\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 73--81.\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}}
\setcounter{page}{73}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Leray Lions degenerated problem]
{Leray Lions degenerated problem with general growth condition}

\author[Y. Akdim, A. Benkirane, M. Rhoudaf \hfil EJDE/Conf/14 \hfilneg]
{Youssef Akdim, Abdelmoujib Benkirane, Mohamed Rhoudaf}

\address{Youssef Akdim \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas F\`es, Maroc}
\email{akdimyoussef@yahoo.fr}

\address{Abdelmoujib Benkirane \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas F\`es, Maroc}
\email{abenkirane@fsdmfes.ac.ma}

\address{Mohamed Rhoudaf \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas F\`es, Maroc}
\email{rhoudaf\_mohamed@yahoo.fr}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J15, 35J70, 35J85}
\keywords{Weighted Sobolev spaces; truncations;
$L^1$-version of Minty's lemma; \hfill\break\indent
Hardy inequality}

\begin{abstract}
 In this paper, we study the existence of solutions for the
 nonlinear degenerated elliptic problem
 $$
 -{\mathop{\rm div}}(a(x,u,\nabla u)) = F\quad \text{in }  \Omega,
 $$
 where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 2$,
 $a:\Omega\times\mathbb{R}\times\mathbb{R}^N\to\mathbb{R}^N $ is
 a Carath\'eodory  function satisfying  the coercivity condition,
 but they verify  the general growth condition and only the
 large monotonicity. The  second term $F$ belongs to
 $W^{-1, p'}(\Omega, w^*)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\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 (i.e., every component $w_i(x)$ is
a measurable function which is  positive a.e. in $\Omega$)
satisfying some integrability conditions.
 The Objective of this paper is to study the following problem,
in the weighted Sobolev space,
\begin{equation}
\begin{gathered}
Au=F \quad \mbox{in } \Omega,\\
u=0 \quad \mbox{on }  \partial \Omega,
 \end{gathered} \label{e1.1}
\end{equation}
where $A$ is a Leray-Lions operator from $W_0^{1,p}(\Omega,w)$
to its dual $W^{-1,p'}(\Omega,w^*)$. The principal part $A$ is a differential
operator of second order in divergence form defined as,
$$
Au=- \mathop{\rm div}(a(x,u,\nabla u))
$$
where $a: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N$ is a
Carath\'eodory function (that is,
measurable with respect to $x$ in $\Omega $ for every $(x, \xi)$
in $\mathbb{R} \times \mathbb{R}^N$ and continuous with respect to $(s, \xi)$ in
$\mathbb{R} \times \mathbb{R}^N$ for almost every $x$ in $\Omega$) satisfying
the coercivity condition. But, on the one hand, they verify the
general growth condition in this form
$$
|a_i(x, s, \xi)|\leq \beta w_i^{1/p}(x)
 [ k(x) + |s|^{p-1} +  {\sum_{j=1}^N}w_j^{1/p'}(x)
[\gamma (s)|\xi_j|]^{p-1}]
$$
  instead the classical growth
condition, where we introduce some continuous function
$\gamma(s)$. This type of the growth condition can not guaranteed
the existence of the weak solution (See Remark \ref{rem4}), for
that we overcame this difficulty by introduce an other type of
solution so-called T-solution. On the other hand, they verify only
the large monotonicity, that is
$$
[a(x,s,\xi) - a(x,s,\eta)](\xi-\eta) \geq 0 \quad
\mbox{for all } (\xi,\eta) \in \mathbb{R}^N\times\mathbb{R}^N.
$$
We overcome this difficulty
of the not strict monotonicity  thanks to a technique (the
$L^1$-version of Minty's lemma) similar to the one used in
\cite{boccardo}. Recently in \cite{bocardo} Boccardo has studied
the problem
 \eqref{e1.1} in the classical Sobolev space $W_0^{1,p}(\Omega)$. For
that the author has proved the existence of the T-solution. Other
works in this direction can be found in \cite{boccardo} (where the
right hand side $f \in L^1$ and $F \in L^{p'}(\Omega)$) and in
\cite{Abdellaou} (where the existence and nonexistence results for
some quasilinear elliptic equations involving the P-Laplaces have
proved).

