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

\markboth{Pseudo-monotonicity and degenerate elliptic operators}
{ Youssef Akdim \&  Elhoussine Azroul }

\begin{document}
\setcounter{page}{9}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline 
Electronic Journal of Differential Equations, 
Conference 09, 2002, pp 9--24. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Pseudo-monotonicity and degenerate elliptic operators  
  of second order
%
\thanks{ {\em Mathematics Subject Classifications:} 35J25, 35J70.
\hfil\break\indent
{\em Key words:} Weighted Sobolev spaces, pseudo-monotonicity, \hfil\break\indent
  nonlinear degenerate elliptic operators.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002.} }

\date{}
\author{Youssef Akdim \&  Elhoussine Azroul} 
\maketitle

\begin{abstract} 
 Extending the theory of pseudo-monotone mappings in
 weighted Sobolev spaces, we prove  some existence results for
 degenerate or singular elliptic equations generated by the
 second-order differential operator
 $$
 Au(x)=-\mathop{\rm div}a(x,u,\nabla u))+a_0(x,u,\nabla u),
 $$
 (in particular, when only large monotonicity is satisfied)
\end{abstract}

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


\section{Introduction}

Let $\Omega$ be a open subset of $\mathbb{R}^N$  ($N\geq 1$) and $p>1$ be
a real number and $\omega=\{\omega_0, \omega_1,\dots,\omega_N\}$ be a
collection of weight functions on $\Omega$, i.e, each $\omega_i$ is
a measurable and positive almost everywhere in $\Omega$, and
satisfying some integrability condition (see section 2 below).

Let us consider the second-order differential operator
$$
Au(x)=A_1u(x)+A_0u(x) \eqno{(1.1)} $$
where
$$
A_1u(x)=-\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i(x,u,\nabla
u)\eqno{(1.2)}
$$
is the top order part of $A$ and where
$$
A_0 u(x)=a_0(x,u,\nabla u)\eqno{(1.3)}
$$
is the lower order part of $A$ and where $\{ a_i(x,\eta,\zeta)$,
$0\leq i\leq N\}$ are functions defined on 
$\Omega\times\mathbb{R}\times\mathbb{R}^N$ and
satisfy a suitable regularity and growth assumptions.

Our objective in this paper, is to extend the theory of pseudo-monotone
mappings in weighted Sobolev spaces.
It's well known that, the essential condition which allows to do this,
is the so-called Leray-Lions condition,
$$
\sum_{i=1}^N(a_i(x,\eta, \zeta)-a_i(x,\eta,
\bar\zeta))(\zeta_i-\bar\zeta_i)>0,    \eqno{(1.4)}
$$
for a.e.\ $x\in \Omega$, all $\eta\in \mathbb{R} $ and all
$\zeta\neq\bar\zeta \in \mathbb{R}^N$
(resp. the so-called weak Leray-Lions condition,
$$
\sum_{i=1}^N(a_i(x,\eta,\zeta)-a_i(x,\eta,
\bar\zeta))(\zeta_i-\bar\zeta_i)\geq 0, \eqno{(1.5)} 
$$
for a.e.\ $x\in \Omega$, all $(\eta,\zeta, \bar\zeta)\in
\mathbb{R}\times\mathbb{R}^N\times\mathbb{R}^N)$.
Let us state the following assumptions:
\begin{enumerate}
\item [(H1)]  The expression
$$
\||u|\|_X=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}{\partial
x_i}|^p\omega_i(x)\,dx\Big)^{1/p}  $$
is a norm on $X=W_0^{1,p}(\Omega,\omega)$  equivalent to the usual norm
(2.3)(see section 2).
There exist a weight function $\bar\omega$ on $\Omega$ and a parameter
$q$, $1<q<\infty$, such that the (Hardy) inequality
$$
\Big(\int_\Omega|u(x)|^q\bar\omega(x)\Big)^{1/q}\leq
c\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}{\partial
x_i}|^p\omega_i(x)\,dx\Big)^{1/p} \eqno{(1.6)}
$$
holds for every $u\in W_0^{1,p}(\Omega,\omega)$ with a constant $c>0$
independent of $u$,  and moreover, the imbedding expressed by (1.6)
is compact, i.e.
$$
W_0^{1,p}(\Omega,\omega)\hookrightarrow\hookrightarrow L^q(\Omega,
\bar\omega). \eqno{(1.7)}
$$
\item [(H2)] Each $a_i(x,\eta,\zeta)$ $(1\leq i\leq N)$ is a
Carath\'eodory function  and
$$
|a_i(x,\eta,\zeta)|\leq C_i\omega_i^{1/p}(x)[g_i(x)
+\bar\omega^{\frac{1}{p'}}|\eta|^{q/p'}
+\sum_{j=1}^N\omega_j^{1/p'}(x)|\zeta_j|^{p-1}],\eqno{(1.8)}
$$
for  a.e. $x\in \Omega$, some constants $C_i>0$, some functions
$g_i(x)\in L^{p'}(\Omega)$, all $(\eta,\zeta)\in \mathbb{R}^{N+1}$ and all
$i=1,\dots,N$.
\end{enumerate}
Recently, Drabek, Kufner and Mustonen \cite{dr-ku-mu}
proved that the mapping $T_1$ defined from $X$ to its dual $X^*$ associated to
the
top order part $A_1$ is pseudo-monotone in $X$, under the weak
conditions (1.5), (H1), (H2).
Hence, the authors obtained the existence
result for the Dirichlet problem associated to the $A_1u=f\in X^*$ by
assuming some degeneracy.

