\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 63, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/63\hfil Existence of solutions]
{Existence of solutions for some nonlinear elliptic equations}
\author[A. Anane, O. Chakrone, M. Chehabi\hfil EJDE-2006/63\hfilneg]
{Aomar Anane, Omar Chakrone, Mohammed Chehabi}  % in alphabetical order

\address{D\'epartement de Math\'ematiques et Informatique,
 Facult\'e des Sciences,
 Universit\'e Mohammed 1er, Oujda, Maroc}
  \email[Aomar Anane]{anane@sciences.univ-oujda.ac.ma}
  \email[Omar Chakrone]{chakrone@sciences.univ-oujda.ac.ma}
  \email[Mohammed Chehabi]{chehb\_md@yahoo.fr}

\date{}
\thanks{Submitted January 23, 2006. Published May 19, 2006.}
\subjclass[2000]{35J15, 35J70, 35J85}
\keywords{Boundary value problem; truncation; $L^1$; p-Laplacian; spectrum}

\begin{abstract}
 In this paper, we study the existence of solutions to the following
 nonlinear elliptic problem in a bounded subset $\Omega$ of
 $\mathbb{R}^{N}$:
 \begin{gather*}
 -\Delta _{p}u  =  f(x,u,\nabla u)+\mu  \quad \hbox{in } \Omega ,\\
 u  =  0 \quad \mbox{on }\partial \Omega ,
 \end{gather*}
 where $\mu $ is a Radon measure on $\Omega $ which is zero on
 sets of $p$-capacity zero,
 $f:\Omega \times \mathbb{R}\times \mathbb{R} ^{N}\to \mathbb{R}$
 is a Carath\'{e}odory function that satisfies certain
 conditions with respect to the one dimensional spectrum.
\end{abstract}

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

\section{Introduction}

We consider the quasilinear elliptic problem
\begin{equation}\label{eP}
\begin{gathered}
-\Delta _{p}u=f(x,u,\nabla u)+\mu \quad\text{in } \Omega , \\
u=0 \quad  \text{on }\partial \Omega ,
\end{gathered}
\end{equation}
where $\Omega $ is a bounded open set in $\mathbb{R}^{N}$, $N\geq 2$,
 $1<p<+\infty $,  $\mu $ is a Radon measure on $\Omega $ and
 $f:\Omega \times \mathbb{R}\times \mathbb{R}^{N}\to
\mathbb{R}$ is a Carath\'{e}odory function. We are interested in
the existence of solutions
to this problem. More precisely, we will prove the existence of a solution
$u\in W_{0}^{1,p}(\Omega )$, if and only if the signed measure
$\mu $ is zero on  sets of capacity zero in $\Omega $.
(i.e $\mu (E)=0$ for every set $E$ such that
$\mathop{\rm cap}_{p}(E,\Omega )=0$).

Boccardo, Gallouet and Orsina have proved in \cite{Boc} the
existence of a solution to the problem
\begin{gather*}
Au+g(x,u,\nabla u)= \mu \quad{in } \Omega , \\
u = 0 \quad\text{on }\partial \Omega ,
\end{gather*}
where $A(u)=-\mathop{\rm div}(a(x,\nabla u))$,
$a: \Omega \times \mathbb{R}^{N}\to \mathbb{R}$ and
$g:\Omega \times \mathbb{R}\times \mathbb{R}^{N}\to \mathbb{R}$ are
Carath\'{e}odory functions such that for almost every
$x\in \Omega $, for every $\xi \in \mathbb{R}^{N}$ and for every
$s\in \mathbb{R}$,
\begin{gather*}
a(x,\xi ).\xi \geq \alpha |\xi |^{p}, \\
|a(x,\xi )|\leq l(x)+\beta |\xi |^{p-1}, \\
|g(x,s,\xi )|\leq b(|s|)[|\xi |^{p}+d(x)],
\end{gather*}
where $\alpha $ and $\beta $ are two positive constants,
$l\in L^{p'}(\Omega )$, $b$ a real-valued, positive,
increasing, continuous function, and $d$ a nonegative function in
$L^{1}(\Omega )$.
They assume that for almost every $x\in \Omega $, for every $\xi $ and
$\eta $ in $\mathbb{R}^{N}$, with $\xi \neq \eta $,
\begin{equation*}
[ a(x,\xi )-a(x,\eta )].(\xi -\eta )>0,
\end{equation*}
They require also that for almost every $x\in \Omega $, for every $\xi $
in $\mathbb{R}^{N}$, for every $s$ in $\mathbb{R}$ such that
$|s|\geq \sigma $,
\begin{equation*}
g(x,s,\xi )\mathop{\rm sgn}(s)\geq \rho |\xi |^{p},
\end{equation*}
where $\rho $ and $\sigma $ are two positive real numbers and
$\mathop{\rm sgn}(s)$ is the sign of $s$.

Let $(\beta ,\alpha ,u)\in \mathbb{R}^{N}\mathbb{\times R\times}
W_{0}^{1,p}(\Omega )\backslash \{0\}$. If $(\beta ,\alpha ,u)$ is a
solution of the problem
\begin{gather*}
-\Delta _{p}u=\alpha m(x)|u|^{p-2}u+\beta .
|\nabla u|^{p-2}\nabla u \quad \text{in }\Omega , \\
u =0\quad\text{on } \partial \Omega ,
\end{gather*}
where $1<p<\infty $ and $m\in M=\{m\in L^{\infty }(\Omega ):
\mathop{\rm meas}\{x\in \Omega : m(x)>0\}\neq 0\}$. In this case,
the pair $(\beta ,\alpha )$ is said to be a one dimensional eigenvalue
and $u$  the associated eigenfunction. We designate by
$\sigma _{1}(-\Delta _{p},m)\subset \mathbb{R}^{N}\mathbb{\times R}$
the set of one dimensional eigenvalues $(\beta ,\alpha )$ with
 $\alpha \geq 0$.

\begin{proposition} \label{prop1}
(1) $\sigma _{1}(-\Delta _{p},m)$ contains the union of the
sequence of graphs of the functions
$\Lambda _{n}:\mathbb{R}^{N}\to \mathbb{R}^{+}$, $n=1,2,\dots $,
where $\Lambda _{n}(\beta )$ is defined for
every $\beta \in \mathbb{R}^{N}$ by
\begin{equation*}
\frac{1}{\Lambda _{n}(\beta )}
=\sup_{K\in A_{n}^{\beta }} \min_{u\in K}
\int_{\Omega }e^{\beta .x}m(x)|u|^{p}dx.
\end{equation*}
with $A_{n}^{\beta }=\{K\subset S_{\beta }$, $K$ compact
symmetrical; $\gamma (K)\geq n\}$,
$$
S_{\beta }=\big\{ u\in W_{0}^{1,p}(\Omega ):\Big( \int_{\Omega
}e^{\beta .x}m(x)|\nabla u|^{p}dx\Big)
^{1/p}=1\big\}
$$
and $\gamma(K)$ indicates the genus of $K$.