\section{Preliminaries}
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, i.e., every component $w_i(x)$ is
a measurable function which is strictly positive a.e. in $\Omega$.
Further, we suppose in all our considerations that
(for each $w_i \neq 0$.)
\begin{gather}
w_i \in L^1_{\rm loc}(\Omega),\label{2.1} \\
w_i^{\frac{-1}{p-1}} \in L^1_{\rm loc}(\Omega),\label{2.2}
\end{gather}
for any $0\leq i\leq N$.

 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 fulfill
$$
\frac{\partial u}{\partial x_i} \in L^p(\Omega, w_i)\quad
\mbox{for } i=1, \dots, N.
$$
Which is a Banach space under the norm
 \begin{equation}
\|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}|^p
w_i(x)\,dx \Big]^{1/p}.\label{2.3}\end{equation} The
condition  \eqref{2.1}  implies that $C_0^{\infty}(\Omega)$ is a space of
$W^{1,p}(\Omega,w)$ and consequently, we can introduce the
subspace $W_0^{1,p}(\Omega,w)$ of $W^{1,p}(\Omega,w)$ as the
closure of  $C_0^{\infty}(\Omega)$ with respect to the norm
 $(\ref{2.3})$. Moreover, condition  $(\ref{2.2})$ implies that
$W^{1,p}(\Omega,w)$ as well as $W_0^{1,p}(\Omega,w)$ are reflexive
Banach spaces.

 We recall that the dual space of weighted Sobolev spaces
$W_0^{1,p}(\Omega,w)$ is equivalent to $W^{-1, p'}(\Omega, w^{*})$,
 where $w^{*} = \{w_i^{*} = w_i^{1-p'}$, $i = 0, \dots, N \}$ and
where $p'$ is the conjugate of $p$ i.e. $p' = \frac{p}{p-1}$.

\section{Basic assumptions and statement of results}

\subsection*{Assumption (H1)}  The expression
\begin{equation}
\||u|\|_X=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}
{\partial x_i}|^pw_i(x)\,dx\Big)^{1/p}\label{h2.6}
\end{equation}
is a norm defined on $X$ and is equivalent to the norm (\ref{2.3}).
\\
There exist a weight function $\sigma$ on $\Omega$ and a
parameter $q$, $1<q<\infty$, such that the Hardy inequality,
\begin{equation}
\Big(\int_\Omega|u(x)|^q\sigma \,dx\Big)^{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},\label{h2.7}
\end{equation}
holds for every $u\in X$ with a constant $c>0$ independent of $u$,
and moreover, the imbedding
\begin{equation}
X\hookrightarrow\hookrightarrow L^q(\Omega, \sigma), \label{h2.8}
\end{equation}
expressed by the inequality \eqref{h2.7} is compact.

Note that $(X,\||.|\|_X)$ is a uniformly convex (and thus
reflexive) Banach space.

\begin{remark} \label{rmk3.1} \rm
If we assume that  $w_0(x)\equiv 1$ and in addition the
integrability condition: There exists
$ \nu \in ]\frac{N}{P},{+\infty}\, [\cap[\frac{1}{P-1},{+\infty}[$
such that
\begin{equation}
w_i^{-\nu} \in L^1(\Omega)
 \quad \mbox{for all } i=1,\dots,N. \label{2.4}
\end{equation}
 Note that the assumptions  \eqref{2.1} and  \eqref{2.4}
 imply that,
 \begin{equation}
\||u\|| =  \Big({ \sum_{i=1}^N}{\int_{\Omega}|\frac{\partial u}
 {\partial x_i}|^p w_i(x)\,dx}\Big)^{1/p},\label{2.5}
\end{equation}
 is a norm defined on $W_0^{1,p}(\Omega,w)$ and its equivalent to
$(\ref{2.3})$  and that, the imbedding
 \begin{equation}
W_0^{1,p}(\Omega,w) \hookrightarrow  L^p(\Omega), \label{2.6}
\end{equation}
is compact  for all $1\leq q\leq p_1^*$  if
 $p.\nu<N(\nu+1)$ and for all $ q \geq 1 $  if
 $p.\nu \geq N(\nu+1)$ where $p_1=\frac{p\nu}{\nu+1}$ and $p_1^*$ is the
Sobolev conjugate of $p_1$ [see \cite{drabek}, pp 30-31].
\end{remark}