Our first purpose in this paper, is to extend the previous result
$\cite{dr-ku-mu}$ in the operator $A$ from (1.1) where the lower
order part $A_0$ is affine with respect to the gradient, i.e., $A_0$ is of
the form
$$
A_0u(x)=c_0(x,u(x))+\sum_{i=1}^Nc_i(x,u(x))\frac{\partial
u(x)}{\partial x_i},\eqno{(1.9)}
$$
where $c_i(x,\eta)$, $0\leq i\leq N$ are some Carath\'eodory
functions defined on $\Omega\times\mathbb{R}$ and satisfy
$$
\begin{gathered}
|c_0(x,\eta)|\leq
C_0\bar\omega^{1/q}(x)[g_0(x)+\bar\omega^{\frac{1}{q'}}(x)
|\eta|^{\frac{q}{q'}}]\\
|c_i(x,\eta)|\leq C_i\omega_i^{1/p}(x)\bar\omega^{1/q}(x)[\gamma_i(x)+
\bar\omega^{\frac{1}{r}}(x)|\eta|^{\frac{q}{r}}]\quad
\mbox{for all }i=1,\dots,N,
\end{gathered}
\eqno{(1.10)}
$$
for a.e.\ $x\in \Omega $,  some constants $C_0>0,\ C_i>0,$
some functions $g_0\in L^{q'}(\Omega)$ and  $ \gamma_i(x)\in
L^r(\Omega)$
with
$$\frac{1}{r}+\frac{1}{p}+\frac{1}{q}<1\eqno{(1.11)}$$
and where $\bar\omega(x)$ and $q$ are from (1.6).
More precisely, we prove the following theorem,

\begin{theorem} \label{thm1.1}
Assume that (H1), (H2), (1.10), (1.5) hold.
Then the mapping $T$ associated to the operator $A$ from (1.1) and
(1.9) is pseudo-monotone in $X$.
\end{theorem}

\begin{remark}  \label{rm1.1}
Theorem \ref{thm1.1} is obviously a consequence of the more general result
(Theorem \ref{thm3.1},  it suffices to take $I=\emptyset$).
\end{remark}

\begin{remark}  \label{rm1.2}
About the existence of such $r$ satisfying (1.11) see Remarks \ref{rm2.1} and
\ref{rm4.2} below.
\end{remark}

The second aim of this paper, is to prove the same result of the
preceding
without restriction on $A_0$ and where (1.4) is applied. This is done
in Theorem \ref{thm3.1}, if we take $I^c=\emptyset$.

This paper is divided into four sections. In section 2, we start our
basic assumptions and we prove some preliminaries lemmas concerning some
convergence and generalized H\"older's inequality  in weighted Sobolev
space. In section 3, we give our general main result and its proof and
we study an example which illustrate our abstract hypotheses. The section
4, is devoted to the study of some particular case where $\omega_0\equiv 1$
on $\Omega$ and where some of our hypotheses (imbedding) are satisfied.

In our work, we shall adopt many ideas from $\cite{go-mu}$
(where the authors have studied the non-degenerated elliptic case).
But the results are generalized and improved.
concerning the existence results for higher order nonlinear degenerated
(or singular) elliptic equations, we refer the reader to
\cite{dr-ku-ni1,dr-ku-ni 2,az} (where the degree theory is used in
the two first papers and where the pseudo-monotonicity is used in the last
but under some restrictions on the weighted).
Finally, not that our approach based on the theory of pseudo-monotone
mappings can be applied in the case of non reflexive Banach space, for
example in weighted Orlicz-Sobolev spaces (see \cite{az} for related
topics).

\section{Preliminaries and basic assumptions}

\paragraph{1) Weighted Sobolev spaces.}
Let $\Omega$ be a open subset of $\mathbb{R}^N\ (N\geq 1)$, with finite
measure, let $1<p<\infty$, and let
$\omega= \{\omega_i(x)\; 0\leq i\leq N\}$ be a vector of weight
functions, i.e. every component $\omega_i(x)$ is a measurable
function which is positive a.e. in $\Omega$. Further, we
suppose that
$$
 \omega_i\in L_{\rm loc}^1(\Omega) \eqno{(2.1)}
$$
and
$$
\omega_i^{-\frac{1}{p-1}}\in L_{\rm loc}^1(\Omega)\eqno{(2.2)}
$$
for any $0\leq i\leq N$ hold in all our considerations.

Now, we denote by $W^{1,p}(\Omega,\omega)$ the space of all real-valued
functions $u\in L^p(\Omega,\omega_0)$ such that the derivatives in the
sense of distributions fulfil
$$
\frac{\partial u}{\partial x_i}\in L^p(\Omega,\omega_i) \quad
\mbox{for all } i=1,\dots,N,
$$
which is a Banach space under the norm,
$$\|u\|_{1,p,\omega}=\Big(\int_\Omega
|u(x)|^p\omega_0(x)\,dx+\sum_{i=1}^N\int_\Omega |\frac{\partial
u(x)}{\partial x_i}|^p\omega_i(x)\,dx\Big)^{1/p}.
\eqno{(2.3)}
 $$
The condition (2.1) implies that $C_0^{\infty}(\Omega)$ is a subspace
of
$W^{1,p}(\Omega,\omega)$ and consequently, we can introduce the
subspace
$W_0^{1,p}(\Omega,\omega)$ of $W^{1,p}(\Omega,\omega)$ as the closure
of
$C_0^{\infty}(\Omega)$ with respect to the norm (2.3).
Moreover, the condition (2.2) implies that $W^{1,p}(\Omega,\omega)$ as
well as $W_0^{1,p}(\Omega,\omega)$
are reflexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces
$W_0^{1,p}(\Omega,\omega)$ is equivalent to
$W^{-1,p'}(\Omega,\omega^*)$,
where $\omega^*=\{\omega_i^*=\omega_i^{1-p'}\; \forall i=0,\dots,N
\}$, with $p'=\frac{p}{p-1}$.
We shall suppose that the expression
$$
\||u|\|_{1,p,\omega}=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial
u(x)}{\partial x_i}|^p\omega_i(x)\,dx\Big)^{1/p}
$$
is a norm defined on $W_0^{1,p}(\Omega,\omega)$ and it's equivalent to
the norm (2.3). The reader can find conditions on the weight
$\omega$ which guarantee this fact in \cite{dr-ku-ni1}.
Notice that $(X,\||.|\|_X)$ is a
uniformly convex (and thus reflexive) Banach space.

