\int_\Omega a(x, u_n, \nabla u_n)\cdot \nabla T_k(u_n^+)(1-(1-\psi_\delta^+)^s)dx\\
&\geq \frac{\alpha_1}{(1+k)^{\theta (p-1)}} \int_\Omega |\nabla T_k(u_n^+)|^p(1-(1-\psi_\delta^+)^s)dx,
 \end{split}
\end{equation}
and
\begin{equation}\label{3.20}
\begin{split}
&\int_\Omega a(x, u_n, \nabla u_n)\nabla \psi_\delta^+ 
 (k-T_k(u_n^+))(1-\psi_\delta^+)^{s-1}dx\\
&\leq k\int_\Omega |T_k(u_n^+)|^{p-1}|\nabla\psi_\delta^+
 |(1-\psi_\delta^+)^{s-1}dx\leq C(\delta) k,
 \end{split}
\end{equation}
here we have used \eqref{3.15} and the fact that $k-T_k(u_n^+)\leq k$.

It can be easily seen that
\begin{gather}\label{3.21}
\begin{split}
&\int_\Omega|u_n|^{q-1}u_n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx\\
&\leq  \int_{\{0\leq u_n\leq k\}}|u_n|^{q-1}u_n (k-T_k(u_n^+))
 (1-(1-\psi_\delta^+)^s)dx\\
&\leq  C(\delta) k^{q+1},
 \end{split} \\
\label{3.22}
0\leq \int_\Omega f^\oplus_n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx\leq C(\delta)k, \\
\label{3.23}
0\leq\int_\Omega f^\ominus _n(k-T_k(u_n^+))(1-(1-\psi_\delta^+)^s)dx\leq C(\delta)k.
\end{gather}
From \eqref{3.18}-\eqref{3.23}, we obtain
\begin{equation}\label{3.24}
 \int_\Omega|\nabla T_k(u_n^+)|^p (1-(1-\psi_\delta^+)^s)dx 
\leq C( \delta)k^{q+1+\theta (p-1)}.
\end{equation}
Similarly, choosing $(k+T_k(u_n^-))(1-(1-\psi_\delta^-)^s)$ as a test 
function in \eqref{1.7}, we find
\begin{equation}\label{3.25}
\begin{split}
 \int_\Omega|\nabla T_k(u_n^-)|^p (1-(1-\psi_\delta^-)^s)dx \leq C( \delta)k^{q+1+\theta (p-1)}.
 \end{split}
\end{equation}

Combining  \eqref{3.16} with
\eqref{3.24} and \eqref{3.25}, and then choosing $\delta=1$ (for example), we have
\begin{equation}\label{3.26}
 \int_\Omega|\nabla T_k(u_n)|^p dx \leq Ck^{q+1+\theta (p-1)},
\end{equation}
which shows that \eqref{3.3} holds.
Consequently, $T_k(u_n)$ is bounded in $W^{1,p}_0(\Omega)$ independently of $n$. 
This implies  that there exists a subsequence of $u_n$ (still denoted by $u_n$) 
which is almost everywhere convergent in $\Omega$ to a measurable function $u$ 
such that $T_k(u)$ belongs to $W^{1,p}_0(\Omega)$ for
every $k>0$ \cite{BBGGPV1995}.
\end{proof}

The next step of the proof is to state some propositions of limit function $u$.

\begin{lemma}\label{lem3.2}
There exists a constant $C$ such that
\begin{gather}
 \int_\Omega|\nabla u|^{(p-1)r'} dx \leq C,\label{3.27}\\
 \int_\Omega| u|^{q} dx \leq C.\label{3.28}
\end{gather}
\end{lemma}