\subsection*{Assumption (H2)}
\begin{gather}
|a_i(x, s, \xi)|\leq \beta w_i^{1/p}(x) [ k(x) + |s|^{p-1} +  {
\sum_{j=1}^N}w_j^{1/p'}(x)[\gamma (s)|\xi_j|]^{p-1}],
\label{2.7} \\
[a(x,s,\xi) - a(x,s,\eta)](\xi-\eta) \geq 0 \quad  \mbox{for  all }
 (\xi,\eta) \in \mathbb{R}^N\times\mathbb{R}^N, \label{2.8} \\
a(x, s, \xi).\xi \geq \alpha {\sum_{i=1}^N}w_i |\xi_i|^p, \label{2.9}
\end{gather}
where $k(x)$ is a positive function in $L^{p'}(\Omega)$, $\gamma (s)$ is a
continuous function  and $\alpha$, $\beta$ are strictly positive
constants.

 We recall that, for $k>1$ and $s$ in $\mathbb{R}$, the truncation is
defined as,
$$
T_k(s)=\begin{cases}
s & \mbox{if }  s \leq k\\
k \frac{s}{|s|} & \mbox{if } |s|>k.
\end{cases}
$$

\section{Existence results}
Consider the  problem
\begin{equation} \label{P}
\begin{gathered}
u \in W_0^{1,p}(\Omega,w),\quad  F \in W^{-1,p'}(\Omega,w^*) \\
  - \mathop{\rm div} (a(x,u,\nabla u)) = F\quad \text{in }  \Omega  \\
u  =  0\quad \mbox{on }  \partial \Omega.
\end{gathered}
\end{equation}

\begin{definition} \label{def3.1} \rm
 A function $u$ in $W_0^{1,p}(\Omega,w)$ is a $T$-solution of
\eqref{P} if
\[
\int_{\Omega} a(x,u,\nabla u) \nabla T_k[u-\varphi] \,dx
= \langle F, T_k[u-\varphi] \rangle \quad
\forall \varphi \in W_0^{1,p}(\Omega,w) \cap L^{\infty}(\Omega).
\]
\end{definition}

\begin{theorem} \label{thm1}
 Assume that  (H1) and(H2). Then the problem \eqref{P}
has at least one $T$-solution $u$.
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
Recall that an existence result for the problem \eqref{P} can be found
 in \cite{drabekk} by using the approach of pseudo monotonicity with
some particular growths condition, that is $\gamma (s) = 1$.
\end {remark}

\begin{remark} \label{rmk3.3} \rm
In \cite{drabek} the authors  study the problem \eqref{P}
 under the  strong hypotheses
\begin{gather*}
[a(x,s,\xi)-a(x,s,\eta)](\xi-\eta)>0,\quad
\mbox{for all }  \xi \neq \eta \in \mathbb{R}^N, \label{2.10}\\
|a_i(x, s,\xi)|\leq \beta w_i^{1/p}(x)  [ k(x) + |s|^{p-1} +
{ \sum_{j=1}^N}w_j^{1/p'}(x)|\xi_j|^{p-1}],
 \label{2.11}
\end{gather*}
 instead of $(\ref{2.8})$ and  $(\ref{2.7})$ (respectively ).
Then the operator $A$  associated to the problem \eqref{P}
 verifies the $(S^+)$ condition and is  coercive.
Hence $A$ is surjective from $W_0^{1,p}(\Omega,w)$ into its dual
  $W^{-1, p'}(\Omega, w^{*})$.
\end{remark}

 \begin{proof}[Proof of Theorem \ref{thm1}]
 Consider the approximate problem
\begin{equation} \label{Pn}
\begin{gathered}
u_n \in W_0^{1,p}(\Omega,w) \\
 - \mathop{\rm div} (a(x,T_n(u_n),\nabla u_n)) = F.
\end{gathered}
\end{equation}
under the following assumptions:

\noindent {\bf Assertion (a): A priori estimates}
The problem \eqref{Pn} has a solution by a classical
result in \cite{drabekk}. Moreover, by using  $u_n$ as test function
in \eqref{Pn} we have,
$$
\int_{\Omega} a(x,T_n(u_n),\nabla u_n).\nabla u_n \,dx =
\int_{\Omega} Fu_n \,dx.
$$
Thanks to assumption $(\ref{2.9})$, we have
$$
\int_{\Omega} a(x,T_n(u_n),\nabla u_n).\nabla u_n
\,dx \geq \alpha {\sum_{i=1}^N}{\int_{\Omega}|\frac{\partial
u_n}{\partial x_i}|^p w_i(x) \,dx} = \alpha \||u_n\||^p
$$
i.e.,
$$
\alpha \||u_n\||^p \leq  \langle F,u_n \rangle  \leq
\|F\|_{-1,p',w^*} \||u_n\||,
$$
which implies $\alpha \||u_n\||^p \leq C_1 \||u_n\||$ for $p >1$,
with $C_1$ is a constant positive,
then the sequence ${u_n}$ is bounded in $W_0^{1,p}(\Omega,w)$,
thus, there exists a function $u \in W_0^{1,p}(\Omega,w)$ and a
subsequence $u_{n_j} $ such that $u_{n_j}$ converges weakly to $u$
in $W_0^{1,p}(\Omega,w)$.

\noindent{\bf Assertion (b)}
 We shall prove that for $\varphi$ in
$W_0^{1,p}(\Omega,w) \cap L^{\infty}(\Omega)$,
 we have
 \begin{equation}
 \int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla T_k[u_{n_j}-\varphi] \,dx
 \leq \langle F, T_k[u_{n_j}-\varphi] \rangle. \label{3.1}
\end{equation}
Let $n_j$ large enough  $(n_j>k+\| \varphi \|_{L^{\infty}(\Omega)})$,
we have by choosing $T_k[u_{n_j}-\varphi]$
as test function in \eqref{Pn}
$$
\int_{\Omega} a(x,u_{n_j},\nabla u_{n_j})
\nabla T_k[u_{n_j}-\varphi] \,dx= \langle F, T_k[u_{n_j}-\varphi] \rangle,
$$
i.e.,
\begin{align*}
&\int_{\Omega} a(x,u_{n_j},\nabla u_{n_j}) \nabla T_k[u_{n_j}-\varphi] \,dx
+ \int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla
T_k[u_{n_j}-\varphi] \,dx \\
&- \int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla T_k[u_{n_j}-\varphi] \,dx\\
&= \langle F, T_k[u_{n_j}-\varphi] \rangle,
\end{align*}
which implies
\begin{equation}
\begin{aligned}
\int_{\Omega} [a(x,u_{n_j},\nabla u_{n_j})-a(x,u_{n_j},\nabla \varphi)]
\nabla T_k[u_{n_j}-\varphi] \,dx& \\
 + \int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla T_k[u_{n_j}-\varphi] \,dx
&= \langle F, T_k[u_{n_j}-\varphi] \rangle.
\end{aligned}\label{3.2}
\end{equation}
Thanks to assumption  $(\ref{2.8})$ and the definition of truncating function,
we have,
\begin{equation}
\int_{\Omega} [a(x,u_{n_j},\nabla u_{n_j})-a(x,u_{n_j},
\nabla \varphi)]\nabla T_k[u_{n_j}-\varphi] \,dx \geq 0. \label{3.3}
\end{equation}
 Combining  $(\ref{3.2})$ and  $(\ref{3.3})$, we obtain  $(\ref{3.1})$.