\paragraph{2) Basic assumptions.}
Let $I$ be a subset of $\{1,2,\dots,N\}$ and $I^c$ its complement,
and
let introduce the  following modified versions of (1.4) and
(1.5),
$$\sum_{i\in
I}(b_i(x,\eta,\zeta_I)-b_i(x,\eta,\bar\zeta_I))(\zeta_i-\bar\zeta_i)>0,
\eqno{(2.4)}
$$
for a.e. $\ x\in\Omega$, all $\eta \in \mathbb{R}$ and all
$\zeta\neq\bar\zeta\in\mathbb{R}^N$ and
$$
\sum_{i\in I^c}(b_i(x,\eta,\zeta_{I^c})-b_i(x,\eta,\bar\zeta_{I^c}))
(\zeta_i-\bar\zeta_i)\geq 0, \eqno{(2.5)}
$$
for a.e. $\ x\in\Omega$, all $\eta \in \mathbb{R}$ and all
$\zeta,\bar\zeta\in\mathbb{R}^N$ where $\zeta_J$ denoted $\zeta_J=\{\zeta_i,\
\ i
\in J\}$ and where $a_i(x,\eta,\zeta)$ are Carath\'eodory functions
such
that,
$$
\begin{gathered}
a_i(x,\eta,\zeta)=b_i(x,\eta,\zeta_I)\quad\mbox{for all } i\in I,\\
a_i(x,\eta,\zeta)=b_i(x,\eta,\zeta_{I^c})\quad\mbox{for all }
 i\in I^c,\\
a_0(x,\eta,\zeta)=c_0(x,\eta,\zeta_I)+{\sum_{i\in
I^c}}c_i(x,\eta,\zeta_I)\zeta_i,
\end{gathered} \eqno{(2.6)}
$$
for a.e.\ $x\in \Omega$, all $(\eta,\zeta)\in \mathbb{R}^{N+1}$ and where
$b_i$ $(i=1,\dots,N)$, $c_0$ and $c_i$  $(i\in I^c)$ are  functions
satisfying the Carath\'eodory conditions (i.e. measurable in $x$ for
 any fixed $\xi=(\eta,\zeta)\in \mathbb{R}^{N+1}$ and continuous in $\xi$
 for almost all fixed $x\in \Omega)$.

We assume the following growth conditions:
\begin{enumerate}
\item[(H2')] Each $a_i(x,\eta ,\zeta)$ is a Carath\'eodory function  and,
that there exists some positives constants $C_i$, and some functions
$g_i(x)\in L^{p'}(\Omega)$ $i=1,\dots,N,$ and $g_0\in L^{q'}(\Omega)$
and some $\gamma_i(x)\in L^r(\Omega)$ for all $i\in I^c)$ such that
\begin{gather*}
|b_i(x,\eta,\zeta_I)|\leq
C_i\omega_i^{1/p}(x)[g_i(x)+\bar\omega^{\frac{1}{p'}}|\eta|^{\frac{q}{p'}}+\sum_{j\in
I}\omega_j^{\frac{1}{p'}}(x)|\zeta_j|^{p-1}]\quad\mbox{for } i\in I
\\
|b_i(x,\eta,\zeta_{I^c})|\leq
C_i\omega_i^{1/p}(x)[g_i(x)+\bar\omega^{\frac{1}{p'}}(x)|\eta|
^{\frac{q}{p'}}+\sum_{j\in I^c}\omega_j^{\frac{1}{p'}}(x)|\zeta_j|^{p-1}],\\
\mbox{for } i\in I^c\\
|c_0(x,\eta,\zeta_I)|\leq
C_0\bar\omega^{1/q}[g_0(x)+\bar\omega^{\frac{1}{q'}}(x)|\eta|^{\frac{q}{q'}}+\sum_{j\in
I}\omega_j^{\frac{1}{q'}}(x)|\zeta_j|^{\frac{p}{q'}}]\\
|c_i(x,\eta,\zeta_I)|\leq
C_i\omega_i^{1/p}(x)\bar\omega^{1/q}(x)[\gamma_i(x)+
\bar\omega^{\frac{1}{r}}(x)|\eta|^{\frac{q}{r}}+\sum_{j\in
I}\omega_j^{\frac{1}{r}}(x)|\zeta_j|^{\frac{p}{r}}],\\
\mbox{for } i\in I^c,
\end{gather*}
for a.e.\ $x\in \Omega $,  all $\eta\in \mathbb{R},\ \  \zeta\in \mathbb{R}^N,$ with
$$ \frac{1}{r}+\frac{1}{p}+\frac{1}{q}<1.\eqno{(2.7)}
$$
\end{enumerate}

\begin{remark}  \label{rm2.1}
\begin{enumerate}
\item[1)] The such $r$ satisfying (2.7), exists when $q>p'$ (it suffices
to
choose $r>\frac{pq}{pq-p-q}>1$).
\item[2)] If $q\leq p'$, we can not found any $r$ satisfying (2.7)
(since $\frac{1}{p}+\frac{1}{p'}=1\leq \frac{1}{p}+\frac{1}{q}$).
\end{enumerate}
\end{remark}

Before to give main general result, let us give and prove the following
lemmas which are needed below.

\begin{lemma} \label{lm2.1}
Let $\Omega$ be a subset of $\mathbb{R}^N$ with finite measure and let $f\in
L^p(\Omega,\sigma_1)$ $(1<p<\infty)$,
$g\in L^q(\Omega,\sigma_2)$ $(1<q<\infty)$ where $\sigma_1$ and $\sigma_2$
are weight functions in $\Omega$ and let
$h\in L^r(\Omega,\sigma_1^{-\frac{r}{p}}\sigma_2^{-\frac{r}{q}})$
$(1<r<\infty)$
with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\leq 1$, then $fgh\in
L^1(\Omega)$.
\end{lemma}