(2)  $\Lambda _{1}(.)$ is the first eigensurface of the spectrum of
$\sigma _{1}(-\Delta _{p},m)$ in the sense
\begin{equation*}
\sigma _{1}(-\Delta _{p},m)\subset
\{ (\beta ,\alpha )\in \mathbb{R}^{N}
\mathbb{\times R}\text{; }\Lambda _{1}(\beta )\leq \alpha \}
\end{equation*}
\end{proposition}

The proof of the above proposition can be found in \cite{Anan}.
When $\mu =h\in W^{-1,p'}(\Omega )$, Anane, Chakrone and
Gossez have proved in \cite{Anan} the existence of a solution
to \eqref{eP}, in the sense
\begin{equation*}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla
v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\langle h,v\rangle
\end{equation*}
for every $v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$.
This is done under the hypotheses of non-resonance
with respect to the spectrum of one dimensional
$\sigma_{1}(-\Delta _{p},1)$:
There exists ($\beta ,\alpha )\in \mathbb{R}^{N}\times \mathbb{R}$
with $\alpha <\Lambda _{1}(\beta ,-\Delta _{p},1)$ where
$\Lambda_{1}(.,-\Delta_{p},1)$ is the first eigensurface of the
spectrum of one dimensional $\sigma _{1}(-\Delta _{p},1)$, such that
for all $\delta >0$ there exists
$a_{\delta }\in L^{p'}(\Omega)$  such that
\begin{equation} \label{eP1}
f(x,s,\xi )s\leq \alpha |s|^{p}+\beta |\xi |^{p-2}\xi s
+\delta (|s|^{p-1}+|\xi |^{p-1}+a_{\delta }(x))|s|
\end{equation}
for almost every $x\in \Omega$  and
for all $(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$;
and for all $k>0$ there exist $\phi _{k}\in L^{1}(\Omega )$ and
$ b_{k}\in \mathbb{R}$ such that
\begin{equation} \label{eP2}
\max_{|s|\leq k} |f(x,s,\xi )|\leq b_{k}|\xi |^{p}+\phi _{k}(x)
\end{equation}
for almost every $x\in \Omega $ and for all $\xi \in \mathbb{R}^{N}$.


\begin{remark} \rm \begin{enumerate}
\item If $f(x,u,\nabla u)=\alpha m(x)|u|^{p-2}u
+\beta .|\nabla u|^{p-2}\nabla u$, then  \eqref{eP} has
a solution for every $\mu \in W^{-1,p'}(\Omega )$, in the
usual sense
\begin{equation*}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla
v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx
+\langle h,v\rangle_{W^{-1,p'}(\Omega ),W_{0}^{1,p}(\Omega )}
\end{equation*}
for every $v\in W_{0}^{1,p}(\Omega )$, if and only if
$(\beta,\alpha )\notin \sigma _{1}(-\Delta _{p},m)$.

\item If $\mu \notin W^{-1,p'}(\Omega )$,  problem
\eqref{eP} does not have always a solution. Indeed in the case
$1<p\leq N$, we have that $L^{1}(\Omega )\nsubseteqq $
$W^{-1,p'}(\Omega )=-\Delta _{p}(W_{0}^{1,p}(\Omega ))$.
\end{enumerate}
\end{remark}

In this work, we assume \eqref{eP2} and that $\mu $ is a
measure. We assume also that
for each $\delta >0$ there exists
$a_{\delta }\in L^{p'}(\Omega )$ such that
\begin{equation} \label{eP3}
f(x,s,\xi )s\leq -\rho |\xi |^{p}|s|+\alpha |
s|^{p}+\beta |\xi |^{p-2}\xi s+\delta (|s|
^{p-1}+|\xi |^{p-1}+a_{\delta }(x))|s|
\end{equation}
for almost every $x\in \Omega$ and for all
$(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$,
where ($\beta ,\alpha )\in \mathbb{R}^{N}\times \mathbb{R}$
satisfies the same conditions as in \eqref{eP1} and $\rho $ is a
positive real number.
In the case $\delta =1$,
there exists $a_{1}\in L^{p'}(\Omega )$  such that
\begin{equation} \label{eP4}
f(x,s,\xi )\mathop{\rm sgn}(s)\leq -\rho |\xi |^{p}+\alpha '|s|
^{p-1}+\beta '|\xi |^{p-1}+a_{1}(x)
\end{equation}
for almost every $x\in \Omega$ and for all
$(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$,
where $\alpha '=\alpha +1$  and  $\beta '=|\beta |+1$.

\begin{remark} \rm \begin{enumerate}
\item The conditions of the sign given in \cite{Boc} imply
\eqref{eP3} in the case $\alpha =0$ and $\beta =0$.

\item The hypothesis \eqref{eP2} and \eqref{eP3} are satisfied
for example if
\begin{equation*}
f(x,s,\xi )=-\rho |\xi |^{p}\mathop{\rm sgn}(s)+\alpha |s|
^{p-2}s+\beta |\xi |^{p-2}\xi +g(x,s,\xi )+l(x,s,\xi )
\end{equation*}
where $g$ and $l$ satisfy
\begin{gather*}
g(x,s,\xi )s\leq 0, \\
|g(x,s,\xi )|\leq b(|s|)(|x|^{p}+c(x)), \\
sl(x,s,\xi )\leq C(|s|^{q-1}+|x|^{q-1}+d(x))|s|
\end{gather*}
with $b$ continuous, $c(x)\in L^{1}(\Omega )$, $q<p$, $d(x)\in
L^{p'}(\Omega )$ and $C$ a constant.
\end{enumerate}
\end{remark}

For every compact subset $K$ of $\Omega $, the $p$-capacity of $K$
with respect to $\Omega $ is defined as
$$
\mathop{\rm cap}_{p}(K,\Omega )
=\inf\{\int_{\Omega }|\nabla u|^{p}dx,\;  u\in C_{0}^{\infty
}(\Omega ) \mbox{ and }  u\geq \chi _{K}\}
$$
where $\chi _{K }$ is the characteristic function of $K$;
we will use the convention that $\inf (\emptyset )=+\infty $. The
$p$-capacity of any open subset $U$ of $\Omega $ is defined by
$\mathop{\rm cap}_{p}(U,\Omega )
=\sup \{\mathop{\rm cap}_{p}(K,\Omega ),\; K \mbox{ compact and }
 K\subseteq U\}$.
Also the $p$-capacity of any subset $B\subseteq \Omega $ by
$\mathop{\rm cap}_{p}(B,\Omega )=\inf \{\mathop{\rm cap}_{p}(U,\Omega ),
\; U \text{ open and } B\subseteq U\}$.
We will denote by $\mathcal{M}_{b}(\Omega )$ the
space of all signed measures on $\Omega $ and by
$\mathcal{M}_{0}^{p}(\Omega )$ the space of all measures $\mu $
in $\mathcal{M}_{b}(\Omega )$ such that $\mu (E)=0$ for every
set $E$ such that $\mathop{\rm cap}_{p}(E,\Omega )=0$.