\noindent{\bf Assertion (c)}  We claim that,
$$
 \int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla T_k[u_{n_j}-\varphi] \,dx
 \to \int_{\Omega} a(x,u,\nabla \varphi) \nabla T_k[u-\varphi] \,dx
$$
and that
$$ \langle F, T_k[u_{n_j}-\varphi] \rangle \to \langle F, T_k[u-\varphi]
\rangle.
$$
Indeed, first, by virtue of  $u_{n_j} \rightharpoonup u$ weakly in
$W_0^{1,p}(\Omega,w)$, and  \cite[Lemma 2.4]{Akdimaz}, we have
\begin{equation}
T_k(u_{n_j}-\varphi) \rightharpoonup
T_k(u-\varphi) \quad \mbox {in }  W_0^{1.p}(\Omega,w). \label{3.4}
\end{equation}
Which gives
\begin{equation}
\frac {\partial T_k}{\partial x_i}(u_{n_j}-\varphi) \rightharpoonup
\frac {\partial T_k}{\partial x_i}(u_{n_j}-\varphi ) \quad
\mbox{in }L^p(\Omega,w_i).\label{3.5}
\end{equation}
Note that $\nabla T_k(u_{n_j}-\varphi )$ is not zero on the subset
$\{ x \in \Omega : |u_{n_j}-\varphi(x)|\leq k \}$
(subset of
$ \{x \in \Omega : |u_{n_j}(x)|\leq k + \|\varphi \|_{L^\infty (\Omega )}\},$).
Thus thanks to assumption  $(\ref{2.7})$, we have
\begin{equation}
\begin{aligned}
|a_i(x,u_{n_j},\nabla \varphi)|^{p'} w_i^{-p'/p}
&\leq [k(x)+|u_{n_j}|^{p-1}+\gamma_0^{p-1} \sum_{k=1}^{N}|
\frac{\partial \varphi }{\partial x_k}|^{p-1}w_k^{1/p'}]^{p'} \\
&\leq \beta [k(x)^{p'}+|u_{n_j}|^{p} + \gamma_0^{p}\sum_{k=1}^{N}|
\frac{\partial \varphi }{\partial x_k}|^{p}w_k].
\end{aligned}\label{3.6}
\end{equation}
where $\{\gamma_0 = \mbox {sup}|\gamma(s)|, |s|\leq k+ \|\varphi\|_\infty$\}.
 Since $u_{n_j} \rightharpoonup u$ weakly in $W_0^{1,p}(\Omega,w)$ and
  $W_0^{1,p}(\Omega,w) \hookrightarrow \hookrightarrow L^q(\Omega,\sigma)$,
it follows that $u_{n_j} \to u$ strongly in $L^q(\Omega,\sigma)$
and $u_{n_j} \to u$  a.e. in $\Omega$. Combining
$(\ref{3.5})$,  $(\ref{3.6})$ and By Vitali's theorem we obtain,
 \begin{equation}
\int_{\Omega} a(x,u_{n_j},\nabla \varphi) \nabla T_k[u_{n_j}-\varphi] \,dx
\to \int_{\Omega} a(x,u,\nabla \varphi) \nabla T_k[u-\varphi] \,dx. \label{3.7}
\end{equation}
Secondly, we show that
\begin{equation}
\langle F,T_k[u_{n_j}- \varphi] \rangle \to \langle F,T_k[u-\varphi] \rangle .
 \label{3.8}
\end{equation}
In view of  $(\ref{3.4})$ and since $F \in W^{-1,p'}(\Omega,w^*)$,
we get
\begin{equation}
\langle F,T_k[u_{n_j}- \varphi] \rangle \to \langle F,T_k[u-\varphi] \rangle .
 \label{3.9}
\end{equation}
The convergence  $(\ref{3.7})$ and  $(\ref{3.9})$ allow to pass to the
limit in the inequality  $(\ref{3.1})$, and to obtain
\begin{equation}
\int_{\Omega} a(x,u,\nabla \varphi) \nabla T_k[u-\varphi] \,dx \leq
\langle F, T_k[u-\varphi] \rangle .\label{3.10}
\end{equation}
Now we introduce the following Lemma which will be proved later
and which is considered as an $L^1$ version of Minty's lemma
(in weighted Sbolev spaces).