Indeed: Let $\frac{1}{s}=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\leq 1$. By
H\"older inequality we have,
$$
\int_\Omega|fgh|^s\leq \Big(\int_\Omega
f^p\sigma_1\Big)^{s/p}\Big(\int_\Omega
g^q\sigma_2\Big)^{s/q}\Big(\int_\Omega
h^r\sigma_1^{-r/p}\sigma_2^{-r/q}\Big)^{s/r}<
\infty,$$
then $fgh\in L^s(\Omega)$ which implies that $fgh \in L^1(\Omega).$

\begin{lemma} \label{lm2.2}
Let $(g_n)_n $ be a sequence of $L^p(\Omega,\sigma)$ and let $g\in
L^p(\Omega,\sigma)$ $(1<p<\infty)$, where $\sigma$ is a weight function
in $\Omega$.
If $g_n \to g$ in measure (in particular a.e.\ in $\Omega $)
and is bounded in $L^p(\Omega,\sigma)$,
then $g_n\to g$ in $L^q(\Omega,\sigma^{q/p})$ for all $q<p$.
\end{lemma}

\paragraph{Proof.}
Let $\varepsilon>0$ and set
$A_n=\{x\in \Omega : |g_n(x)-g(x)|\sigma^{1/p}(x)\leq
(\frac{\varepsilon}{2\mathop{\rm meas}(\Omega)})^{1/q}\}$, we have
\begin{eqnarray*}
\int_\Omega|g_n-g|^q\sigma^{q/p}\,dx
&=&\int_{A_n}|g_n-g|^q\sigma^{q/p}\,dx+\int_{A_n^c}|g_n-g|^q\sigma^{q/p}\,dx
\\
&\leq&
\frac{\varepsilon}{2}+\int_{A_n^c}|g_n-g|^q\sigma^{q/p}\,dx.
\end{eqnarray*}
By H\"older inequality,
\begin{eqnarray*}
\int_{A_n^c}|g_n-g|^q\sigma^{q/p}\,dx&\leq&
\Big(\int_\Omega|g_n-g|^p\sigma\,dx\Big)^{q/p}
\Big(\mathop{\rm meas}(A_n^c)\Big)^{1-\frac{q}{p}}
\\
&\leq& M \left(\mathop{\rm meas}(A_n^c)\right)^{1-\frac{q}{p}},
\end{eqnarray*}
where $M$ is a constant does not depend on $n$.
On the other hand since
$g_n \to g$ in measure we have
$$\mathop{\rm meas}(A_n^c)\to
0 \quad\mbox{as }n\to \infty,
$$
then there exists some $n_0\in \mathbb{N}$  such that for all $n\geq n_0,$
$$\int_{A_n^c}|g_n-g|^q\sigma^{q/p}\,dx\leq
\frac{\varepsilon}{2}.
$$
\begin{remark} \label{rm2.4}
We can also give an other proof of the last lemma, by using the
non-weighted
case, i.e.,
$g_n\sigma^{1/p}$ is bounded in $L^p(\Omega)$ and
$g_n(x)\sigma^{1/p}(x) \to
g(x)\sigma^{1/p}(x)$,
in measure, hence
$g_n\sigma^{1/p} \to g\sigma^{1/p}$ in
$L^q(\Omega)$ for all $q<p$.
\end{remark}

The following lemma is a generalization of
\cite[Lemma 3.2]{le-li} in weighted spaces.

\begin{lemma} \label{lm2.3}
Let $g\in L^q(\Omega,\sigma)$ and let $ g_n\in L^q(\Omega,\sigma)$,
with
$\|g_n\|_{q,\sigma} \leq c$ $(1<q<\infty )$. If
$g_n(x)\to g(x)$ a.e.\ in $\Omega$, then $g_n\rightharpoonup g$ in
$L^q(\Omega,\sigma)$, where $\rightharpoonup$ denotes weak convergence.
\end{lemma}

\paragraph{Proof.}
Since $g_n\sigma^{1/q}$ is bounded in $L^q(\Omega)$ and
$g_n(x)\sigma^{1/q}(x) \to g(x)\sigma^{1/q}(x)$,
a.e.  in $\Omega$, by the \cite[Lemma 3.2]{le-li}, we have
$$g_n\sigma^{1/q} \rightharpoonup g\sigma^{1/q}\quad
 \mbox{in }L^q(\Omega).
$$
Moreover for all $ \varphi \in
L^{q'}(\Omega,\sigma^{1-q'})$, we have $\varphi\sigma^{-1/q}\in
L^{q'}(\Omega)$, then
$$
\int_\Omega g_n\varphi\,dx \to \int_\Omega g\varphi\,dx,\
\mbox{ i.e. } g_n\rightharpoonup g \mbox{ in }L^q(\Omega,\sigma).
$$

\begin{lemma} \label{lm2.4}
Let $g_n\in L^p(\Omega,\sigma_1)$ and let $g\in L^p(\Omega,\sigma_1)$
$(1<p<\infty)$.
If $g_n\rightharpoonup g$ in $ L^p(\Omega,\sigma_1)$,  then
$$
g_nv\rightharpoonup gv \quad\mbox{in }L^s(\Omega,\sigma_1^{s/p}\sigma_2^{s/q})
\mbox{ for any }v\in L^{q}(\Omega,\sigma_2),
$$
with $q>1$ and $\frac{1}{s}=\frac{1}{p}+\frac{1}{q}$.
\end{lemma}