Our main result is stated as follows.

\begin{theorem} \label{thm1}
Assume \eqref{eP2}, \eqref{eP3} and that $\mu $ is a measure in
$\mathcal{M}_{b}(\Omega )$. Then, there exists a solution $u$ of
\begin{equation} \label{eP5}
\begin{gathered}
-\Delta _{p}u=f(x,u,\nabla u)+\mu \quad\text{in }\Omega , \\
u=0\quad\text{on } \partial \Omega
\end{gathered}
\end{equation}
in the sense that $u\in W_{0}^{1,p}(\Omega )$,
$f(x,u,\nabla u)\in L^{1}(\Omega )$, and
\begin{equation*}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla
v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }v\,d\mu ,
\end{equation*}
for every $v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$,
if and only if $\mu \in \mathcal{M}_{0}^{p}(\Omega )$.
\end{theorem}

\section{Proof of Main result}

The notation $\langle .,.\rangle $ stands hereafter for
the duality pairing between $W^{-1,p'}(\Omega )$ and
$W_{0}^{1,p}(\Omega )$. We define, for $s$ and $k$ in $\mathbb{R}$,
with $k>0$,
\begin{equation*}
T_{k}(s)=\begin{cases}
k\mathop{\rm sgn}(s) & \text{if }|s|>k, \\
s & \text{if } |s|\leq k,
\end{cases}
\end{equation*}
and
$G_{k}(s)=s-T_{k}(s)$.

\begin{lemma} \label{lem1}
Let $g\in L^{\infty }(\Omega )$ and $F\in (L^{p'}(\Omega ))^{N}$.
Under the hypotheses \eqref{eP2} and \eqref{eP3}, the problem
\begin{equation} \label{eP6}
\begin{gathered}
-\Delta _{p}u=f(x,u,\nabla u)+g- \mathop{\rm div}F \quad \text{in }\Omega, \\
u=0 \quad\text{on }\partial \Omega ,
\end{gathered}
\end{equation}
admits a solution $u\in W_{0}^{1,p}(\Omega )$ in the sense that
$f(x,u,\nabla u)$ and $f(x,u,\nabla u)u$ are in $L^{1}(\Omega )$,
and that
$$
 \int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla
v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }gv
+\int_{\Omega }F\nabla v
$$
for every $v\in W_{0}^{1,p}(\Omega )\cap L^{\infty}(\Omega )$ and
for $v=u$.
\end{lemma}

\begin{proof}
Letting $l=g-\mathop{\rm div}F$, we have $l\in W^{-1,p'}(\Omega )$.
Then  \eqref{eP3} implies \eqref{eP1},
and Lemma \ref{lem1} is a particular case of a result in \cite{Anan}.
\end{proof}

\begin{lemma} \label{lem2}
$\mathcal{M}_{0}^{p}(\Omega )=L^{1}(\Omega )+W^{-1,p'}(\Omega )$
for every $1<p<+\infty $.
\end{lemma}

For the proof of the above lemma see \cite{Bocc}.

\begin{lemma} \label{lem3}
Let $a$, $b$ be two nonnegative numbers, and let
$\varphi (s)=se^{\theta s^{2}}$ with $\theta =b^{2}/(4a^{2})$.
Then for all $s\in \mathbb{R}$,
$a\varphi '(s)-b|\varphi (s)|\geq a/2$.
\end{lemma}

\begin{proof}
For $s\in \mathbb{R}$ let $\psi (s)=a\varphi '(s)-b|\varphi (s)|$.
Then
\[
\psi (s) =e^{\theta s^{2}}[a(1+2\theta s^{2})-b|s|] = ae^{\theta
s^{2}}[(1+2\theta s^{2})-2\sqrt{\theta }|s|],
\]
Then $\psi $ is even, and assuming that $s\geq 0$, we obtain that
for every $s\geq 0$,
\[
\psi (s) = 2ae^{\theta s^{2}}\big[ (\sqrt{\theta }s-\frac{1}{2})^{2}
+\frac{1}{4}\big]
\geq \frac{a}{2}.
\]
\end{proof}

\begin{remark} \label{rmk2.4} \rm
Let $\mu \in \mathcal{M}_{0}^{p}(\Omega )$. If $p>N$, then
$L^{1}(\Omega )\subset W^{-1,p'}(\Omega )$; therefore,
$\mathcal{M}_{0}^{p}(\Omega )=W^{-1,p'}(\Omega )$.
Then the existence of a solution of \eqref{eP5} is a consequence of
 \cite[Theorem 7.1]{Anan}. That is why, we assume that
$1<p\leq N$.
\end{remark}

\begin{proof}[Proof of the Theorem \ref{thm1}]
Note that if $u\in W_{0}^{1,p}(\Omega )$ is a solution of
\eqref{eP5}, then
\[
\mu =-\Delta _{p}u-f(x,u,\nabla u)
\]
 with $\Delta _{p}u \in W^{-1,p'}(\Omega)$ and
 $f(x,u,\nabla u)\in L^{1}(\Omega )$;
So by Lemma \ref{lem2}, $\mu \in \mathcal{M}_{0}^{p}(\Omega )$.

Conversely, suppose that $\mu \in \mathcal{M}_{0}^{p}(\Omega )$,
so by Lemma \ref{lem2} there exists $g\in L^{1}(\Omega )$ and
$F\in (L^{p'}(\Omega ))^{N}$ such that $\mu =g-\mathop{\rm div}F$.
There exists a sequence $(g_{n})_{n}$ of $L^{\infty }(\Omega )$
that converges strongly to $g$ in $L^{1}(\Omega )$ and
$\widetilde {g}\in L^{1}(\Omega )$ such that
$|g_{n}(x)|\leq |\widetilde {g}(x)|$
for every $n\in \mathbb{N}$  and for almost every
$x\in \Omega$.