Result  \eqref{3.10} and the following lemma complete the proof of
Theorem \ref{thm1}.
\end{proof}

\begin{lemma} \label{lem3.1}
Let $u$ be a measurable function such that $T_k(u)$ belongs to
$W_0^{1,p}(\Omega,w)$ for every $k>0$. Then the following two statements
are equivalent:
\begin{itemize}
\item[(i)] For every $\varphi $ in $W_0^{1,p}(\Omega,w)\cap L^\infty (\Omega)$
and  every $k>0$,
 $$
\int_{\Omega} a(x,u,\nabla \varphi ) \nabla T_k[u-\varphi ] \,dx \leq
\int_{\Omega}F\nabla T_k(u-\varphi )\,dx\,.
$$
\item[(ii)] For every $\varphi $ in
$W_0^{1,p}(\Omega,w)\cap L^\infty (\Omega)$ and every $k>0$,
 $$
\int_{\Omega} a(x,u,\nabla u) \nabla T_k[u-\varphi] \,dx
=  \int_{\Omega}F\nabla T_k(u-\varphi )\,dx\,.
$$
\end{itemize}
\end{lemma}

\begin{proof}
 Note that (ii) implies (i) is easily proved adding and subtracting
$$
\int_{\Omega} a(x,u,\nabla \varphi) \nabla T_k[u-\varphi] \,dx,
$$
and then using assumption
$(\ref{2.8})$. Thus, it only remains to prove that (i) implies
(ii).
Let $h$ and $k$ be  positive real numbers, let $\lambda \in  ]-1,1[$
and $\psi \in W_0^{1,p}(\Omega,w) \cap L^{\infty}(\Omega)$.
Choosing, $\varphi = T_h(u-\lambda T_k(u-\psi)) \in
W_0^{1,p}(\Omega,w) \cap L^{\infty}(\Omega)$ as test function in
$(\ref{3.10})$, we have,
\begin{equation}
I \leq J,\label{3.11}\end{equation}  with
\begin{gather*}
I = \int_{\Omega} a(x,u,\nabla T_h(u-\lambda T_k(u-\psi))
\nabla T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx,\\
J= \langle F,T_k(u-T_h(u-\lambda T_k(u-\psi)) \rangle .
\end{gather*}
Put  $A_{hk} =\{x \in \Omega:  |u-T_h(u-\lambda T_k(u-\psi))| \leq k \}$
and
 $B_h = \{x \in \Omega: |u-\lambda T_k(u-\psi)| \leq h \}$.
 Then, we have
\begin{align*}
I &= \int_{A_{kh} \cap B_h}
a(x,u,\nabla T_h(u-\lambda T_k(u-\psi)) \nabla T_k(u-T_h(u-\lambda
T_k(u-\psi)) \,dx\\
&\quad + \int_{A_{kh} \cap B_h^C}
a(x,u,\nabla T_h(u-\lambda T_k(u-\psi)) \nabla T_k(u-T_h(u-\lambda
T_k(u-\psi)) \,dx\\
&\quad + \int_{A_{kh}^C} a(x,u,\nabla
T_h(u-\lambda T_k(u-\psi)) \nabla T_k(u-T_h(u-\lambda
T_k(u-\psi))) \,dx.
\end{align*}
 Since $\nabla T_k(u-T_h(u-\lambda
T_k(u-\psi))$ is  zero in $A_{kh}^C$, we obtain
  \begin{equation}
\int_{A_{kh}^C} a(x,u,\nabla T_h(u-\lambda
T_k(u-\psi)) \nabla T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx = 0.
\label{3.12}
\end{equation}
Moreover, if $x \in B_h^C$, we have
$\nabla T_h(u- \lambda T_k(u- \psi) = 0$ which implies,
\begin{align*}
&\int_{A_{kh} \cap B_h^C} a(x,u,\nabla T_h(u-\lambda
T_k(u-\psi)) \nabla T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx\\
