\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath} % used for R in Real numbers
\pagestyle{myheadings}

\markboth{Strongly nonlinear degenerated unilateral problems}
{ Elhoussine Azroul, Abdelmoujib Benkirane \&  Ouidad Filali }

\begin{document}
\setcounter{page}{49}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations,
Conference 09, 2002, pp 49--64. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Strongly nonlinear degenerated unilateral problems with $L^1$ data
%
\thanks{ {\em Mathematics Subject Classifications:} 35J15, 35J70, 35J85.
\hfil\break\indent
{\em Key words:} Weighted Sobolev spaces, Hardy inequality, \hfil\break\indent
quasilinear degenerated elliptic operators.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002.} }

\date{}
\author{Elhoussine Azroul, Abdelmoujib Benkirane \&  Ouidad Filali}
\maketitle

\begin{abstract}
 In this paper, we study the existence of solutions for strongly nonlinear
 degenerated unilateral problems associated to nonlinear operators
 of the form $Au+g(x,u,\nabla u)$. Here $A$ is a Leray-Lions operator
 acting from $W_0^{1,p}(\Omega,w)$ into its dual,
 while $g(x,s,\xi)$ is a nonlinear term which has a growth condition
 with respect to $\xi$ and no growth condition with respect to $s$,
 the second term belongs to $L^{1}(\Omega )$.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{coro}[theorem]{Corollary}
\numberwithin{equation}{section}


\section{Introduction}

Let $\Omega$ be a bounded open set of $\mathbb{R}^N$, $p$ be a real
number such that $1<p<\infty$ and $w= \{w_i(x)$, $0\leq i\leq N\}$
be a vector of weight functions on $\Omega$, i.e. each $w_i(x)$ is
a measurable a.e. strictly positive on $\Omega$. Let
$W_0^{1,p}(\Omega,w)$ be the weighted Sobolev space associated
with the vector $w$. Let $A$ be a nonlinear operator from
$W_0^{1,p}(\Omega,w)$ into its dual $W^{-1,p'}(\Omega,w^*)$, i.e.
$$
Au=-{\rm div} (a(x,u,\nabla u)). 
$$
In the non-degenerated case, Bensoussan, Boccardo and Murat have proved in [4],
the existence of a solution for the quasilinear equation of the form,
$$
Au+g(x,u,\nabla u)=f.
$$
They assume that $g$ is a nonlinearity having natural growth with respect to
$|\nabla u|$ (of order $p$), and which satisfies the sign-condition and
$f\in W^{-1,p'}(\Omega)$. Recently, in weighted case, Akdim, Azroul and
Benkirane have first in [2] extended the last result to weighted Sobolev
spaces and in  [3] the authors have studied the following degenerated
unilateral problem:
\begin{gather*}
\langle Au,v-u\rangle +\int_\Omega g(x,u,\nabla u)(v-u)\,dx\geq
\langle f,v-u\rangle \quad \forall\,  v\in K_\psi\cap L^\infty(\Omega)\\
u\in W_0^{1,p}(\Omega,w)\quad  u\geq\psi\mbox{ a.e. in }\Omega\\
g(x,u,\nabla u)\in L^1(\Omega)\quad  g(x,u,\nabla u)u\in L^1(\Omega),
\end{gather*}
where $K_\psi=\{v\in W_0^{1,p}(\Omega,w),\; v\geq \psi\mbox{ a.e. in }\Omega\}$,
 with $\psi$  a measurable function on $\Omega$ such that
 $\psi^+\in W_0^{1,p}(\Omega,w)\cap L^{\infty}(\Omega)$ and
 $f\in W^{-1,p'}(\Omega,w^*)$.
The purpose of this paper, is to study the previous problems for
$f\in L^1(\Omega)$. More precisely, we prove the existence theorem for the
following degenerated unilateral problem:
\begin{equation} \label{1.1}
\begin{gathered}
\langle Au,T_{k}(v-u)\rangle +\int_\Omega g(x,u,\nabla u)T_{k}(v-u)\,dx\geq
\langle f,T_{k}(v-u)\rangle \\
\mbox{ for all }\ v\in K_\psi \quad \mbox{ and all } k>0\\
u\in W_0^{1,p}(\Omega,w)\quad  u\geq\psi\mbox{ a.e. in }\Omega\\
g(x,u,\nabla u)\in L^1(\Omega).
\end{gathered} 
\end{equation}
For that, we assume in addition that the nonlinearity $g$ satisfies some
coercivity conditions (see (2.10)).

Concerning the existence result for the degenerated elliptic equations where
the second member lies in the dual $ W^{-1,p'}(\Omega,w^*)$
(resp. for the quasilinear equation where the second member is
in $L^1(\Omega)$), we refer the reader to [6-7-8](resp. [1-2]).

\paragraph{Remarks}
\begin{enumerate}
\item[1)] Note that the use of the truncation operator in \eqref{1.1} is
justified by the fact that the solution does not in general belong to
$L^{\infty}(\Omega)$ for $f\in L^{1}(\Omega)$.
\item[2)] An other work in this direction can be found in [5] in non-weighted
case.
\end{enumerate}

The paper is organized as follows: Section 2 contains some preliminaries and
is concerned with the basic assumptions and some technical lemmas.
In section 3, we state and prove main results.
The last section is devoted to an example which illustrates our abstract
 conditions.

\section{Preliminaries}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^N\ (N\geq 1)$,
let $1<p<\infty$, and let $w= \{w_i(x),\ 0\leq i\leq N\}$ be a vector of
weight functions, i.e. every component $w_i(x)$ is a measurable
 function which is strictly positive $a.e$. in $\Omega$ satisfying the
integrability conditions
$$ w_i\in L_{\rm loc}^1(\Omega), \quad
w_i^{-\frac{1}{p-1}}\in L_{\rm loc}^1(\Omega) \eqno{(2.1)}
$$
for $0\leq i\leq N$.

We define the weighted space $L^p(\Omega,\gamma)$, where $\gamma$ is a
weight function on $\Omega$ by
$$L^p(\Omega,\gamma)=\{u=u(x),\ u\gamma^{1/p}\in L^p(\Omega)\}
$$
with the norm
$$
\|u\|_{p,\gamma}=\Big(\int_\Omega|u(x)|^p \gamma(x)\,dx\Big)^{1/p}.
$$
We denote by $W^{1,p}(\Omega,w)$ the space of all real-valued functions
$u\in L^p(\Omega,w_0)$ such that the derivatives in the sense of
distributions satisfy
$$
\frac{\partial u}{\partial x_i}\in L^p(\Omega,w_i) \quad \mbox{for all }
i=1,\dots ,N.
$$
This set of functions defines a Banach space under the norm
$$
\|u\|_{1,p,w}=\Big(\int_\Omega |u(x)|^pw_0(x)\,dx+\sum_{i=1}^N
\int_\Omega |\frac{\partial u(x)}{\partial x_i}|^pw_i(x)\,dx\Big)^{1/p}.
\eqno{(2.2)}
$$
Since we shall deal with the Dirichlet problem, we shall use the space
$$X=W_0^{1,p}(\Omega,w)$$
defined  as the closure of $C_0^{\infty}(\Omega)$ with respect to the norm
(2.2). Note that, $C_0^{\infty}(\Omega)$ is dense in $W_0^{1,p}(\Omega,w)$
and $(X,\|.\|_{1,p,w})$ is a reflexive Banach space.