\paragraph{Proof.} Let$\varphi \in L^{s'}(\Omega,
\sigma_1^{\frac{s}{p}(1-s')}\sigma_2^{\frac{s}{q}(1-s')})$.
For any $v\in L^{q}(\Omega,\sigma_2)$ we have, $v\varphi\in
L^{p'}(\Omega,\sigma_1^{1-p'})$. Indeed,  since
$\frac{1}{p'}=\frac{1}{s'}+\frac{1}{q},$ we have by H\"older's
inequality,
\begin{eqnarray*}
\lefteqn{\int_\Omega |v\varphi|^{p'}\sigma_1^{1-p'}(x)\,dx}\\
&=&\int_\Omega
|v\sigma_2^{1/q}(x)|^{p'}|\varphi|^{p'}\sigma_1^{1-p'}(x)
\sigma_2^{-p'/q}(x)\,dx\\
&\leq &\Big(\int_\Omega|v|^{q}\sigma_2(x)\,dx\Big)^{p'/q}
\Big(\int_\Omega|\varphi|^{s'}\sigma_1^{\frac{s'}{p'}(1-p')}(x)
\sigma_2^{-s'/q}(x)\,dx\Big)^{p'/s'}\\
&= &\Big(\int_\Omega|v|^{q}\sigma_2\,dx\Big)^{p'/q}
\Big(\int_\Omega|\varphi|^{s'}\sigma_1^{\frac{s}{p}(1-s')}(x)
\sigma_2^{\frac{s}{q}(1-s')}(x)\,dx\Big)^{p'/s'}<\infty.
\end{eqnarray*}
Finally, since $g_n\rightharpoonup g$ in $ L^p(\Omega,\sigma_1)$, then
$$\int_\Omega g_nv\varphi\,dx\to \int_\Omega gv\varphi\,dx
\mbox{ i.e. }g_nv\rightharpoonup gv \mbox{ in }
L^s(\Omega,\sigma_1^{s/p}\sigma_2^{s/q})\;\forall v\in
L^{q}(\Omega,\sigma_2).$$

\begin{lemma} \label{lm2.5}
Let $\Omega$ be a subset of $\mathbb{R}^N$ with finite measure and let $1\leq
p\leq
q$ then, we have the continuous imbedding
$L^q(\Omega,\sigma) \hookrightarrow L^p(\Omega,\sigma^{p/q})$
where $\sigma$ is a weight function in $\Omega$.
\end{lemma}

The proof of this lemma can be deduced easily from H\"older's inequality.

\section{Main general result}

Under the previous assumptions, the differential operator (1.1)
(with coefficients satisfying (2.6), generates a mapping $T$ from
$X=W_0^{1,p}(\Omega,\omega)$ to its dual $X^*$ through the formula,
$$
\begin{aligned}
\langle Tu,v\rangle =&\int_\Omega \sum_{i\in I}b_i(x,u,\zeta_I(\nabla
u))\frac{\partial
v}{\partial x_i}\,dx+\int_\Omega \sum_{i\in
I^c}b_i(x,u,\zeta_{I^c}(\nabla
u))\frac{\partial v}{\partial x_i}\,dx
\\
&+\int_\Omega c_0(x,u,\zeta_I(\nabla
u))v\,dx+\int_\Omega
\sum_{i\in I^c}c_i(x,u,\zeta_I(\nabla u))\frac{\partial u}{\partial
x_i}v\,dx,
\end{aligned} \eqno{(3.1)}
$$
for all  $u,v\in X$ and where $\langle,\rangle$ denotes the duality pairing
between $X^*$ and $X$. When we have adopted the notation $\zeta_J(\nabla
u)=\{\frac{\partial u}{\partial x_i},\; i\in J\}$.

We recall that the mapping $T$ is well defined and bounded, this
can be
easily seen by Lemma \ref{lm2.1} and H\"older's inequality.

\paragraph{Definition} %3.1
A bounded mapping $T$ from $X$ to $X^*$ is called pseudo-monotone if
for any
sequence $u_n \in X$ with $u_n \rightharpoonup u$ in $X$ and
${\limsup_{n\to\infty}} \langle T\,u_n,u_n-u\rangle \leq 0$, one
has
$$\liminf_{n\to\infty}\langle Tu_n,u_n-v\rangle
\geq \langle Tu,u-v\rangle \quad \mbox{for all } v\in X.
$$

\begin{theorem} \label{thm3.1}
Assume that (H1), (H2'), (2.4) and (2.5) hold.
Then the corresponding mapping $T$ defined by  (3.1) is pseudo-monotone in
$X=W_0^{1,p}(\Omega,\omega)$.
\end{theorem}

\begin{remark} \label{rm3.1} \rm
1) When $I=\emptyset$, the previous theorem applies in particular to
operators like $(1.1)$ with $A_0$ affine with respect to the gradient
variable, this gives from (1.5) a sufficient condition (theorem 1.1
in the introduction).\\
2)  When $I=\emptyset$ and $A_0\equiv 0$, we immediately obtain
\cite[Proposition 1]{dr-ku-mu}.\\
3) When  $I^c=\emptyset$, we obtain  \cite[Theorem 7.4]{az} and
when
$A_0\equiv 0$, $I=\emptyset$,  we give  in \cite[Theorem 7.2]{az}.\\
4) Theorem \ref{thm3.1} generalizes \cite[Theorem 3.1]{go-mu} in the weighted
case.
\end{remark}

Applying the previous theorem, we obtain the following existence
results, which generalizes the corresponding (cf. \cite{az,dr-ku-mu}).

\begin{coro} \label{coro3.1}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ and assume the
hypotheses in Theorem \ref{thm3.1}. Also assume the degenerate ellipticity condition
$$
\sum_{i=0}^N a_i(x,\xi)\xi_i\geq C_0\sum_{i=1}^N\omega_i(x)|\xi_i|^p
$$
for a.e.\ $x\in \Omega$, some $C_0>0$ and all $\xi\in \mathbb{R}^{N+1}$.
Then for any $f\in X^*$ the Dirichlet associated problem
\begin{align*}
\int_\Omega \sum_{i\in I} b_i(x,u,\zeta_I(\nabla u))\frac{\partial
v}{\partial x_i}\,dx+\int_\Omega \sum_{i\in
I^c}b_i(x,u,\zeta_{I^c}(\nabla
u))\frac{\partial v}{\partial x_i}\,dx&\\
+\int_\Omega c_0(x,u,\zeta_I(\nabla u))v\,dx+\int_\Omega
\sum_{i\in I^c}c_i(x,u,\zeta_I(\nabla u))\frac{\partial u}{\partial
x_i}v\,dx&=\int_\Omega fv\,dx
\end{align*}
for all  $v\in X$ has at least one  solution $u\in X$.
\end{coro}