\begin{proof}
Firstly, we show that $u_n$ is a Cauchy sequence in measure. To do this, we define
\begin{align*}
\Phi(t)=\int_0^t\frac{1}{\left(1+|s|\right)^{\gamma}}ds,
\end{align*}
where $\gamma=1+(p-1)(1-\theta)$. It can be easily seen that
\begin{align*}
|\Phi(t)|\leq  \frac{1}{(p-1)|1-\theta|}.
\end{align*}
Choose $\Phi(u_n)(1-\psi_\delta)^s$ as a test function in \eqref{1.7}, we obtain
\begin{equation}\label{3.29}
\begin{split}
 &\int_\Omega \frac{a(x, u_n, \nabla u_n)}{(1+|u_n|)^{\gamma}}\cdot
  \nabla u_nd\mu_\delta+\int_\Omega|u_n|^{q-1}u_n\Phi(u_n)d\mu_\delta\\
& = s\int_\Omega a(x, u_n, \nabla u_n)\nabla \psi_\delta 
 \Phi(u_n)(1-\psi_\delta)^{s-1}dx
 +\int_\Omega f^\oplus_n\Phi(u_n)d\mu_\delta \\
&\quad -\int_\Omega f^\ominus _n\Phi(u_n)d\mu_\delta.
 \end{split}
\end{equation}
Obviously, by \eqref{1.2},
\begin{gather}\label{3.30}
 \int_\Omega \frac{a(x, u_n, \nabla u_n)}{(1+|u_n|)^{\gamma}}\cdot 
 \nabla u_nd\mu_\delta
 \geq\alpha_1\int_\Omega \frac{|\nabla u_n|^p}{(1+|u_n|)^{p}}d\mu_\delta, \\
\label{3.31}
\int_\Omega|u_n|^{q-1}u_n\Phi(u_n)d\mu_\delta\geq 0,
\end{gather}
Consider  the first terms of the right-hand side of \eqref{3.29}, 
using \eqref{1.3}, we have
\begin{equation}\label{3.32}
\begin{split}
&\int_\Omega a(x, u_n, \nabla u_n)\nabla \psi_\delta 
 \Phi(u_n)(1-\psi_\delta)^{s-1}dx\\
&\leq C \int_\Omega|\nabla u_n|^{p-1}(|\nabla \psi_\delta^+
 |+|\nabla \psi_\delta^-|)(1-\psi_\delta)^{s-1}dx\\
&\leq C (\delta+I_2(n,\delta)).
 \end{split}
\end{equation}
Therefore, using \eqref{3.29}--\eqref{3.32} and \eqref{3.12}, we have
\begin{equation}\label{3.33}
 \int_\Omega \frac{|\nabla u_n|^p}{(1+|u_n|)^{p}}d\mu_\delta\leq C(\delta).
\end{equation}
Similar arguments  as the proof of Lemma \ref{lem3.1}, choose 
$\Phi(k-T_k(u_n^+))(1-(1-\psi_\delta)^s)$ as a test function, show that
\begin{equation}\label{3.34}
\int_\Omega \frac{|\nabla u_n|^p}{(1+|u_n|)^{p}}(1-(1-\psi_\delta)^s)dx
\leq C(\delta).
\end{equation}
Inequalities \eqref{3.33} and \eqref{3.34} yield
\begin{equation}\label{3.35}
 \int_\Omega \frac{|\nabla u_n|^p}{(1+|u_n|)^{p}}dx\leq C(\delta).
\end{equation}

Split $\{| u_n|\geq k\}$ as  $\{| u_n|\geq k\}\cap A_\delta$ and  
$\{| u_n|\geq k\}\cap A_\delta^c$, where $A_\delta=A_\delta^++A_\delta^-$ 
and $A_\delta^+$, $A_\delta^-$ appear in Proposition \ref{pro2.3}.
In view of \eqref{2.6}, we have
\begin{equation}\label{3.36}
\operatorname{meas}(\{| u_n|\geq k\}\cap A_\delta)
\leq\operatorname{meas}( A_\delta)\leq 2\delta.
\end{equation}
As for $\{| u_n|\geq k\}\cap A_\delta^c$, using \eqref{3.13}, \eqref{3.35} 
and Poincar\'{e} inequality, we have
\begin{equation}\label{3.37}
\begin{split}
 &\operatorname{meas}(\{| u_n|\geq k\}\cap A_\delta^c)\\
 &\leq \frac{1}{(\ln(1+ k))^p}\int_{\{| u_n|\geq k\}\cap A_\delta^c}
 \big( \ln (1+|T_k(u_n)|)\big)^{p} dx\\
 &=\frac{1}{(\ln(1+ k))^p}\int_{\{| u_n|\geq k\}\cap A_\delta^c}
 \big(\ln (1+|T_k(u_n)|)\big)^{p} (1-\psi_\delta)^sdx\\
 &=\frac{C}{(\ln(1+ k))^p}\int_{\{| u_n|\geq k\}\cap A_\delta^c}
 \big(\ln (1+|T_k(u_n)|) (1-\psi_\delta)^\frac{s}{p}\big)^{p}dx\\
 &\leq \frac{C}{(\ln(1+ k))^p}\int_{\Omega}
 \frac{\nabla T_k(u_n)| ^{p}}{(1+ |T_k(u_n)|)^p} d\mu_\delta\\
 &\quad +\frac{C}{(\ln(1+ k))^p}\int_{\{| u_n|\geq k\}}|\nabla \psi_\delta| ^{p} 
 (1-\psi_\delta)^{s-p}( \ln (1+|T_k(u_n)|))^{p}dx\\
&\leq \frac{C}{(\ln(1+ k))^p}\int_{\Omega}
 \frac{\nabla T_k(u_n)| ^{p}}{(1+ |T_k(u_n)|)^p} d\mu_\delta
 +C\int_{\Omega}|\nabla \psi_\delta| ^{p} (1-\psi_\delta)^{s-p}dx\\
&\leq \frac{C(\delta)}{(\ln(1+ k))^p}
 +C\Big(\int_{\Omega}|\nabla \psi_\delta| ^{r}dx\Big)^{p/r}\\
&\leq \frac{C}{(\ln(1+ k))^p}+C\delta^{p/r},
 \end{split}
\end{equation}
here we have  used  that $1-\psi_\delta\equiv 1$ on $A_\delta^c$
 by Proposition \ref{pro2.3}.