By Lemma \ref{lem1}, the problem
\begin{equation}
\begin{gathered}
-\Delta _{p}u_{n}= f(x,u_{n},\nabla u_{n})+g_{n}-\mathop{\rm div}F \quad
\text{in } \Omega , \\
u_{n}=0\quad \text{on } \partial \Omega ,
\end{gathered}  \label{eP7}
\end{equation}
admits a solution $u_{n}\in W_{0}^{1,p}(\Omega )$ in the sense
that
\begin{equation}
 f(x,u_{n},\nabla u_{n}),\ f(x,u_{n},\nabla u_{n})u_{n}\in
L^{1}(\Omega ),  \label{eP8}
\end{equation}
and
\begin{equation}
\int_{\Omega }|\nabla u_{n}|^{p-2}\nabla
u_{n}\nabla v\,dx=\int_{\Omega }f(x,u_{n},\nabla
u_{n})v\,dx+\int_{\Omega }g_{n}v+\int_{\Omega }F\nabla v,
\label{eP9}
\end{equation}
for every $v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ and for
$v=u_{n}$.

\begin{lemma} \label{lem4}
The sequence $(u_{n})_{n}$ is bounded in $W_{0}^{1,p}(\Omega )$.
\end{lemma}

\begin{proof}
Let us choose $v=\varphi (T_{1}(u_{n}))$ as a test function in
\eqref{eP9}, where $\varphi (s)=se^{\theta s^{2}}$ with
 $\theta=\frac{b^{2}}{4a^{2}}$, $a=1$ and $b=b_{1}$
  ($b_{1}\geq 0$ is given for $k=1$ by \eqref{eP2}). Setting
\begin{gather*}
a(\xi )=|\xi |^{p-2}\xi \quad \forall \xi \in \mathbb{R}^{N}, \\
\varphi _{1}=\varphi (T_{1}(u_{n})), \quad
\varphi_{1}'=\varphi '(T_{1}(u_{n})),
\end{gather*}
we have
\begin{equation}
\begin{aligned}
\int_{\Omega }a(\nabla u_{n})\nabla [ \varphi (T_{1}(u_{n}))]dx
&= \int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi (T_{1}(u_{n}))dx \\
&\quad +\int_{\Omega }g_{n}\varphi (T_{1}(u_{n}))dx+\int_{\Omega
}F\nabla [ \varphi (T_{1}(u_{n}))]dx.
\end{aligned} \label{eP10}
\end{equation}
On the other hand,
\begin{align*}
\int_{\Omega }\ a(\nabla u_{n})\nabla [ \varphi (T_{1}(u_{n}))]dx
& =\int_{\Omega }a(\nabla u_{n})\varphi _{1}'\nabla (T_{1}(u_{n}))dx \\
& =\int_{\Omega }\varphi _{1}'|\nabla (T_{1}(u_{n}))|^{p}dx.
\end{align*}
Since $\varphi '$ is an even function in $\mathbb{R}$, $\varphi '$
is increasing in $\mathbb{R}^{+}$ and $|T_{1}(u_{n})|\leq 1$, we
have
\begin{align*}
\int_{\Omega }F\nabla [ \varphi (T_{1}(u_{n}))]dx
& \leq  \| F\| _{L^{p'}}\|
\varphi (T_{1}(u_{n}))\| _{1,p} \\
& \leq \| F\| _{L^{p'}}(\int_{\Omega }|
\varphi _{1}'\nabla (T_{1}(u_{n}))|^{p}dx)^{1/p} \\
& \leq \| F\| _{L^{p'}}\varphi '(1)\| T_{1}(u_{n})\| _{1,p}.
\end{align*}
Since $\varphi $ is increasing in $\mathbb{R}$, we get
\[
\int_{\Omega }g_{n}\varphi (T_{1}(u_{n}))dx
 \leq \varphi (1)\int_{\Omega }|g_{n}|dx
 \leq \varphi (1)\| \widetilde {g}\| _{L^{1}}.
\]
Writing
\begin{align*}
&\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi (T_{1}(u_{n}))dx\\
&=\int_{\{|u_{n}|\leq  1\}}\varphi_{1}f(x,u_{n},\nabla
u_{n})dx
+\int_{\{|u_{n}|>1\}}\varphi _{1}f(x,u_{n},\nabla u_{n})dx.
\end{align*}
By \eqref{eP2}, we have
\begin{align*}
|\int_{\{|u_{n}|\leq  1\}}\varphi
_{1}f(x,u_{n},\nabla u_{n})dx|
& \leq \int_{\{|u_{n}|\leq  1\}}|
\varphi _{1}||f(x,u_{n},\nabla u_{n})|dx \\
& \leq \int_{\{|u_{n}|\leq  1\}}|\varphi_{1}|
[ b_{1}|\nabla u_{n}|^{p}+\phi _{1}(x)]dx \\
& \leq b_{1}\int_{\{|u_{n}|\leq 1\}}|
\varphi _{1}||\nabla u_{n}|^{p}dx+\varphi (1)\|
\phi_{1}\| _{L^{1}} \\
& \leq b_{1}\int_{\Omega }|\varphi _{1}||\nabla
(T_{1}(u_{n}))|^{p}dx+\varphi (1)\| \phi _{1}\| _{L^{1}}.
\end{align*}
On the other hand, on $\{|u_{n}|>1\}$,
$T_{1}(u_{n})=\mathop{\rm sgn}(u_{n})$, so
$\varphi (T_{1}(u_{n}))=\mathop{\rm sgn}(u_{n})$
$e^{\theta }$ and by \eqref{eP4}, we get
\begin{align*}
&\int_{\{|u_{n}|>1\}}\varphi _{1}f(x,u_{n},\nabla u_{n})dx\\
& = \int_{\{|u_{n}|>1\}}e^{\theta }f(x,u_{n},\nabla
u_{n})\mathop{\rm sgn}(u_{n})dx\\
& \leq   e^{\theta }\int_{\{|u_{n}|>1\}}[-\rho |
\nabla u_{n}|^{p}+\alpha '|u_{n}|^{p-1}
+ \beta '|\nabla u_{n}|^{p-1}+a_{1}(x)]dx.
\end{align*}
Adding  the above inequalities, by \eqref{eP10}, we obtain
\begin{equation}
\begin{aligned}
& \int_{\Omega }[\varphi _{1}'-b_{1}|\varphi
_{1}|]|\nabla (T_{1}(u_{n}))|^{p}dx+\rho e^{\theta
}\int_{\{|u_{n}|>1\}}|\nabla u_{n}|^{p}dx \\
&\leq \| F\| _{L^{p'}}\varphi '(1)\| T_{1}(u_{n})\| _{1,p}
+\varphi (1)\| \widetilde{g}\| _{L^{1}}+\varphi (1)\| \phi
_{1}\| _{L^{1}}
\\
&\quad +e^{\theta }\int_{\{|u_{n}|>1\}}[\alpha '|
u_{n}|^{p-1}+\beta '|\nabla u_{n}|^{p-1}+a_{1}(x)]dx.
\end{aligned}  \label{eP11}
\end{equation}
Using H\"{o}lder's inequality, we have
\begin{gather*}
\int_{\{|u_{n}|>1\}}|\nabla u_{n}|^{p-1}dx
\leq \| u_{n}\| _{1,p}^{p-1}(\mathop{\rm meas}(\Omega ))^{1/p},
\\
\int_{\{|u_{n}|>1\}}|u_{n}|^{p-1}dx
\leq \|u_{n}\| _{p}^{p-1}(\mathop{\rm meas}(\Omega ))^{1/p}.
\end{gather*}
By Poincar\'{e}'s inequality, there exists $c>0$ such that
\begin{equation*}
\| u_{n}\| _{p}\leq c\| \nabla u_{n}\| _{p}.
\end{equation*}
So
\begin{equation*}
\int_{\{|u_{n}|>1\}}|u_{n}|^{p-1}dx
\leq c^{p-1}\| u_{n}\| _{1,p}^{p-1}(\mathop{\rm meas}
(\Omega ))^{1/p}.
\end{equation*}
Replacing this in (\ref{eP11}) and using  that
$\varphi _{1}'-b_{1}|\varphi _{1}|\geq \frac{1}{2}$, we
obtain
\begin{equation*}
\frac{1}{2}\int_{\Omega }|\nabla (T_{1}(u_{n}))|
^{p}dx+\rho e^{\theta }\int_{\{|u_{n}|>1\}}|\nabla
u_{n}|^{p}dx\leq c_{1}\| u_{n}\|
_{1,p}+c_{2}\| u_{n}\| _{1,p}^{p-1}+c_{3},
\end{equation*}
where $c_{1}=\| F\| _{L^{p'}}\varphi '(1)$,
$c_{2}=e^{\theta }[\alpha 'c^{p-1}+\beta '](\mathop{\rm meas}(\Omega ))^{
\frac{1}{p}}$ and
$c_{3}=\varphi (1)\| \widetilde {g}\| _{L^{1}}+\varphi (1)\| \phi _{1}\|
_{L^{1}}+e^{\theta }\| a_{1}(x)\| _{L^{1}}$.
Set
$c_{4}=\min (\frac{1}{2},\rho e^{\theta })$, we have
\begin{equation*}
c_{4}\| u_{n}\| _{1,p}^{p}\leq c_{1}\|
u_{n}\| _{1,p}+c_{2}\| u_{n}\|_{1,p}^{p-1}+c_{3},
\end{equation*}
since $p>1$, $(u_{n})_{n}$ is a bounded sequence in
$W_{0}^{1,p}(\Omega )$.
\end{proof}