\paragraph{Proof of Theorem \ref{thm3.1}.}
Let $(u_n)_n$ be a sequence in $X$ such that:
$$u_n\rightharpoonup u \mbox{ in }X\eqno{(3.2)}$$
and
$$\limsup_{n\to\infty}\langle Tu_n,u_n-u\rangle \leq 0,\eqno{(3.3)}
$$
i.e.
\begin{eqnarray*}
\limsup_{n\to\infty}&\{&\int_\Omega \sum_{i\in
I}b_i(x,u_n,\zeta_I(\nabla u_n))(\frac{\partial u_n}{\partial
x_i}-\frac{\partial u}{\partial x_i})\,dx\\
& &+\int_\Omega \sum_{i\in I^c}b_i(x,u_n,\zeta_{I^c}(\nabla
u_n))(\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial
x_i})\,dx\\
& &+\int_\Omega c_0(x,u_n,\zeta_I(\nabla u_n))(u_n-u)\,dx\\
& &+\int_\Omega \sum_{i\in I^c}c_i(x,u_n,\zeta_I(\nabla
u_n))\frac{\partial
u_n}{\partial x_i}(u_n-u)\,dx\}\leq 0. \\
\end{eqnarray*}
{\bf a)} We shall prove that
$$\langle Tu_n,v\rangle \to \langle Tu,v\rangle \quad \mbox{as }n\to \infty
\mbox{ for all } v\in X.\eqno{(3.4)}
$$
\textbf{First step.} We show that
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I}(b_i(x,u_n,\zeta_I(\nabla u_n))-b_i(x,u_n,\zeta_I(\nabla
u)))(\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial
x_i})\,dx=0\eqno{(3.5)}$$
and $$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}(b_i(x,u_n,\zeta_{I^c}(\nabla u_n))-b_i(x,u_n,\zeta_{I^c}(\nabla
u)))(\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial
x_i})\,dx=0.\eqno{(3.6)}$$
Indeed:
First, we can choose $q_1$  such that $1<q_1<r,$ and
$\frac{1}{q_1}+\frac{1}{p}+\frac{1}{q}<1$ (due to
$\frac{1}{r}+\frac{1}{p}+\frac{1}{q}<1$).
It follows from the compact imbedding (1.7) that, for a subsequence,
$$
\begin{gathered}
u_n\to u \mbox{ in }L^{q}(\Omega,\bar\omega)\\
u_n(x)\to u(x)\ a.e. \mbox{ in }\Omega.
\end{gathered}\eqno{(3.7)}
$$
By (H2'), the sequences $\{ c_0(x,u_n,\zeta_I(\nabla u_n))\}$
(resp.
$\{ c_i(x,u_n,\zeta_I(\nabla u_n))\frac{\partial u_n}{\partial x_i}\
(i\in I^c)\}$)
remains bounded in $L^{q'}(\Omega,\bar\omega^{1-q'})$ (resp. $L^{\tilde
s}(\Omega,\bar\omega^{\frac{-\tilde s}{q}})$ with $\frac{1}{\tilde
s}=\frac{1}{p}+\frac{1}{r}$).\\
Indeed,
\begin{eqnarray*}
\lefteqn{\int_\Omega|\bar\omega^{-1/q}c_i(x,u_n,\zeta_I(\nabla
u_n))\frac{\partial u_n}{\partial x_i}|^{\tilde s}}\\
&\leq&
\Big(\int_\Omega \omega_i^{-r/p}\bar\omega^{-r/q}
|c_i(x,u_n,\zeta_I(\nabla u_n))|^{r}\Big)^{\tilde s/r}
\Big(\int_\Omega |\frac{\partial u_n}{\partial x_i}|^{p}\omega_i\Big)
^{\tilde s/p}<c.
\end{eqnarray*}
Thanks to Lemma \ref{lm2.5} and since $q'\leq \tilde s$, we have
$$
L^{\tilde s}(\Omega,\bar\omega^{-\tilde s/q})\hookrightarrow
L^{q'}(\Omega,\bar\omega^{-q'/q}),
$$
then  $\{ c_i(x,u_n,\zeta_I(\nabla u_n))\frac{\partial u_n}{\partial
x_i}\
(i\in I^c)\}$) is bounded in  $L^{q'}(\Omega,\bar\omega^{1-q'})$).
Hence, using (3.7) we conclude that
$$\lim_{n\to \infty} \int_\Omega c_0(x,u_n,\zeta_I(\nabla
u_n))(u_n-u)\,dx=0\eqno{(3.8)}$$
and
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}c_i(x,u_n,\zeta_I(\nabla u_n))\frac{\partial u_n}{\partial
x_i}(u_n-u)\,dx=0.\eqno{(3.9)}
$$
On the other hand, in virtue of (3.7) and continuity of the Nemytskii
operators (see \cite{dr-ku-ni1}), we have
\begin{gather*}
b_i(x,u_n,\zeta_I(\nabla u))\to b_i(x,u,\zeta_I(\nabla u))
\quad\mbox{in }L^{p'}(\Omega,\omega_i^*), \ i\in I\\
b_i(x,u_n,\zeta_{I^c}(\nabla u))\to b_i(x,u,\zeta_{I^c}(\nabla
u))\quad\mbox{in }L^{p'}(\Omega,\omega_i^*), \ i\in I^c,
\end{gather*}
which implies
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I}b_i(x,u_n,\zeta_I(\nabla u))(\frac{\partial u_n}{\partial
x_i}-\frac{\partial u}{\partial x_i})\,dx=0\eqno{(3.10)}
$$
and
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u))(\frac{\partial u_n}{\partial
x_i}-\frac{\partial u}{\partial x_i})\,dx=0.\eqno{(3.11)}
$$
Combining  (2.4), (2.5), (3.3), (3.8), (3.9), (3.10) and
(3.11) we conclude the assertions (3.5) and (3.6).