We recall that the dual space of weighted Sobolev space 
$W_0^{1,p}(\Omega,w)$ is equivalent to $W^{-1,p'}(\Omega,w^*)$,
where $w^*=\{w_i^*=w_i^{1-p'},\ \forall i=0,\dots ,N \}$, where $p'$ is the
conjugate of $p$ i.e. $p'=p/(p-1)$ (for more details we refer to [7]).

Now we state the following assumptions:
\paragraph{Assumption (H1)}
The expression
$$
\||u|\|=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}
{\partial x_i}|^pw_i(x)\,dx\Big)^{1/p}
$$
is a norm defined on $W_0^{1,p}(\Omega,w)$ and is equivalent to the 
norm (2.2). There exists a weight function $\sigma$ on $\Omega$ such that 
$\sigma \in L^{1}(\Omega)$ and $\sigma^{1-q'} \in L^{1}(\Omega)$ 
for some parameter $q$, so that $ 1<q<p+p'$ such that the Hardy inequality,
$$
\Big(\int_\Omega|u(x)|^q\sigma(x) \,dx\Big)^{\frac{1}{q}}
\leq c\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}
{\partial x_i}|^pw_i(x)\,dx\Big)^{1/p}, \eqno{(2.3)}
$$
holds for every $u\in W_0^{1,p}(\Omega,w)$ with a constant $c>0$ independent
of $u$, and moreover, the imbedding
$$
W_0^{1,p}(\Omega,w)\hookrightarrow\hookrightarrow L^q(\Omega,\sigma),
\eqno{(2.4)}
$$
determined by the inequality $(2.3)$ is compact.
Note that $(W_0^{1,p}(\Omega,w),\||.|\|)$ is a uniformly convex and thus
reflexive Banach space. Let $A$ be a nonlinear operator from
$W_0^{1,p}(\Omega, w)$ into its dual $W^{-1,p'}(\Omega, w^*)$ defined by
$$ Au=-{\rm div}(a(x,u,\nabla u)),
$$
where  $a:\Omega\times\mathbb{R}\times\mathbb{R}^N\to \mathbb{R}^N$ is a Carath\'eodory
vector-function  satisfying for a.e $x\in \Omega$,
for all $s\in \mathbb{R} $ and $\xi$, $\eta $ in $\mathbb{R}^N$ with $\xi\not=\eta$.

\paragraph{Assumption (H2)}
\begin{gather*}
|a_i(x,s,\xi)|\leq \beta w_i^{1/p}(x)[k(x)+\sigma^{1/p'}|s|^{\frac{q}{p'}}
+\sum_{j=1}^Nw_j^{1/p'}(x)|\xi_j|^{p-1}],\quad i=1,\dots ,N,\tag{2.4}\\
[a(x,s, \xi)-a(x,s,\eta)](\xi-\eta)>0,\tag{2.5}\\
a(x,s, \xi).\xi\geq\alpha \sum_{i=1}^Nw_i|\xi_i|^{p},\tag{2.6}
\end{gather*}
where $k(x)$ is a positive  function in $L^{p'}(\Omega)$ and
$\alpha,\ \beta$ are strictly positive constants.

\paragraph{Assumption (H3)}
Let $g(x,s,\xi)$ be a Carath\'eodory function satisfying
\begin{gather*}
g(x,s,\xi)s\geq 0\tag{2.7}\\
|g(x,s,\xi)|\leq b(|s|)\Big(\sum_{i=1}^Nw_i|\xi_i|^{p}+c(x)\Big),\tag{2.8}\\
|g(x,s,\xi)|\geq \rho _{2}\sum_{i=1}^Nw_i|\xi_i|^{p}
\quad \mbox{for }|s|>\rho _{1}\tag{2.9}
\end{gather*}
where $b:\mathbb{R}^+\to \mathbb{R}^+$ is a continuous increasing function and $c(x)$
is a positive function which lies in $L^1(\Omega)$, $c\geq 0$
and $\rho _{1}>0$, $\rho _{2}>0$.
We consider,
$$f\in L^{1}(\Omega).\eqno{(2.10)}$$
Now we recall some lemmas which will be used later.

\begin{lemma}[{cf. [2]}] \label{lemcvfe}
Let $g\in L^r(\Omega,\gamma)$ and let $ g_n\in L^r(\Omega,\gamma)$, with
$\|g_n\|_{r,\gamma} \leq c$ ($1<r<\infty$).
 If $g_n(x)\to  g(x)$  a.e. in $\Omega$, then $g_n\rightharpoonup g$
 weakly in $L^r(\Omega,\gamma)$, where $\gamma $ is a weight function
on $\Omega$.
\end{lemma}

\begin{lemma}[{cf. [1]}]\label{lemcvfotr}
Assume that (H1) holds. Let $u\in W_0^{1,p}(\Omega,w)$, and let
$T_k(u)$, $k\in \mathbb{R}^+$, be the usual truncation then
$T_k(u)\in W_0^{1,p}(\Omega,w)$. Moreover, we have
$$T_k(u)\to  u \mbox{ strongly in } W_0^{1,p}(\Omega,w).
$$
\end{lemma}

\begin{lemma}[{cf. [2]}]\label{lemcvfo1}
Assume that (H1) and (H2) are satisfied, and let
 $(u_n)$ be a sequence of $ W_0^{1,p}(\Omega,w)$ such that
$u_n\rightharpoonup u$ weakly in $W_0^{1,p}(\Omega,w)$
and
$$\int_\Omega[a(x,u_n,\nabla u_n)-a(x,u_n,\nabla u)]\nabla(u_n-u)\; dx\to  0.$$
Then,
$u_n\to  u$  strongly in $W_0^{1,p}(\Omega,w)$.
\end{lemma}

\begin{lemma}[{cf. [1]}] \label{lemcvfo2}
Assume that (H1) holds, let $(u_n)\in W_0^{1,p}(\Omega,w)$ such that
$u_n\rightharpoonup u$ weakly in $W_0^{1,p}(\Omega,w)$,then
$T_{k}u_n\rightharpoonup T_{k}u$ weakly in $W_0^{1,p}(\Omega,w)$.
\end{lemma}