For a subsequence, still denoted by $(u_{n})_{n}$, we have
\begin{equation}
\begin{gathered}
u_{n}\rightharpoonup u \quad \text{weakly in }W_{0}^{1,p}(\Omega ), \\
u_{n}\to u \quad \text{strongly in }\ L^{p}(\Omega ), \\
u_{n}(x)\to u(x)\quad \text{for almost every }x\in \Omega .
\end{gathered}  \label{eP12}
\end{equation}

\begin{lemma} \label{lem5}
For every $k>0$, the sequence $(T_{k}(u_{n}))_{n}$
converges strongly to $T_{k}(u)$ in $W_{0}^{1,p}(\Omega )$.
\end{lemma}

\begin{proof}
 Let $k>0$. Consider $\varphi (s)=se^{\theta s^{2}}$ with
$\theta =\frac{b^{2}}{4a^{2}}$, $a=1$ and $b=a_{k}$ ($a_{k }\geq 0$ is
given by \eqref{eP2}.
Setting
\begin{gather*}
a(\xi )=|\xi |^{p-2}\xi,\quad \forall \xi \in \mathbb{R}^{N},
\varphi _{n}=\varphi (T_{k}(u_{n})-T_{k}(u)), \quad
\varphi_{n}'=\varphi '(T_{k}(u_{n})-T_{k}(u)).
\end{gather*}
By \eqref{eP12}, the continuity of $\varphi $ and
$\varphi'$, and the dominated convergence theorem, we have
\begin{equation}
\begin{gathered}
\varphi _{n}\rightharpoonup 0 \quad\text{and}\quad
\varphi _{n}'\rightharpoonup 1 \quad
\text{weak-$\ast$ in $L^{\infty }(\Omega )$ and a. e.
$x\in \Omega $}, \\
\varphi _{n}\to 0\quad\text{and}\quad
\varphi _{n}'\to 1 \quad\text{in $L^{q}(\Omega )$ for every $q\geq 1$}.
\end{gathered}  \label{eP13}
\end{equation}
We will denote by $\varepsilon _{n}$\ any quantity which converges
to zero as $n$ tends to infinity.
Let $v=\varphi _{n}$, be a test function in
(\ref{eP9}). Then
\begin{equation}
\begin{aligned}
&\int_{\Omega }a(\nabla u_{n})\nabla
(T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx \\
& = \int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx
+\int_{\Omega }g_{n}\varphi _{n}dx
+\int_{\Omega }F\nabla (T_{k}(u_{n})-T_{k}(u))\varphi_{n}' \\
&:= A+B+C+D
\end{aligned}  \label{eP14}
\end{equation}

For the third term on the right-hand side:
Since $\varphi _{n}\rightharpoonup 0$ weak-$\ast$ in
$L^{\infty}(\Omega )$ and $g_{n}\to g\ $in $L^{1}(\Omega )$,
we have $\int_{\Omega }g_{n}\varphi _{n}dx\to 0$ so that
\begin{equation}
C=\varepsilon _{n}.  \label{eP15}
\end{equation}

For the forth  term on the right-hand side:
It is clear that $F\varphi _{n}'\to F$ in
$(L^{p'}(\Omega ))^{N}$ and $T_{k}(u_{n})\rightharpoonup T_{k}(u)\ $
weakly in $W_{0}^{1,p}(\Omega )$, so that
\begin{equation}
 D=\varepsilon _{n}.  \label{eP16}
\end{equation}

For the second term on the right-hand side:
\begin{align*}
&\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx\\
&=\int_{\{|u_{n}|>k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx
+ \int_{\{|u_{n}|\leq  k\}}f(x,u_{n},\nabla
u_{n})\varphi _{n}dx
:= B_1+B_2.
\end{align*}
On the set $\{|u_{n}|>k\}$, $\varphi _{n}$ has the same sign as
$u_{n}$, so by \eqref{eP4},
\begin{align*}
&f(x,u_{n},\nabla u_{n})\varphi _{n} \\
&\leq -\rho |\nabla u_{n}|^{p}|\varphi _{n}|+\alpha '|
u_{n}|^{p-1}|\varphi _{n}|+\beta '|\nabla
u_{n}|^{p-1}|\varphi_{n}|+a_{1}(x)|\varphi _{n}|\\
&\leq [\alpha '|u_{n}|^{p-1}+\beta
'|\nabla u_{n}|^{p-1}+a_{1}(x)]|\varphi _{n}|.
\end{align*}
By Lemma \ref{lem4} and \eqref{eP13}, we have
$B_{1}\leq \varepsilon _{n}$,
so that
\begin{equation*}
\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx
\leq \int_{\{|u_{n}|\leq  k\}}f(x,u_{n},\nabla
u_{n})\varphi _{n}dx+\varepsilon _{n}.
\end{equation*}
By \eqref{eP2}, we have
\begin{align*}
|\int_{\{|u_{n}|\leq  k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx|
& \leq \int_{\{|u_{n}|\leq  k\}}|
 f(x,u_{n},\nabla u_{n})||\varphi _{n}|dx \\
& \leq \int_{\{|u_{n}|\leq  k\}}[b_{k}|
 \nabla u_{n}|^{p}+\phi _{k}(x)]|\varphi _{n}|dx \\
& \leq b_{k}\int_{\Omega }|\nabla T_{k}(u_{n})| ^{p}|
\varphi _{n}|dx+\int_{\Omega }\phi _{k}(x)|\varphi _{n}|dx,
\end{align*}
and
\begin{align*}
\int_{\Omega }|\nabla T_{k}(u_{n})|^{p}|\varphi _{n}|dx
&= \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla T_{k}(u_{n})|
 \varphi_{n}|dx \\
& = \int_{\Omega }(a(\nabla T_{k}(u_{n}))-a(\nabla
T_{k}(u)))(\nabla
T_{k}(u_{n})-\nabla T_{k}(u))|\varphi _{n}|dx \\
& \quad + \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla T_{k}(u)|
\varphi_{n}|dx \\
& \quad + \int_{\Omega }a(\nabla T_{k}(u))(\nabla
T_{k}(u_{n})-\nabla T_{k}(u))|\varphi _{n}|dx.
\end{align*}
By \eqref{eP13}, since $(T_{k}(u_{n}))_{n}$ is bounded in
$W_{0}^{1,p}(\Omega )$, we have
\begin{equation}
\begin{aligned}
&\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx\\
&\leq \varepsilon _{n}+b_{k}\int_{\Omega }\big(a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))
\big)\big(\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big)|
\varphi _{n}|dx.
\end{aligned}  \label{eP17}
\end{equation}