\paragraph{Second step.}
For to prove of the relation (3.4) it suffices to show the following
assertions:\\
(i) For every  $v\in X$,
$$\lim_{n\to \infty}\int_\Omega c_0(x,u_n,\zeta_I(\nabla
u_n))v\,dx=\int_\Omega c_0(x,u,\zeta_I(\nabla u))v\,dx.\eqno{(3.12)}
$$
(ii) For every  $v\in X$,
$$ \lim_{n\to \infty}\int_\Omega\sum_{i\in
I}b_i(x,u_n,\zeta_I(\nabla
u_n))\frac{\partial v}{\partial x_i}\,dx=\int_\Omega\sum_{i\in
I}b_i(x,u,\zeta_I(\nabla u))\frac{\partial v}{\partial
x_i}\,dx.\eqno{(3.13)}
$$
(iii) For every  $v\in X$,
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}c_i(x,u_n,\zeta_I(\nabla u_n))\frac{\partial u_n}{\partial
x_i}v\,dx=\int_\Omega \sum_{i\in I^c}c_i(x,u,\zeta_I(\nabla
u))\frac{\partial u}{\partial x_i}v\,dx.\eqno{(3.14)}
$$
(iv) For every  $v\in X$,
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u_n))\frac{\partial v}{\partial
x_i}\,dx=\int_\Omega \sum_{i\in I^c}b_i(x,u,\zeta_{I^c}(\nabla
u))\frac{\partial v}{\partial x_i}\,dx.\eqno{(3.15)}
$$
\textbf{Proof of (i)and (ii).}
Invoking Landes \cite[Lemma 6]{la}, we obtain from (3.5) and the
strict
monotonicity (2.4) that,
$$\frac{\partial u_n}{\partial x_i}\to
\frac{\partial u}{\partial x_i}\quad\mbox{a.e. in }\Omega
\mbox{ for each } i\in I,\eqno{(3.16)}
$$
which gives
\begin{gather*}
c_0(x,u_n,\zeta_I(\nabla u_n))\to c_0(x,u,\zeta_I(\nabla u))
\quad \mbox{a.e.  in }\Omega,\\
b_i(x,u_n,\zeta_I(\nabla u_n))\to b_i(x,u,\zeta_I(\nabla
u))\quad\mbox{a.e. in }\Omega\  \forall  i\in I.
\end{gather*}
The growth conditions (H2') imply that, the sequences
$\{ c_0(x,u_n,\zeta_I(\nabla u_n))\}$ \\ (resp. $\{
b_i(x,u_n,\zeta_I(\nabla
u_n))\ \ i\in I\})$
remains bounded in $L^{q'}(\Omega,\bar\omega^{1-q'})$
(resp.  $L^{p'}(\Omega,\omega_i^*)$).
Hence by Lemma \ref{lm2.3} we conclude (i) and (ii).
\\
\textbf{Proof of (iii).}
Similarly, by (3.7) and (3.16) we can write,
$$
c_i(x,u_n,\zeta_I(\nabla u_n))\to c_i(x,u,\zeta_I(\nabla
u))\quad\mbox{a.e.  in }\Omega \mbox{ for all } i\in I^c.
$$
And by the growth conditions (H2') also
$\{c_i(x,u_n,\zeta_I(\nabla u_n)), \ i\in I^c\}$ is bounded in
$L^{r}(\Omega,\omega_i^{-\frac{r}{p}}\bar\omega^{-\frac{r}{q}})$,
then in virtue of Lemma \ref{lm2.2}, we have
$$ c_i(x,u_n,\zeta_I(\nabla
u_n))\to c_i(x,u,\zeta_I(\nabla u))\quad \mbox{in }
L^{q_1}(\Omega,\omega_i^{\frac{-q_1}{p}}\bar\omega^{\frac{-q_1}{q}})
\quad \forall i\in I^c.
$$
Let $s>1$ such that $\frac{1}{s}=\frac{1}{p}+\frac{1}{q}$. Since
$\frac{1}{s'}+\frac{1}{s}=1>\frac{1}{s}+\frac{1}{q_1}$ i.e.\
$s'<q_1$, we have (as in the proof of Lemma \ref{lm2.5}),
$$\int_\Omega
|v|^{s'}\omega_i^{-s'/p}\bar\omega^{-s'/q}\,dx\leq
\Big(\int_\Omega
|v|^{q_1}\omega_i^{-q_1/p}\bar\omega^{\frac{-q_1}{q}}\,dx\Big)
^{s'/q_1}( \mathop{\rm meas}(\Omega))^{1-\frac{s'}{q_1}}
$$
for all $v\in L^{q_1}(\Omega,\omega_i^{-q_1/p}
\bar\omega^{-q_1/q})$.
Then
$$L^{q_1}(\Omega,\omega_i^{\frac{-q_1}{p}}\bar\omega^{\frac{-q_1}{q}})
\hookrightarrow L^{s'}(\Omega,\omega_i^{\frac{-s'}{p}}
\bar\omega^{\frac{-s'}{q}}),
$$
which implies
$$c_i(x,u_n,\zeta_I(\nabla u_n))\to
c_i(x,u,\zeta_I(\nabla u))\quad\mbox{in }
L^{s'}(\Omega,\omega_i^{-s'/p}\bar\omega^{-s'/q})\ \
\forall i\in I^c. $$
On the other hand, from Lemma \ref{lm2.4} we obtain,
 $$\frac{\partial u_n}{\partial
x_i}v\rightharpoonup \frac{\partial u}{\partial x_i}v \quad
\mbox{in } L^s(\Omega,\omega_i^{s/p}\bar\omega^{s/q}),
$$
for any $v\in L^{q}(\Omega,\bar\omega)$ and so for any $v\in X$,
$$
\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}c_i(x,u_n,\zeta_I(\nabla u_n))\frac{\partial u_n}{\partial
x_i}v\,dx=\int_\Omega \sum_{i\in I^c}c_i(x,u,\zeta_I(\nabla
u))\frac{\partial u}{\partial x_i}v\,dx
$$
for any $v\in X$.
\\
\textbf{Proof of (iv).}
As before, the growth conditions (H2') implies that, the sequence
$\{ b_i(x,u_n,\zeta_{I^c}(\nabla u_n))\ \ i\in I^c\}$
is bounded in $L^{p'}(\Omega,\omega_i^*)$.
Next, we show that,
$$\int_\Omega \sum_{i\in
I^c}\{b_i(x,u,\zeta_{I^c}(v))-h_i\}(v_i-\frac{\partial u}{\partial
x_i})\,dx\geq 0\ \ \mbox{ for all } v=(v_i)\in
{\prod_{i=1}^NL^p(\Omega,\omega_i)}, \eqno{(3.17)}
$$
here $h_i$ stands for the weak limit of $\{b_i(x,u_n,\zeta_{I^c}(\nabla
u_n)),\ i\in I^c\}$ in $L^{p'}(\Omega,\omega_i^{1-p'})$. Indeed by
(3.6)
we have
$$\limsup_{n\to \infty}\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u_n))\frac{\partial u_n}{\partial
x_i}\,dx\leq \int_\Omega \sum_{i\in I^c}h_i\frac{\partial u}{\partial
x_i}\,dx\eqno{(3.18)}$$
and from (2.5), we obtain for any $v=(v_i)\in
{\prod_{i=1}^NL^p(\Omega,\omega_i)}$,
\begin{eqnarray*}
\lefteqn{ \int_\Omega\sum_{i\in I^c}b_i(x,u_n,\zeta_{I^c}(\nabla
u_n))\frac{\partial
u_n}{\partial x_i}\,dx}\\
&\geq&\int_\Omega\sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u_n))v_i\,dx
+\int_\Omega\sum_{i\in I^c}b_i(x,u_n,\zeta_{I^c}(v))(\frac{\partial
u_n}{\partial x_i}-v_i)\,dx.
\end{eqnarray*}
Letting $n\to\infty$ we conclude by (3.18) that,
$$\int_\Omega \sum_{i\in I^c}h_i\frac{\partial u}{\partial
x_i}\,dx\geq\int_\Omega \sum_{i\in I^c}h_iv_i\,dx+\int_\Omega\sum_{i\in
I^c}b_i(x,u,\zeta_{I^c}(v))(\frac{\partial u}{\partial x_i}-v_i)\,dx$$
and hence (3.17) follows.
Choosing $v=\nabla u+t\tilde w$ with $t>0, \tilde w=(\tilde w_i) \in
{\prod_{i=1}^NL^p(\Omega, \omega_i)}$ and letting
$t\to
0$ we obtain,
$$h_i=b_i(x,u,\zeta_{I^c}(\nabla u))\quad\mbox{a.e. in }\Omega,
$$ which gives,
$$\lim_{n\to \infty}\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u_n))\frac{\partial v}{\partial
x_i}\,dx=\int_\Omega \sum_{i\in I^c}b_i(x,u,\zeta_{I^c}(\nabla
u))\frac{\partial v}{\partial x_i}\,dx
$$
for all $v\in X$. \hfil$\square$