&=\int_{A_{kh} \cap B_h^C} a(x,u,0) \nabla
T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx.
\end{align*}
 Now, thanks to assumption $(\ref{2.9})$, we have $a(x,u,0) = 0$.
Then
\begin{equation}
\int_{A_{kh} \cap B_h} a(x,u,0)
\nabla T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx = 0.
\label{3.13}
\end{equation}
Combining  $(\ref{3.12})$ and $(\ref{3.13})$, we obtain
$$
I = \int_{A_{kh} \cap B_h} a(x,u,\nabla T_h(u-\lambda T_k(u-\psi)) \nabla
T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx,
$$
letting $h \to + \infty$, we have
\begin{equation}
A_{kh} \to \{x , |T_k(u- \psi)| \leq k \} = \Omega, \label{3.14}
\end{equation}
and $B_h \to \Omega$ which implies
\begin{equation}
 A_{kh} \cap B_h \to \Omega. \label{3.15}
\end{equation}
Then
 \begin{equation}
\begin{aligned}
&\lim_{h\to +\infty}\int_{A_{kh}
\cap B_h} a(x,u,\nabla T_h(u-\lambda T_k(u-\psi)) \nabla
T_k(u-T_h(u-\lambda T_k(u-\psi)) \,dx\\
&=\lambda \int_ \Omega a(x,u, \nabla(u- \lambda T_k(u- \psi)
\nabla T_k(u- \psi) \,dx.
\end{aligned} \label{3.16}
\end{equation}
On the other hand, we have
$$
J = \langle F,T_k[u-T_h(u- \lambda T_k(u- \psi)] \rangle .
$$
Then
\begin{equation}
\lim_{h\to +\infty} \langle F,T_k(u-T_h(u-
\lambda T_k(u- \psi)) \rangle = \lambda \langle F,T_k[u- \psi]
\rangle .\label{3.17}
\end{equation}
Together  $(\ref{3.16})$, $(\ref{3.17})$ and passing to the limit
in  $(\ref{3.11})$, we obtain
$$
\lambda \int_ \Omega a(x,u, \nabla(u-
\lambda T_k(u- \psi) \nabla T_k(u- \psi) \,dx \leq \lambda \langle
F,T_k[u- \psi] \rangle
$$
for every $\psi \in
W_0^{1,p}(\Omega,w)\cap L^{\infty}(\Omega)$, and for $k > 0 $.
Choosing $\lambda > 0 $ dividing by $\lambda$, and then letting
$\lambda$ tend to zero , we obtain
\begin{equation}
\int_ \Omega a(x,u, \nabla u) \nabla T_k(u- \psi)
\,dx \leq  \langle F,T_k[u- \psi] \rangle.
\label{3.18}
\end{equation}
 For $\lambda < 0 $ , dividing by
$\lambda$, and then letting $\lambda$ tend to zero , we obtain
\begin{equation}
\int_ \Omega a(x,u, \nabla u) \nabla
T_k(u- \psi) \,dx \geq  \langle F,T_k[u- \psi] \rangle ,
\label{3.19}
\end{equation}
Combining  $(\ref{3.18})$ and
$(\ref{3.19})$, we conclude that
$$
\int_ \Omega a(x,u, \nabla u) \nabla T_k(u- \psi)
\,dx = \langle F,T_k[u- \psi] \rangle .
$$
This completes the proof of Lemma.
\end{proof}

\begin{remark}\label{rem4}\rm
(1) The fact that the terms $T_n(u_n)$ is introduced in
\eqref{Pn} and also $\gamma(s)$ is a continuous function,
allow to have a weak solution for the a  approximate problem.