For the firs term on the right-hand side (A):
We verify easily that $a(\nabla T_{k}(u_{n}))+a(\nabla
G_{k}(u_{n}))=a(\nabla u_{n})$, so that
\begin{align*}
&\int_{\Omega }a(\nabla u_{n})\nabla
(T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx \\
& = \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla
(T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx
+\int_{\Omega }a(\nabla G_{k}(u_{n}))\nabla
(T_{k}(u_{n})\\
&\quad -T_{k}(u))\varphi _{n}'dx := A_1+A_2.
\end{align*}
We have $\nabla (T_{k}(u_{n}))=0$ if $\nabla (G_{k}(u_{n}))\neq
0$, so
\begin{align*}
A_{2} &= -\int_{\Omega }a(\nabla G_{k }(u_{n}))\nabla
(T_{k}(u))\varphi _{n}'dx \\
& = -\int_{\Omega }a(\nabla G_{k }(u_{n}))\nabla
(T_{k}(u))\chi_{\{|u_{n}|\geq k\}}\varphi _{n}'dx.
\end{align*}
Since $\nabla T_{k}(u)=0$ on the set $\{|u|\geq k\}$,
$\nabla T_{k}(u)\chi _{\{|u_{n}|\geq k\}}\to 0$ for almost
every $x\in \Omega $, so, by Lebesgue theorem
$A_{2}=\varepsilon _{n}$.
For $(A_1)$, we have
\begin{align*}
&\int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla (T_{k}(u_{n})-
T_{k}(u))\varphi _{n}'dx\\
&=\int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla
T_{k}(u))]\nabla
(T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx\\
&\quad +\int_{\Omega }a(\nabla T_{k}(u))\nabla
(T_{k}(u_{n})-T_{k}(u))\varphi
_{n}'dx := A_{1.1}+A_{1.2}
\end{align*}
By (\ref{eP13}) , since $T_{k}(u_{n})\rightharpoonup T_{k}(u)$
weakly in $W_{0}^{1,p}(\Omega )$, we have
$A_{1.2}=\varepsilon _{n}$.
Thus
\begin{equation}
A=\int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))
]\nabla \big(T_{k}(u_{n})-T_{k}(u)\big)\varphi
_{n}'dx+\varepsilon _{n}. \label{eP18}
\end{equation}
By \eqref{eP15}, \eqref{eP16}, \eqref{eP17}, \eqref{eP18}
 and from \eqref{eP14}, we obtain
\begin{equation*}
\int_{\Omega }[a\big(\nabla T_{k}(u_{n})\big)-a\big(
\nabla T_{k}(u)\big)]\nabla \big(T_{k}(u_{n})-T_{k}(u)
\big) [\varphi _{n}'-b_{k}|\varphi _{n}|]
dx\leq \varepsilon _{n}.
\end{equation*}
Since $\varphi _{n}'-b_{k}|\varphi _{n}|\geq \frac{1}{2}$
with $a=1$ and $b=b_{k})$ and
\begin{gather*}
[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))]\nabla \big(T_{k}(u_{n})
-T_{k}(u)\big)\geq 0, \\
\int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))]
\nabla \big(T_{k}(u_{n})-T_{k}(u)\big)dx=\varepsilon _{n};
\end{gather*}
therefore,
\begin{equation*}
\langle -\Delta _{p}(T_{k}(u_{n}))+\Delta
_{p}(T_{k}(u)),T_{k}(u_{n})-T_{k}(u)\rangle \to 0.
\end{equation*}
Since $T_{k}(u_{n})\rightharpoonup T_{k}(u)$ weakly in
$W_{0}^{1,p}(\Omega )$,
\begin{gather*}
\langle -\Delta _{p}(T_{k}(u)),T_{k}(u_{n})-T_{k}(u)\rangle \to 0,\\
\langle -\Delta _{p}(T_{k}(u_{n})),T_{k}(u_{n})-T_{k}(u)\rangle
\to 0.
\end{gather*}
Since $-\Delta _{p}$ belongs to the class $(S^{+})$
(see \cite{Mu}), $T_{k}(u_{n})\to T_{k}(u)$ strongly in
$W_{0}^{1,p}(\Omega )$.
\end{proof}

\begin{lemma} \label{lem6}
The following to limit hold:
\begin{equation}
\begin{gathered}
  \lim_{k\to +\infty } [\sup_{n\in \mathbb{N}}
 \int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}dx] =0, \\
\lim_{k\to +\infty }[\sup_{n\in \mathbb{N}}
 \int_{\{|u_{n}|\geq k\}}|f(x,u_{n},\nabla u_{n})|dx]=0.
\end{gathered} \label{eP19}
\end{equation}
\end{lemma}