\section{Main result}

Let $\psi$ be a measurable function with values in $\mathbb{R}$ such that,
$$
\psi^+\in W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega).\eqno{(3.1)}
$$
and let
$$
K_\psi=\{v\in W_0^{1,p}(\Omega,w)\; v\geq \psi\mbox{ a.e. in }\Omega\}.
$$
Consider the nonlinear problem with Dirichlet boundary condition,
$$ % \label{p}
\begin{gathered}
\langle Au,T_{k}(v-u)\rangle +\int_\Omega g(x,u,\nabla u)T_{k}(v-u)\,dx
\geq \langle f,T_{k}(v-u)\rangle \\
\mbox{for all }v\in K_\psi \;\;\mbox{and all }k>0\\
u\in W_0^{1,p}(\Omega,w)\quad  u\geq\psi\mbox{ a.e. in }\Omega\\
g(x,u,\nabla u)\in L^1(\Omega),
\end{gathered} \eqno{(3.2)}
$$
where $u$ is the solution of this problem.

\begin{theorem}\label{thm3.1}
Under the assumptions (H1)-(H3), (2.10) and (3.1), there exists at least
one solution of (3.2).
\end{theorem}

\paragraph{Remarks}  %3.2
\begin{enumerate}
\item[1)] Theorem \ref{thm3.1} generalizes to weighted case the analogous in [5].
\item[2)] In the particular case when $w_0(x)\equiv 1$, we can replace
(H1) by the conditions: There exists
$s \in ]\frac{N}{p},\infty[\cap [\frac{1}{p-1},\infty[$
such that $w_i^{-s}\in L^1(\Omega)$ for all $i=1,\dots,N$,
(which is an integrability condition, stronger than (2.1)), since
$$
\||u|\|_X=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}
{\partial x_i}|^p w_i(x)\,dx\Big)^{1/p}
$$
is a norm defined on $W_0^{1,p}(\Omega,w)$ and equivalent to (2.2) and
also the following imbeddings hold:
$$
W_0^{1,p}(\Omega,w)\hookrightarrow L^q(\Omega)
$$
for  $1\leq q<p_1^*$, if $ps<N(s+1)$, and $q\geq 1$ is arbitrary if
$ps \geq N(s+1)$, where $p_1=ps/(s+1)$ and $p_1^*$ is devoted
the Sobolev conjugate of $p_1$ (for more details see [2,7]).
Hence the hypotheses (H1) is satisfied for $\sigma \equiv 1$.
\end{enumerate}

\paragraph{Proof of Theorem \ref{thm3.1}}
Consider the sequence of approximate problems:
$$
\begin{gathered}
\langle Au_n,T_{k}(v-u_n)\rangle +\int_\Omega g(x,u_n,\nabla u_n)T_{k}(v-u_n)
\,dx\geq \langle f_n,T_{k}(v-u_n)\rangle \\
\mbox{for all }v\in K_\psi \cap L^\infty(\Omega)\quad\mbox{and all }k>0\\
u_n\in W_0^{1,p}(\Omega,w)\quad  u_n\geq\psi\quad \mbox{a.e.  in }\Omega\\
g(x,u_n,\nabla u_n)\in L^1(\Omega),
\end{gathered} \eqno{(3.3)}
$$
where $f_n$ is a sequence of smooth functions which converges strongly to $f$
in $L^1(\Omega)$ with $\|f_n\|_{L^1(\Omega)}\leq C$. For some constant $C$.
By Theorem 6.1 and Lemma 6.2 of [1] or via Theorem 4.1 of [3] there exists
at least one solution $u_n$ of (3.3).
In order to pass to the limit in the approximate problem (3.3),
we claim that:
{\bf Assertion(1)}
$$(u_n)_n \mbox { is bounded in }W_0^{1,p}(\Omega,w)\eqno{(3.4)}$$
{\bf Assertion(2)}
$$ \nabla T_k(u_n) \to  \nabla T_k(u)\quad \mbox {strongly in }
\prod_{i=1}^{N}L^{p}(\Omega,w_i)\eqno{(3.5)}
$$
which implies $ \nabla u_n \to  \nabla u $ a.e in $\Omega $\\
{\bf Assertion(3)}
$$g(x,u_n,\nabla u_n)\to  g(x,u,\nabla u)\quad\mbox {strongly in }
L^{1}(\Omega )\eqno{(3.6)}
$$
We can pass to the limit in the approximate problems (3.3), indeed,
\begin{multline*}
\int_\Omega a(x,u_n,\nabla u_n)\nabla T_k(v-u_n)\,dx
+ \int_\Omega g(x,u_n,\nabla u_n)T_k(v-u_n)\,dx \\
\geq \int_\Omega f_n T_k(v-u_n)\,dx
\end{multline*}
For all $v\in K_{\psi }\cap L^\infty(\Omega)$ and $k>0$.
From (2.5) and (3.4) we deduce that $a(x,u_n,\nabla u_n)$ is bounded
in $\prod_{i=1}^{N}L^{p'}(\Omega,w^*_i)$.
Using (3.5) we obtain
$$
\nabla u_n \to  \nabla u  \quad\mbox{a.e in } \Omega. \eqno(3.7)
$$
Hence, we get
$$
a(x,u_n,\nabla u_n)\to  a(x,u,\nabla u)\quad\mbox{a.e in } \Omega \eqno(3.8)
$$
which implies with Lemma \ref{lemcvfe} that
$$
a(x,u_n,\nabla u_n)\rightharpoonup a(x,u,\nabla u)\quad\mbox{weakly in }
\prod_{i=1}^{N}L^{p'}(\Omega,w^*_i).\eqno(3.9)
$$
On the other hand, let $ v \in L^\infty(\Omega)$ and set
$h= k+||v||_\infty$, then
\begin{eqnarray*}
|\frac{\partial T_k(v-u_n)}{\partial x_i}|w_i^{1/p}& = & (\chi_{|v-u_n|\leq k}|\frac{\partial (v-u_n)}{\partial x_i}|)w_i^{1/p}\\
& \leq & \chi_{|u_n|\leq k+||v||_\infty}(|\frac{\partial v}{\partial x_i}|+|\frac{\partial u_n}{\partial x_i}|)w_i^{1/p}\\
& \leq & |\frac{\partial v}{\partial x_i}|w_i^{1/p}+ |\frac{\partial T_h(u_n)}{\partial x_i}|w_i^{1/p}
\end{eqnarray*}
for $i=1,\dots ,N$.
Which implies by using the Vitali's theorem with (3.5) and (3.7) that
$$
\nabla T_k(v-u_n)\to  \nabla T_k(v-u)\quad\mbox{strongly in }
\prod_{i=1}^{N}L^{p}(\Omega,w_i) \eqno(3.10)
$$
for any $ v\in W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega)$.
From (3.9) and (3.10) we can pass to the limit in the first term of (3.3).
Since $g(x,u_n,\nabla u_n)\to  g(x,u,\nabla u)$ strongly in $L^1(\Omega)$
and $f_n\to  f$ strongly in $L^1(\Omega)$, then we can pass to the limit in
$$
\langle A(u_n),T_k(v-u_n)\rangle + \int_\Omega g(x,u_n,\nabla{u_n})T_k(v-u_n)
\,dx \geq \int_\Omega f_n T_k(v-u_n)\,dx.
$$
This allows to consider the problem
$$
\begin{gathered}
\langle Au,T_{k}(v-u)\rangle +\int_\Omega g(x,u,\nabla u)T_{k}(v-u)\,dx
\geq \langle f,T_{k}(v-u)\rangle \\
\mbox{for all }v\in K_\psi \cap L^\infty (\Omega )\mbox{ and all } k>0\\
u\in W_0^{1,p}(\Omega,w)\quad  u\geq\psi\quad \mbox{a.e.  in }\Omega\\
g(x,u,\nabla u)\in L^1(\Omega),
\end{gathered} \eqno{(3.11)}
$$
Set $\phi = T_m(v)$ as a test function, where $m\geq \|\psi^+\|_\infty $ and
$v\in K_\psi$, then $\phi \in K_\psi \cap L^\infty(\Omega)$.
Multiplying (3.11) by $\phi$, we obtain
$$
\begin{aligned} % \label{Pm}
\langle Au,T_{k}(T_m(v_m)-u)\rangle
+\int_\Omega g(x,u,\nabla u)T_{k}(T_m(v_m)-u)\,dx&\\
\geq \int_\Omega f T_{k}(T_m(v_m)-u)\,dx&
\end{aligned} \eqno(3.12)
$$
From Lemma \ref{lemcvfotr} and using the Vitali's theorem, we have
$$
\nabla T_{k}(T_m(v)-u)\to  \nabla T_k(v-u)
\quad\mbox{strongly in }\prod_{i=1}^{N}L^{p}(\Omega,w_i) .
$$
Finally, passing to the limit in (3.12) as $m$ tends to infinity, we obtain:
$$
\langle Au,T_{k}(v-u)\rangle +\int_\Omega g(x,u,\nabla u)T_{k}(v-u)\,dx
\geq \int_\Omega f T_{k}(v-u)\,dx
$$
for any $ v\in W_0^{1,p}(\Omega,w)$, $v\geq\psi$ a.e.  in $\Omega$.