\noindent (2) Since in the formulation of the problem \eqref{P},
 we have  $a(x,u,\nabla u)$ instead of
$a(x,T_n(u_n),\nabla u_n)$, then the term $a(x,u,\nabla u)$ may
not belongs in $L^{p'}(\Omega,w^*)$ and not in $L^1(\Omega)$, thus
the problem \eqref{P} can have a T-solutions but, not a
weak solution.

For example if $w_i\equiv 1$, $i=1,\dots,N$ and
$a(x,u,\nabla u)=e^{|u|}|\nabla u|^{p-2}\nabla u$, with
$\gamma(s)=e^{|s|}$ then
\begin{gather*}
u \in W_0^{1,p}(\Omega,w), F \in W^{-1,p'}(\Omega,w^*) \\
-\mathop{\rm div}(e^{|u|}|\nabla u|^{p-2}\nabla u)=F\quad
\text{in }  \Omega  \\
 u  =  0\quad \mbox{on }  \partial \Omega.
\end{gather*}
our simple problem has a $T$-solutions, but
not a weak solution
\end{remark}

 \begin{example} \label{exa4} \rm
Let us consider the special case:
$$
a_i(x,\eta,\xi) = e^{|s|}w_i(x)|\xi_i|^{p-1}
\mathop{\rm  sgn}(\xi_i)  \quad i=1,\dots,N ,
$$
with $w_i(x)$ is a weight function $(i=1,\dots,N)$.
 For simplicity, we shall suppose that
$w_i(x)=w(x)$, for $i=1,\dots,N-1$, and
 $w_N(x) \equiv 0$  it is easy to show that the $a_i(x,s,\xi)$
 are Caracth\'eodory function satisfying the
 growth condition  $(\ref{2.7})$ and the coercivity  $(\ref{2.8})$.
 On the other hand  the monotonicity condition is verified. In fact,
 \begin{align*}
 &\sum_{i=1}^{N}(a_i(x,s,\xi)-a_i(x,s,\hat \xi))(\xi_i-\hat \xi_i)\\
 &= e^{|s|}w(x)\sum_{i=1}^{N-1}(|\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) \geq 0
\end{align*}
for almost all $x\in \Omega$ and for all $\xi, \hat \xi \in \mathbb{R}^N$.
This last inequality can not be strict, since for
$\xi \neq \hat \xi$ with $\xi_N \neq \hat \xi_N$ and
$\xi_i = \hat \xi_i$, $i=1,\dots,N-1$. The corresponding expression is zero.
In particular, let us use special weight functions $w$  expressed
in terms of the distance to the bounded $\partial \Omega$. Denote
$d(x)=\mathop{\rm dist}(x,\partial \Omega)$ and set
$w(x)=d^\lambda(x),$  such that,
 \begin{equation}
\lambda <\min(\frac{p}{N},p-1)\label{4.1}
\end{equation}
\end{example}

\begin{remark} \rm
Condition  $(\ref{4.1})$ is sufficient for  \eqref{2.4} to
hold [see \cite{Kufner},pp 40-41].
 \end{remark}

 Finally, the hypotheses of Theorem \ref{thm1} are  satisfied. Therefore, for all
 $ F \in { \prod_{i=1}^N}L^{p'}(\Omega,w_i^*) $ the following problem
has at last one  solution:
\begin{gather*}
 T_k(u) \in W_0^{1,p}(\Omega,w), \\
  \int_{\Omega} { \sum_{i=1}^N} w_i(x)e^{|u|}|
  \frac{\partial u}{\partial x_i}|^{p-1} \mathop{\rm sgn}
\big(\frac{\partial u}{\partial x_i}\big)
\frac{\partial T_k (u - \varphi)}{\partial x_i} \,dx
=  \int_{\Omega}F  T_k(u- \varphi) \,dx \\
\ \forall \varphi \in W_0^{1,p}(\Omega,w) \cap L^\infty(\Omega)\,.
\end{gather*}

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\end{document}