\begin{proof}
For the first limit, we define $\psi : \mathbb{R}\to \mathbb{R}^{+}$
by $\psi (-s)=-\psi (s)$ for all $s\in \mathbb{R}$ and
\[
\psi (s)=\begin{cases}
0 & \text{if }0\leq s\leq k-1, \\
s-(k-1) & \text{if } k-1\leq s\leq k, \\
1 & \text{if } s\geq k,
\end{cases}
\]
where $k>1$,
so that $\psi $ is continuous, bounded in $\mathbb{R}$ and
$\psi (u_{n})\in W_{0}^{1,p}(\Omega )$. We choose
$v=\psi (u_{n})$, as a test function in \eqref{eP9} we have
\begin{align*}
&\int_{\Omega }|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi
(u_{n})dx\\
&=\int_{\Omega }f(x,u_{n},\nabla u_{n})\psi
(u_{n})dx+\int_{\Omega }g_{n}\psi (u_{n})dx+\int_{\Omega }F\nabla
\psi (u_{n})dx.
\end{align*}
Using  Young's inequality, we obtain
\begin{align*}
\int_{\Omega }|\nabla \psi (u_{n})|^{p}dx
& \leq \int_{\Omega }f(x,u_{n},\nabla u_{n})\psi (u_{n})dx
+\int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx \\
&\quad + c\int_{\{ k-1<|u_{n}|<k\}}|F|
^{p'}dx+\frac{1}{2}\int_{\Omega }|\nabla \psi (u_{n})|^{p}dx.
\end{align*}
So that
\begin{equation}
\begin{aligned}
0 & \leq \frac{1}{2}\int_{\Omega }|\nabla \psi
(u_{n})|^{p}dx \\
& \leq  \int_{\Omega }f(x,u_{n},\nabla u_{n})\psi
(u_{n})dx+\int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx \\
&\quad +c\int_{\{ k-1<|u_{n}|<k\}}|F|^{p'}dx .
\end{aligned}
 \label{eP20}
\end{equation}
Using \eqref{eP4} and  that $\psi (s)$ has the same sign
as $s$, and that is zero if $|s|\leq k-1$, we get
\begin{align*}
\int_{\Omega }f(x,u_{n},\nabla u_{n})\psi (u_{n})dx
 &=\int_{\{|u_{n}|>k-1\}}f(x,u_{n},\nabla u_{n})\psi (u_{n})dx \\
& \leq \int_{\{|u_{n}|>k-1\}}[-\rho |\nabla
u_{n}|^{p}|\psi (u_{n})|+\alpha '| u_{n}|^{p-1}|\psi (u_{n})| \\
&\quad + \beta '|\nabla u_{n}|^{p-1}|\psi
(u_{n})|+a_{1}(x)|\psi (u_{n})|]dx.
\end{align*}
 From (\ref{eP20}), we have
\begin{equation}
\begin{aligned}
&\rho \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}| \psi (u_{n})|dx \\
&\leq \int_{\{|u_{n}|\geq k-1\}}|g_{n}|dx+c\int_{\{ k-1<|u_{n}|
<k\}}|F|^{p'}dx
  + \alpha '\int_{\{|u_{n}|>k-1\}}|u_{n}|^{p-1}|\psi (u_{n})|dx \\
&\quad + \beta '\int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p-1}|\psi (u_{n})|dx
+ \int_{\{|u_{n}|>k-1\}}a_{1}(x)|\psi (u_{n})|dx.
\end{aligned} \label{eP21}
\end{equation}
Since $u_{n}\to u$ in $L^{p}(\Omega )$, there exists
$v\in L^{p}(\Omega )$ such that $|u_{n}|\leq |v|$.
Since $|g_{n}|\leq |\widetilde {g}|$,
$|\widetilde {g}|\in L^{1}(\Omega )$ and $|\psi (s)|\leq 1$, we have
\begin{align*}
&\rho \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}| \psi (u_{n})|dx\\
&\leq \int_{\Omega }[|\widetilde {g}|+c|F|^{p'}+\alpha '|v|^{p-1}
 +a_{1}(x)]\chi_{\{|v|\geq k-1\}}dx
 + \beta '\int_{\{|v|>k-1\}}|\nabla u_{n}|^{p-1}dx \\
&\leq \int_{\Omega }r(x)\chi _{\{|v|\geq k-1\}}dx
 +\beta '\| u_{n}\|_{1,p}^{p-1}(\int_{\Omega }\chi _{\{|
v|\geq k-1\}}dx)^{1/p},
\end{align*}
where $r(x)=|\widetilde {g}|+c|F|^{p'}+\alpha '|v|^{p-1}+a_{1}(x)$.
We have $r\in L^{1}(\Omega )$ and
$(u_{n})_{n}$ is bounded in $W_{0}^{1,p}(\Omega )$, so that
\begin{equation*}
\lim_{k\to +\infty }[\sup_{n\in \mathbb{N}}
\int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx  ]=0.
\end{equation*}
Since
\begin{align*}
\int_{\{|u_{n}|\geq k\}} |\nabla u_{n}|^{p}dx
&=\int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx \\
&\leq \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx,
\end{align*}
it follows that
\begin{equation*}
\lim_{k\to +\infty }[\sup_{n\in \mathbb{N}}
\int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}dx]=0.
\end{equation*}

For the second limit, we let $l:\Omega \times \mathbb{R\times R}^{N}\to
\mathbb{R} $ defined by
\begin{equation*}
l(x,s,\xi )=f(x,s,\xi )-\alpha |s|^{p-1}\mathop{\rm sgn}(s)-\beta |
\xi |^{p-2}\xi -\big(|s|^{p-1}+|\xi |^{p-1}
+a_{1}(x)\big)\mathop{\rm sgn}(s).
\end{equation*}
 From \eqref{eP3}, we get
$l(x,s,\xi )s\leq -\rho |\xi |^{p}|s|$
for almost every $x\in \Omega $, and
for all $(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$.

By \eqref{eP20} and using  that $\psi (s)$ has the same
sign as $s$ and that it is zero if $|s|\leq k-1$, we have
\begin{align*}
0 & \leq \int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx \\
& \quad + c\int_{\{ k-1<|u_{n}|<k\}}|F|
^{p'}dx+\int_{\Omega }l(x,u_{n},\nabla u_{n})\psi (u_{n})dx \\
& \quad + \int_{\Omega }[\alpha '|u_{n}|^{p-1}+\beta
'|\nabla u_{n}|^{p-1}+a_{1}(x)]|\psi (u_{n})|dx.
\end{align*}
Since $l(x,u_{n},\nabla u_{n})\psi (u_{n})\leq -|l(x,u_{n},\nabla
u_{n})|\chi _{\{|u_{n}|\geq \ k\}}$, we have
\begin{align*}
\int_{\{|u_{n}|\geq \ k\}}|l(x,u_{n},\nabla u_{n})|dx
&\leq \int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx+c\int_{\{
 k-1<|u_{n}|<k\}}|F|^{p'}dx \\
&\quad + \int_{\{|u_{n}|\geq \ k-1\}}\alpha '|u_{n}|
^{p-1}|\psi (u_{n})|dx \\
&\quad + \int_{\{|u_{n}|\geq \ k-1\}}\beta '|\nabla u_{n}|^{p-1}|
\psi (u_{n})|dx \\
&\quad + \int_{\{|u_{n}|\geq \ k-1\}}a_{1}(x)|\psi (u_{n})| dx.
\end{align*}
In the same way as in the first limit, we prove that
\begin{equation*}
\lim_{k\to +\infty }[\sup_{n\in \mathbb{N}}
\int_{\{|u_{n}|\geq k\}}|l(x,u_{n},\nabla u_{n})|dx]=0.
\end{equation*}
Also
\begin{gather*}
|f(x,u_{n},\nabla u_{n})|\leq |l(x,u_{n},\nabla u_{n})|
+\alpha '|u_{n}|^{p-1}+\beta '|\nabla u_{n}|^{p-1}+a_{1}(x),
\\
\lim_{k\to +\infty }[\sup_{n\in \mathbb{N}}
\int_{\{|u_{n}|\geq k\}}|f(x,u_{n},\nabla u_{n})|dx ]=0.
\end{gather*}
\end{proof}