\paragraph{Proof of assertion 1:}
We consider the sequence of approximate problems (3.3).
By Theorem 6.1 and Lemma 6.2 of [1], there exists at least one solution
$u_n$ of (3.3).
Let $v=\psi^+$ as test function in (3.3), then
$$
\langle Au_n,T_{k}(\psi^+-u_n)\rangle
+\int_\Omega g(x,u_n,\nabla u_n)T_{k}(\psi^+-u_n)\,dx
\geq \int_\Omega f_n T_{k}(\psi^+-u_n)\,dx.
$$
 Since $u_n-\psi^+$ and $u_n$ have the same sign, we obtain by using (2.8)
 and $\|f_n\|_{L^1(\Omega)}\leq C $,
$$
\int_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n})\nabla(u_n-\psi^+)\,dx
\leq \int_\Omega f_n T_k(u_n-\psi^+)\,dx\leq kC
\eqno(3.13)
$$
which implies that
$$
\int_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n})\nabla{u_n}\,dx
\leq Ck+\int_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n})|\nabla{\psi^+}|\,dx
$$
using Young's inequality, we obtain
\begin{align*}
\int&_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n})\nabla{u_n}\,dx\\
\leq& Ck  + \sum_{i=1}^N\int_{|u_n-\psi^+|\leq k}\frac{\eta^{p'}}{p'}
|a_i(x,u_n,\nabla{u_n})|^{p'}w_i^{1-p'}\,dx\\
&+ \sum_{i=1}^N\int_{|u_n-\psi^+|\leq k}\frac{1}{p}\frac{1}{\eta^p}w_i
|\frac{\partial {\psi^+}}{\partial x_i}|^p\,dx,
\end{align*}
where $\eta$ is a positive constant. From (2.5) we have
\begin{align*}
\int&_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n}) \nabla{u_n}\,dx \\
\leq& C_1+\frac{\eta^{p'}}{p'}\beta^{p'}N\int_\Omega k^{p'}(x)\,dx
+ \frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n-\psi^+|\leq k}\sigma |u_n|^q\,dx\\
&+\frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n-\psi^+|\leq k}
 \sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p\,dx\\
\leq & C_2+ \frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n|
\leq ||\psi^+||_\infty+k}\sigma |u_n|^q\,dx+\frac{\eta^{p'}}{p'}\beta^{p'}N
\int_{|u_n-\psi^+|\leq k}\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p
\,dx\\
\leq & C_2+ \frac{\eta^{p'}}{p'}\beta^{p'}N(||\psi^+||_\infty+k)^q
\int_\Omega\sigma \,dx+\frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n-\psi^+
|\leq k}\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p\,dx.
\end{align*}
Consequently, using (2.7) and since $\sigma \in L^1(\Omega) $, we have
$$
\int_{|u_n-\psi^+|\leq k}\alpha\sum_{i=1}^N w_i|
\frac{\partial u_n}{\partial x_i}|^p\,dx\leq
C_3+\frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n-\psi^+|
\leq k}\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p\,dx,
$$
we choose $0<\eta<\frac{1}{\beta}(\frac{\alpha p'}{N})^{1/p'}$, this implies
$$
\int_{|u_n-\psi^+|\leq k}\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p\,dx\leq C \eqno(3.14)
$$
On the other hand, from (3.13) and $a(x,s,\xi)\xi\geq 0$
\begin{align*}
\int_\Omega& g(x,u_n,\nabla{u_n})T_k(u_n-\psi^+)\,dx\\
 \leq & k \int_\Omega |f_n|\,dx-\int_{|u_n-\psi^+|\leq k}a(x,u_n,
 \nabla{u_n})\nabla(u_n-\psi^+)\,dx \\
\leq & C_1 + \int_{|u_n-\psi^+|\leq k}a(x,u_n,\nabla{u_n})|\nabla \psi^+|\,dx.
\end{align*}
As in the proof of (3.14) we can show that
\begin{multline*}
\int_\Omega g(x,u_n,\nabla{u_n})T_k(u_n-\psi^+)\,dx \\
\leq  C_2 + \frac{\eta^{p'}}{p'}\beta^{p'}N\int_{|u_n-\psi^+|\leq k}
\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p\,dx
\leq  C_3,
\end{multline*}
where $C_1, C_2$ and $C_3$ are positive constants.
When $|u_n-\psi^+|>k$, we have $T_k(u_n-\psi^+)=+k $(or $-k$),
and since $T_k(u_n-\psi^+), u_n-\psi^+, u_n $ and
$ g(x,u_n,\nabla{u_n})$ have the same sign, we obtain
\begin{multline*}
\int_{|u_n-\psi^+|\leq k} g(x,u_n,\nabla{u_n})T_k(u_n-\psi^+)\,dx\\
+ \int_{|u_n-\psi^+|> k} g(x,u_n,\nabla{u_n})T_k(u_n-\psi^+)\,dx
\leq  C_3
\end{multline*}
which gives
$$
k \int_{|u_n-\psi^+|> k} |g(x,u_n,\nabla{u_n})|\,dx \leq C_3.
$$
From (2.10), we have
$$
|g(x,u_n,\nabla{u_n})|\geq \rho_2 \sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p \;\;\;\mbox{for}\;\;\;|u_n|\geq \rho_1
$$
Choosing $k>\rho_1+||\psi^+||_\infty$, then $|u_n-\psi^+|>k$ implies
$|u_n| > \rho_1$.
We deduce that
$$
\int_{|u_n-\psi^+| > k}\rho_2 \sum_{i=1}^N w_i|\frac{\partial u_n}
{\partial x_i}|^p\,dx\leq C_4 .\eqno(3.15)
$$
Finally, combining (3.14) and (3.15) we have $|||u_n|||\leq C$.