Combining  \eqref{3.36} and \eqref{3.37}, we arrive at
\begin{align*}
\operatorname{meas}(\{| u_n|\geq k\})
&= \operatorname{meas}(\{| u_n|\geq k\}\cap A_\delta)
 +\operatorname{meas}(\{| u_n|\geq k\}\cap A_\delta^c)\\
&\leq 2\delta+\frac{C}{(\ln(1+ k))^p}+C\delta^{p/r},
\end{align*}
which implies that  $u_n$ is a Cauchy sequence in measure.

We thus have that (up to subsequences, still denoted by $u_n$ ) $u_n$ converges
almost everywhere in $\Omega$ to some function $u$ and
\[
\alpha_1\int_\Omega\frac{|\nabla T_k(u)|^p}{(1+|T_k(u)|)^{\theta (p-1)}} d\mu_\delta
+ k^{q+1}\mu_\delta(\{|u|\geq k\})\leq C(\delta)k.
\]
Furthermore,
\[
\int_\Omega|\nabla u|^{(p-1)r'}(1-\psi_\delta)^{(s-1)r'}dx\leq C(\delta).
\]
Letting $\delta$ tend to zero, we find
\[
\int_\Omega|\nabla u|^{(p-1)r'}dx\leq C,
\]
which shows that \eqref{3.27} holds. In a similar way we can prove that
\[
\int_\Omega| u|^{q}dx\leq C,
\]
which is \eqref{3.28}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
By  Lemmas \ref{lem3.1} and \ref{lem3.2}, with
similar arguments as  the proof of \cite{OP2002}, we choose 
$T_k(u_n-T_h(u))(1-\psi_\delta)^s$ as a test function in \eqref{1.7}, 
 and show that
\[
\nabla u_n(1-\psi_\delta)^s\to \nabla u(1-\psi_\delta)^s,
\quad \text{almost everywhere in }\Omega.
\]
We choose
\[
\frac{1}{\varepsilon}T_k(G_{k-\varepsilon}(u_n))(1-\psi_\delta)^s
\]
as a test function in \eqref{1.7},  and arrive at
\[
\lim_{k\to\infty}\sup_{n\in \mathbb{N}}\int_\Omega|\nabla |u_n|^p
 (1-\psi_\delta)^sdx=0.
\]
Then choosing $v(1-\psi_\delta)^s$ as a test function in \eqref{1.7},  
where $v\in C_0^\infty(\Omega)$, we can pass to the limit.
More details can be found in \cite[steps 4,  5, 6]{OP2002}, so 
we omit them here. 
\end{proof}

\subsection*{Acknowledgments}

This work  was partially supported by the NSF of China (No. 11401473),
 NSF of Gansu Province (No.1506RJYA272),  by the
Fundamental Research Funds for the Central Universities (No. 31920160059), 
by the Talent Introduction Scientific Research Foundation of Northwest 
University for Nationalities (No. xbmuyjrc201305), 
by the Science and Humanity Foundation of the Ministry of Education(No.15YJA880085), 
by the Foundation of State Nationalities Affairs Commission (No.14XBZ016) 
and by the Research and Innovation teams of Northwest University for Nationalities.


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\end{document}