\begin{lemma}
The sequence $(u_{n})_{n}$ converges strongly to $u$ in
$W_{0}^{1,p}(\Omega ) $.
\end{lemma}

\begin{proof}
We begin by proving that the sequence $\{|\nabla u_{n}|
^{p}\}$ is equi-integrable in $L^{1}(\Omega )$.
Let $\varepsilon >0$ be fixed. Let now $E$ be a measurable subset
of $\Omega $, we have
\begin{equation*}
\int_{E}|\nabla u_{n}|^{p}dx=\int_{E\cap \{|u_{n}|
\leq k\}}|\nabla u_{n}|^{p}dx+\int_{E\cap \{|u_{n}|
>k\}}|\nabla u_{n}|^{p}dx.
\end{equation*}
By lemma \ref{lem6} there exists $k>0$ such that for all
$n\in \mathbb{N}$,
\[
\int_{\{|u_{n}|>k\}}|\nabla u_{n}|^{p}dx\leq \frac{
\varepsilon }{2}.
\]
For $k$ fixed, we have
\[
\int_{E\cap \{|u_{n}|\leq k\}}|\nabla u_{n}|^{p}dx
\leq \int_{E}|\nabla T_{k}(u_{n})|^{p}dx.
\]
Since $T_{k}(u_{n})$ converges strongly to $T_{k}(u)$ in
$W_{0}^{1,p}(\Omega )$, there exists $\gamma >0$ such that
\begin{equation*}
\mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N}\text{ \
}\int_{E}|\nabla T_{k}(u_{n})|^{p}dx\leq \frac{\varepsilon
}{2},
\end{equation*}
so that
\begin{equation*}
\forall n\in \mathbb{N}\ \ \int_{E\cap \{|u_{n}|\leq
k\}}|\nabla u_{n}|^{p}dx\leq \frac{\varepsilon }{2}.
\end{equation*}
Then, there exists $\gamma >0$ such that
\begin{equation*}
\mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N}\int_{E}|
\nabla u_{n}|^{p}dx\leq \varepsilon .
\end{equation*}
Therefore, the sequence $\{|\nabla u_{n}|^{p}\}$ is
equi-integrable in $L^{1}(\Omega )$. By Lemma \ref{lem5} we have
$\nabla u_{n}\to \nabla u$ for almost every $x\in \Omega $, so,
$|\nabla u_{n}|^{p}\to |\nabla u|^{p}$ strongly in $L^{1}(\Omega )$,
thus the sequence $(u_{n})_{n}$
converges strongly to $u$ in $W_{0}^{1,p}(\Omega )$.
\end{proof}

\begin{lemma}
The sequence $(f(x,u_{n},\nabla u_{n}))_{n}$ converges to
$f(x,u,\nabla u)$ in $L^{1}(\Omega )$.
\end{lemma}

\begin{proof}
We begin by proving that the sequence
$\{|f(x,u_{n},\nabla u_{n})|\}$ is equi-integrable in $L^{1}(\Omega )$.
Let $\varepsilon >0$ be fixed. Let now $E$ be a measurable subset of
$\Omega $, we have
\begin{align*}
&\int_{E}|f(x,u_{n},\nabla u_{n})|dx\\
&=\int_{E\cap \{|u_{n}|\leq k\}}|f(x,u_{n},\nabla u_{n})|dx
+\int_{E\cap\{|u_{n}|>k\}}|f(x,u_{n},\nabla u_{n})|dx.
\end{align*}
By Lemma \ref{lem6}, there exists $k>0$ such that
\begin{equation*}
\forall n\in \mathbb{N},\; \int_{E\cap \{|u_{n}|>k\}}|
f(x,u_{n},\nabla u_{n})|dx\leq \frac{\varepsilon }{2}.
\end{equation*}
When $k$ is fixed,  by \eqref{eP2} we have
\begin{equation*}
\int_{E\cap \{|u_{n}|\leq k\}}|f(x,u_{n},\nabla
u_{n})|
dx\leq \int_{E}[b_{k}|\nabla T_{k}(u_{n})|^{p}+\phi _{k}(x)
]dx.
\end{equation*}
Since $\phi _{k}\in L^{1}(\Omega )$ and
$T_{k}(u_{n})\to T_{k}(u) $  strongly in
$W_{0}^{1,p}(\Omega )$, there exists $\gamma >0$ such that
\begin{equation*}
\mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N}
\; \int_{E}[b_{k}|\nabla T_{k}(u_{n})|^{p}+\phi _{k}(x)]dx
\leq \frac{\varepsilon }{2},
\end{equation*}
so that
\begin{equation*}
\forall n\in \mathbb{N}\; \int_{E\cap \{|u_{n}|\leq
k\}}|f(x,u_{n},\nabla u_{n})|dx\leq \frac{\varepsilon
}{2}.
\end{equation*}
Therefore,  the sequence $\{|f(x,u_{n},\nabla u_{n})|\}_{n}$ is
equi-integrable in $L^{1}(\Omega )$. Since
$f:\Omega \times \mathbb{R} \times \mathbb{R}^{N}\to \mathbb{R}$
is a Carath\'{e}odory function, we have
$f(x,u_{n},\nabla u_{n})\to \ f(x,u,\nabla u)$
for almost every $x\in \Omega $. so
$f(x,u_{n},\nabla u_{n})\to \ f(x,u,\nabla u)$ strongly in
$L^{1}(\Omega )$.
\end{proof}

Going back to the the proof of Theorem 1.1, by
 \eqref{eP9} we have that for every
$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty}(\Omega )$,
\begin{equation*}
\int_{\Omega }|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla
v\,dx=\int_{\Omega }f(x,u_{n},\nabla u_{n})v\,dx+\int_{\Omega
}g_{n}v+\int_{\Omega }F\nabla v.
\end{equation*}
 As $n$ approaches infinity, we get that for every
 $v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$,
\begin{equation*}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx
=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }gv+\int_{\Omega}F\nabla v.
\end{equation*}
Thus the problem
\begin{gather*}
-\Delta _{p}u  = f(x,u,\nabla u)+\mu \quad\text{in } \Omega , \\
u = 0 \quad\text{on }\partial \Omega
\end{gather*}
admits a solution $u\in W_{0}^{1,p}(\Omega )$ in the sense that
$f(x,u,\nabla u)\in L^{1}(\Omega )$, and for every
$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$,
\begin{equation*}
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla
v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }v\,d\mu.
\end{equation*}
\end{proof}

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\end{document}