\paragraph{Proof of assertion 2}
Let $k\geq \|\psi^+\|$ and  $ \delta =(\frac{b(k)}{2\alpha})^2$. Set
$ \varphi (s)= se^{\delta s^2}$, $z_n=T_k(u_n)-T_k(u)$,
$\eta=e^{-4\delta k^2}$, and $v_n=u_n-\eta \varphi (z_n)$.
By the choice of $k$, the above test function is admissible for
(3.3). Multiplying (3.3) by $v_n$, for $h>0$, we obtain
$$
\langle A(u_n),T_h(\eta \varphi (z_n))\rangle  +
 \int_\Omega g(x,u_n,\nabla{u_n})T_h(\eta \varphi (z_n))\,dx
\leq  \int_\Omega f_n T_h(\eta \varphi (z_n))\,dx.
$$
Choosing $h>2k$, we have $|\eta \varphi (z_n)| \leq |z_n|\leq 2k<h$
and
$$
\langle A(u_n), \varphi (z_n) \rangle + \int_\Omega g(x,u_n,\nabla{u_n})
 \varphi (z_n)\,dx\leq \int_\Omega f_n \varphi (z_n)\,dx.
$$
As $n \to  \infty, $ we have $f_n \to  f $ strongly in $L^1(\Omega)$, and $\varphi (z_n)\rightharpoonup 0 $ weak$^*$ in $L^\infty(\Omega)$. Hence we have
$$
\int_\Omega f_n \varphi (z_n)\,dx \to  0.
$$
Since $ g(x,u_n,\nabla{u_n}) \varphi (z_n)\geq 0 $ on the subset
${|u_n(x)|>k}$, this implies
$$
\langle A(u_n), \varphi (z_n) \rangle + \int_{|u_n|\leq k} g(x,u_n,
\nabla{u_n}) \varphi (z_n)\,dx\leq \varepsilon(n), \eqno(3.16)
$$
where $\varepsilon(n)$ is a real number which converge to zero when $n$ tends to infinity.\\
On the other hand
\begin{align*}
\langle A(u_n),\varphi (z_n)\rangle
=& \int_{|u_n|\leq k} a(x,u_n,\nabla{u_n})(\nabla T_k(u_n)-\nabla T_k(u))\varphi'(z_n) \,dx \\
&+ \int_{|u_n|> k} a(x,u_n,\nabla{u_n})(\nabla T_k(u_n)
-\nabla T_k(u))\varphi'(z_n) \,dx\\
=& \int_\Omega a(x,u_n,\nabla{u_n})(\nabla T_k(u_n)
-\nabla T_k(u))\varphi'(z_n) \,dx \\
& - \int_{|u_n|> k} a(x,u_n,\nabla{u_n})\nabla T_k(u)\varphi'(z_n) \,dx\\
=& \int_\Omega (a(x,u_n,\nabla T_k(u_n))- a(x,u_n,\nabla T_k(u)))
(\nabla T_k(u_n)\\
&-\nabla T_k(u))\varphi'(z_n) \,dx\\
&+ \int_\Omega a(x,u_n,\nabla T_k(u)))(\nabla T_k(u_n)
-\nabla T_k(u))\varphi'(z_n) \,dx\\
&- \int_{|u_n|> k} a(x,u_n,\nabla{u_n})\nabla T_k(u)\varphi'(z_n) \,dx.
\end{align*}
Since $\nabla T_{k}(u)\chi_{\{|u_n|> k\}}\to  0 $ strongly in
$ \prod_{i=1}^{N}L^{p}(\Omega,w_i)$, and from (2.3), (2.5) and (3.4)
we have $(a(x,u_n,\nabla{u_n})\varphi'(z_n))_n $ is bounded in
$\prod_{i=1}^{N}L^{p'}(\Omega,w_i^*)$, then
$$
- \int_{|u_n|> k} a(x,u_n,\nabla{u_n})\nabla T_k(u)\varphi'(z_n) \,dx
=\varepsilon(n)\to  0 \quad \mbox{as } n \to  \infty.
$$
Moreover, since $u_n \rightharpoonup u $ weakly in $W_0^{1,p}(\Omega, w)$,
 by Lemma \ref{lemcvfo1} we have
$$
T_k(u_n)\rightharpoonup T_k(u)\quad \mbox{weakly in } W_0^{1,p}(\Omega, w).
$$
Then
$$ \nabla T_k(u_n )\rightharpoonup \nabla T_k(u)\quad \mbox{weakly in }
\prod_{i=1}^{N}L^{p}(\Omega,w_i).
$$
Since the sequence $ (a(x,u_n,\nabla T_k(u))\varphi'(z_n))_n $ converges
strongly in the space $ \prod_{i=1}^{N}L^{p'}(\Omega,w_i^*)$,
$$
\int_\Omega a(x,u_n,\nabla T_k(u))(\nabla T_k(u_n)
-\nabla T_k(u))\varphi'(z_n) \,dx
=\varepsilon(n)\to  0 \quad \mbox{as }n\to  \infty,
$$
and
$$
\begin{aligned}
\langle A(u_n), \varphi (z_n)\rangle
=&\int_\Omega (a(x,u_n,\nabla T_k(u_n))
- a(x,u_n,\nabla T_k(u)))\\
&\times (\nabla T_k(u_n)-\nabla T_k(u))\varphi'(z_n)\,dx +\varepsilon(n).
\end{aligned} \eqno(3.17)
$$
On the other hand
\begin{align*}
\big|\int&_{|u_n|\leq k} g(x,u_n,\nabla u_n) \varphi (z_n)\,dx\big|\\
  \leq& \int_{|u_n|\leq k}b(k)( \sum_{i=1}^N w_i|\frac{\partial u_n}
  {\partial x_i}|^p+c(x)) |\varphi (z_n)|\,dx\\
\leq & \int_{|u_n|\leq k}b(k)c(x)|\varphi (z_n)|\,dx+ \int_{|u_n|\leq k}
\sum_{i=1}^N w_i|\frac{\partial u_n}{\partial x_i}|^p b(k)|\varphi (z_n)|\,dx \\
\leq & \int_{|u_n|\leq k}b(k)c(x)|\varphi (z_n)|\,dx+ \int_{|u_n|\leq k}
\sum_{i=1}^N a_i(x,u_n,\nabla u_n)\frac{\partial u_n}{\partial x_i}
\frac{b(k)}{\alpha}|\varphi (z_n)|\,dx\\
\leq & \int_{|u_n|\leq k}b(k)c(x)|\varphi (z_n)|\,dx+ \int_{|u_n|\leq k}
\sum_{i=1}^N [a_i(x,u_n,\nabla T_k(u_n)) \\
& - a_i(x,u_n,\nabla T_k(u))](\frac{\partial T_k(u_n)}{\partial x_i}
-\frac{\partial T_k(u)}{\partial x_i})|\varphi (z_n)|\frac{b(k)}{\alpha}\,dx \\
& + \int_{|u_n|\leq k} \sum_{i=1}^N a_i(x,u_n,\nabla T_k(u))
(\frac{\partial T_k(u_n)}{\partial x_i}-\frac{\partial T_k(u)}{\partial x_i})
|\varphi (z_n)|\frac{b(k)}{\alpha}\,dx\\
& + \int_{|u_n|\leq k} \sum_{i=1}^N a_i(x,u_n,\nabla T_k(u_n))
\frac{\partial T_k(u)}{\partial x_i}|\varphi (z_n)|\frac{b(k)}{\alpha}\,dx.
\end{align*}
Since
$\nabla T_k(u_n)\rightharpoonup \nabla T_k(u)$ weakly in
$\prod_{i=1}^{N}L^{p}(\Omega,w_i)$  and
$$
a(x,u_n,\nabla T_k(u)) |\varphi (z_n)| \to  0 \quad \mbox{ strongly in }
 \prod_{i=1}^{N}L^{p'}(\Omega,w_i^*),
$$
it follows that
$$
\int_\Omega \sum_{i=1}^N a_i(x,u_n,\nabla T_k(u))(\frac{\partial T_k(u_n)}
{\partial x_i}-\frac{\partial T_k(u)}{\partial x_i})|\varphi (z_n)|
\frac{b(k)}{\alpha}\,dx= \varepsilon(n).
$$
From (2.5) and (3.4), $(a(x,u_n,\nabla T_k(u_n)))_n$  converges
weakly in $\prod_{i=1}^{N}L^{p'}(\Omega,w_i^*)$.
Since $ \nabla T_k(u)|\varphi (z_n)|\frac{b(k)}{\alpha}\to  0 $ strongly
in $\prod_{i=1}^{N}L^{p}(\Omega,w_i)$,
$$
\int_\Omega \sum_{i=1}^N a_i(x,u_n,\nabla T_k(u_n))\frac{\partial T_k(u)}{\partial x_i})|\varphi (z_n)|\frac{b(k)}{\alpha}\,dx= \varepsilon(n).
$$
Moreover, since $\int_{|u_n|\leq k}b(k)c(x)|\varphi (z_n)|\,dx
= \varepsilon(n)$, we have
\begin{align*}
|\int&_{|u_n|\leq k} g(x,u_n,\nabla u_n) \varphi (z_n)\,dx|\\
\leq& \int_\Omega \sum_{i=1}^N
\big[a_i(x,u_n,\nabla T_k(u_n))- a_i(x,u_n,\nabla T_k(u))\big]\\
&\times (\frac{\partial T_k(u_n)}{\partial x_i}
-\frac{\partial T_k(u)}{\partial x_i})|\varphi (z_n)|\frac{b(k)}{\alpha}\,dx
+\varepsilon(n)
\end{align*}
which with (3.16) and (3.17) gives
\begin{multline*}
\int_\Omega \big[a(x,u_n,\nabla T_k(u_n))- a(x,u_n,\nabla T_k(u))\big]\\
\times(\nabla T_k(u_n)-\nabla T_k(u))(\varphi'(z_n)
-\frac{b(k)}{\alpha}|\varphi (z_n)|)\,dx\leq \varepsilon(n).
\end{multline*}
Choosing, $\delta \geq (b(k)/(2\alpha))^2$, we obtain for all $s \in \mathbb{R}$
$$
\varphi'(s)-\frac{b(k)}{\alpha}|\varphi (s)|\geq \frac{1}{2};
$$
thus,
$$
\frac{1}{2}\int_\Omega [a(x,u_n,\nabla T_k(u_n))- a(x,u_n,\nabla T_k(u))](\nabla T_k(u_n)-\nabla T_k(u))\,dx \leq \varepsilon(n).
$$
Then
$$
\int_\Omega \big[a(x,u_n,\nabla T_k(u_n))- a(x,u_n,\nabla T_k(u))\big]
(\nabla T_k(u_n)-\nabla T_k(u))\,dx \to  0
$$
as $n \to  \infty$.
Moreover, $T_k(u_n)\rightharpoonup T_k(u)$ weakly in $ W_0^{1,p}(\Omega,w)
$ and in view of Lemma \ref{lemcvfo1}, we have
$T_k(u_n)\to  T_k(u)$ as $n \to  \infty$ strongly in $W_0^{1,p}(\Omega,w)$;
hence,
$$
\nabla T_k(u_n) \to  \nabla T_k(u) \;\;\mbox{strongly in}\;\; \prod_{i=1}^{N}L^{p}(\Omega,w_i).
$$
Consequently, there exists a subsequence still denoted by $(u_n)_n$ such that,
$\nabla u_n \to  \nabla u \quad \mbox{a.e in } \Omega$.

\paragraph{Proof of assertion 3}
From (3.5) we deduce that
$$
g(x,u_n,\nabla u_n) \to  g(x,u,\nabla u) \;\;\; \mbox{a.e in } \Omega.
\eqno(3.18)
$$
For any measurable subset $E$ of $\Omega$ and any $m>0$ we have
$$
\begin{aligned}
&\int_E |g(x,u_n,\nabla u_n)|\,dx \\
&\leq \int_{E\cap \{|u_n|\leq m\}} |g(x,u_n,\nabla u_n)|\,dx
+\int_{E\cap \{|u_n|> m\}} |g(x,u_n,\nabla u_n)|\,dx.
\end{aligned} \eqno(3.19)
$$
By  (2.9), (3.5) and by using Vitali's theorem, we have for $\varepsilon > 0$
there exists  $\rho(\varepsilon,m)>0$ such that for $\rho(\varepsilon,m)>|E|$
we have
$$
\int_{E\cap \{|u_n|\leq m\}} |g(x,u_n,\nabla u_n)|\,dx
\leq \frac{\varepsilon}{2}\;\;\;\forall n. \eqno(3.20)
$$
Now let $v_n=u_n-S_m(u_n)$ where for $m>1$,
$$
S_m(s)=\begin{cases} 0 & |s|\leq m-1 \\
\frac{s}{|s|}  & |s|\geq m \\
S'_m(s)=1   &  m-1\leq |s| \leq m.
\end{cases}
$$
Note that: If $u_n \leq m-1$, we have $S_m(u_n)\leq 0 $ and
 $ v_n \geq u_n \geq \psi$;
if $u_n \geq m-1 $, we have $ 0\leq S_m(u_n)\leq 1$ and
$$
u_n-S_m(u_n)\geq u_n-1\geq m-2\geq \psi \quad \mbox{for }
m\geq 2+||\psi^+||_\infty.
$$
Then, $v_n$ is admissible for (3.3). So, multiplying (3.3) by $v_n$ we
obtain
$$
\langle A(u_n), T_k(S_m(u_n))\rangle + \int_\Omega g(x,u_n,\nabla u_n)
T_k(S_m(u_n))\,dx \leq \int_\Omega f_n T_k(S_m(u_n))\,dx.
$$
Which by choosing $k\geq 1$ implies
\begin{multline*}
\int_\Omega a(x,u_n,\nabla u_n)\nabla u_n s'_m(u_n)\,dx+\int_\Omega g(x,u_n,
\nabla u_n)S_m(u_n)\,dx \\
\leq \int_\Omega f_n S_m(u_n)\,dx,
\end{multline*}
i.e,
\begin{multline*}
\int_{m-1\leq |u_n|\leq m} a(x,u_n,\nabla u_n)\nabla u_n \,dx
+\int_{|u_n|>m-1} g(x,u_n,\nabla u_n)S_m(u_n)\,dx \\
\leq \int_{|u_n|>m-1} |f_n|\,dx.
\end{multline*}
Since $a(x,u_n,\nabla u_n)\nabla u_n \geq 0$,
$$
\int_{|u_n|>m-1} g(x,u_n,\nabla u_n)S_m(u_n)\,dx
\leq \int_{|u_n|>m-1} |f_n|\,dx.
$$
Since $S_m(u_n)$ and $u_n$ have the same sign,
$$
\int_{|u_n|>m-1} |g(x,u_n,\nabla u_n)| |S_m(u_n)|\,dx
\leq \int_{|u_n|>m-1} |f_n|\,dx
$$
and
$$
\int_{|u_n|> m} |g(x,u_n,\nabla u_n)| \,dx \leq \int_{|u_n|>m-1} |f_n|\,dx.
$$
Since $f_n \to  f $ strongly in $L^1(\Omega)$ and since
$ |\{|u_n|>m-1\}|\to  0 $ uniformly in $n$ when $m \to  \infty$
(due to the fact that $\sigma^{1-q'}\in L^1(\Omega)$), there exists
$m(\varepsilon)>1$ such that
$$
\int_{|u_n|>m-1} |f_n|\,dx \leq \frac{\varepsilon}{2}\quad \forall n.
$$
Then
$$
\int_{|u_n|> m} |g(x,u_n,\nabla u_n)| \,dx\leq
 \frac{\varepsilon}{2}\quad \forall n. \eqno(3.21)
$$
From (3.19), (3.20) and (3.21), we have
$$
\int_E |g(x,u_n,\nabla u_n)| \,dx\leq \varepsilon \quad \forall n. \eqno(3.22)
$$
Then $(g(x,u_n,\nabla u_n))_n $ is equi-integrable.
Thanks to (3.18), (3.22) and Vitali's theorem yields,
$$
g(x,u_n,\nabla u_n)\to  g(x,u,\nabla u) \quad\mbox{strongly in } L^1(\Omega).
$$
\section{Example}
The following example is closely inspired from the one used in [1,2].
Let $\Omega$ be a bounded domain of $\mathbb{R}^N (N \geq 1)$ satisfying the
cone condition and let $\psi$ be a real valued measurable function such that
$\psi^+ \in W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega)$.
Let us consider the Carath\'eodory functions
$$
a_i(x,s,\xi)=w_i|\xi_i|^{p-1}{\rm sgn}(\xi) \quad\mbox{for }i=0,\dots,N
$$
and
$$
g(x,s,\xi)=\rho s |s|^r\sum_{i=1}^Nw_i|\xi_i|^p\quad \mbox{with} \quad
\rho >0,
$$
where $w_i(x$) are a given weight functions on $\Omega$ satisfying:
$$ w_i(x)\equiv \mbox{some weight function $w(x)$ in $\Omega$
for all $i=0,\dots,N$.}
$$
Then, we consider the Hardy inequality (2.3) in the form,
$$\Big(\int_\Omega|u(x)|^q\sigma(x)\,dx\Big)^{1/q}
\leq c\Big(\int_\Omega |\nabla u(x)|^pw\Big)^{1/p}.%\eqno{(5.4)}
$$
It is easy to show that the functions $a_i(x,s,\xi)$ satisfy the growth
condition (2.5) and the coercivity (2.7). Also the Carath\'eodory function
$g(x,s,\xi)$ satisfies
the conditions (2.8), (2.9) and (2.10), in fact, concerning (2.10) we have,
$$
|g(x,s,\xi)|=\rho |s|^{r+1} \sum_{i=1}^Nw_i|\xi_i|^p.
$$
Then
$$
|g(x,s,\xi)|\geq\rho |\rho|^{r+1}\sum_{i=1}^Nw_i|\xi_i|^p
\quad \mbox{for } |s|>\rho_1\geq 1.
$$
Choosing for example $\rho_1=1 $ and $ \rho_2=\rho>0$.
On the other hand, the monotonicity condition is satisfied. In
fact,
\begin{multline*}
 \sum_{i=1}^N(a_i(x,s,\xi)-a_i(x,s,\hat\xi))(\xi_i-\hat\xi_i) \\
 = w(x)\sum_{i=1}^N(|\xi_i|^{p-1}\mathop{\rm sgn}\xi_i-|\hat\xi_i|^{p-1}
\mathop{\rm sgn}\hat \xi_i)(\xi_i-\hat\xi_i)>0
\end{multline*}
for almost all $x\in \Omega$ and for all $\xi,\hat\xi\in \mathbb{R}^N$
with $\xi\neq \hat\xi$, since $w>0$ a.e. in $\Omega$.
In particular, let us use the special weight functions $w$ and
$\sigma$ expressed in terms of the distance to the boundary
$\partial \Omega$. Denote
$d(x)=\mathop{\rm dist}(x,\partial\Omega)$ and set
$$ w(x)=d^\lambda(x),\quad \sigma(x)=d^\mu(x).
$$
In this case, the Hardy inequality reads
$$ \Big(\int_\Omega |u(x)|^q\,d^\mu(x)\,dx\Big)^{1/q}
\leq c\Big(\int_\Omega |\nabla u(x)|^p\,d^\lambda(x)\,dx \Big)^{1/p}\,.%
$$
The corresponding imbedding is compact if:\\
(i) For, $1< p\leq q<\infty$,
$$
\lambda< p-1,\quad \frac{N}{q}-\frac{N}{p}+1\geq 0,\quad
\frac{\mu}{q}-\frac{\lambda}{p}+\frac{N}{q}-\frac{N}{p}+1>0,\eqno(4.1)
$$
(ii) For $1\leq q<p<\infty$,
$$
\lambda<p-1,\quad \frac{\mu}{q}-\frac{\lambda}{p}+\frac{1}{q}
-\frac{1}{p}+1>0,\eqno(4.2)
$$
(iii) For $q>1$,
$$
\frac{1}{q'-1}>\mu >-1.\eqno(4.3)
$$

\begin{coro} \label{coro4.1}
If $f\in L^1(\Omega)$, the following problem:
\begin{gather*}
\int _{|v-u|\leq k}\sum_{i=1}^N d^\lambda(x)|
\frac{\partial u}{\partial x_i}|^{p-1}\mathop{\rm sgn}
(\frac{\partial u}{\partial x_i})\frac{\partial (v-u)}{\partial x_i}\,dx\\
+\int_{\Omega}\rho u|u|^r \sum_{i=1}^N
d^\lambda(x)|\frac{\partial u}{\partial x_i}|^p T_k(v-u)\, dx
\geq \int_{\Omega}fT_k(v-u)\,dx.\\
u\in W_0^{1,p}(\Omega,d^\lambda), u\geq \psi,\quad \rho u |u|^r \sum_{i=1}^N
d^\lambda(x)|\frac{\partial u}{\partial x_i}|^p\in L^1(\Omega) \\
v \in W_0^{1,p}(\Omega,d^\lambda),\quad
\mbox{for all $v\geq \psi$ and all $k>0,$}
\end{gather*}
has at least one solution.

\end{coro}

\paragraph{Remarks}
\begin{enumerate}
\item[1)] Note that conditions (4.1) and (4.2) are sufficient to show the
 compact imbedding (2.4) (cf. [6, example 1], [8, example 1.5]
 and [10, theorem 19.17, 19.22]).
\item[2)] Condition (4.3) is sufficient for (2.3) to hold (cf. [9 p.p 40-41]).
\end{enumerate}


\begin{thebibliography}{00} \frenchspacing

\bibitem{[1]} Y. A{\sc kdim},
{\em Sur certains probl\`emes quasi-lin\'eaires et
unilat\'eraux d\'eg\'en\'er\'es ou singuliers, }
{Th\`ese de Doctorat, Universit\'e de F\`es, (2002).}

\bibitem{[2]} Y. A{\sc kdim}, E. A{\sc zroul} and A. B{\sc enkirane},
{\em Existence of solutions for quasilinear degenerated elliptic equations, }
{Electronic J. Diff. Eqns., vol. {\bf 2001} N 71 (2001) 1-19.}

\bibitem{[3]} Y. A{\sc kdim}, E. A{\sc zroul} and A. B{\sc enkirane},
{\em Existence of solutions for quasilinear degenerated elliptic unilateral
problem, }{to appear in J. Math. Blaise-Pascal}.

\bibitem{[4]} A. B{\sc ensoussan}, L. B{\sc occardo} and F. M{\sc urat},
{\em On a non linear partial differential equation having natural growth
terms and unbounded solution,}{ Ann. Inst. Henri Poincar\'e {\bf 5} $N^°4$
(1988), 347-364.}

\bibitem{[5]} A. B{\sc enkirane} and A. E{\sc lmahi},
{\em Strongly nonlinear elliptic unilateral problems having natural
growth terms and $L^1$ data,}{ Rendiconti di Matematica, Serie VII vol.
{\bf 18},  (1998), 289-303.}

\bibitem{[6]} P. D{\sc rabek}, A. K{\sc ufner} and V. M{\sc ustonen},
 {\em Pseudo-monotonicity and degenerated or singular elliptic operators,}
 { Bull. Austral. Math. Soc. Vol. {\bf 58} (1998), 213-221.}

\bibitem{[7]} P. D{\sc rabek}, A. K{\sc ufner} and F. N{\sc icolosi},
{\em Quasilinear elliptic equations with degenerations and singularities,}
{  De Gruyter Series in Nonlinear Analysis and Applications, Berlin New York,
(1997).}

\bibitem{[8]} P. D{\sc rabek}, A. K{\sc ufner} and F. N{\sc icolosi},
{\em Nonlinear elliptic equations, singular and degenerate cases,}
{University of West Bohemia, (1996).}

\bibitem{[9]} A. Kufner,
{\em Weighted Sobolev Spaces,} John Wiley and Sons, (1985).

\bibitem{[10]} B. Opic and A. Kufner,
{\em Hardy-type inequalities}, Pitman Research Notes in Mathematics Series
{\bf 219} (Longman Scientific and Technical, Harlow, 1990).

\end{thebibliography}
\noindent \textsc{Elhoussine Azroul}  (e-mail: elazroul@caramail.com)\\
\textsc{Abdelmoujib Benkirane} (e-mail: abdelmoujib@iam.net.ma )\\
\textsc{Ouidad Filali} (e-mail: ouidadf@hotmail.com)\\[2pt]
D\'epartement de Math\'ematiques et Informatique, \\
Facult\'e des Sciences Dhar-Mahraz, B.P. 1796 Atlas, F\`es, Maroc.

\end{document}