\paragraph{b)} We shall prove that
$$ \liminf_{n\to \infty}\langle Tu_n,u_n\rangle \geq \langle Tu,u
\rangle \eqno{(3.19)}
$$
by  (2.4) and (2.5) we have
\begin{align*}
&\int_\Omega {\sum_{i\in I}} b_i(x,u_n,\zeta_I(\nabla
u_n))\frac{\partial u_n}{\partial x_i}\,dx+\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u_n))\frac{\partial u_n}{\partial
x_i}\,dx\\
& \geq \int_\Omega \sum_{i\in I}b_i(x,u_n,\zeta_I(\nabla
u_n))\frac{\partial
u}{\partial x_i}\,dx+\int_\Omega \sum_{i\in I}b_i(x,u_n,\zeta_I(\nabla
u))(\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial
x_i})\,dx\\
& + \int_\Omega \sum_{i\in I^c}b_i(x,u_n,\zeta_{I^c}(\nabla
u_n))\frac{\partial u}{\partial x_i}\,dx+\int_\Omega \sum_{i\in
I^c}b_i(x,u_n,\zeta_{I^c}(\nabla u))(\frac{\partial u_n}{\partial
x_i}-\frac{\partial u}{\partial x_i})\,dx,
\end{align*}
then  letting $n\to\infty$, and using (3.8) and (3.9), we
conclude (3.19).

\paragraph{Example} % 3.1. }
Many ideas in this example have adapted from the corresponding examples
1-2 in \cite{dr-ku-mu}.
We shall suppose that the weight functions satisfy:
$\omega_{i_0}(x)\equiv 0$ for some $i_0\in I^c$, and
$\omega_i(x)=\omega(x)$, $x\in \Omega$, for all
$i\in I\cup I^c$ and $i\neq i_0$ with $\omega(x)>0$ a.e. in $\Omega$.
Then, we can consider the Hardy inequality in the form
$$
\Big(\int_\Omega |u(x)|^q\bar\omega(x)\,dx\Big)^{1/q}\leq
c\Big(\sum_{i\neq i_0}\int_\Omega |\frac{\partial u}{\partial
x_i}|^p\omega
\Big)^{1/p}\eqno{(3.20)}
$$
for every $u\in X$ with a constant $c>0$ independent of $u$ and for
some $q\geq p'$.

Let us consider the Carath\'eodory functions:
$$
\begin{gathered}
b_i(x,\eta,\zeta_I)=\omega|\zeta_i|^{p-1}\mathop{\rm sgn}\zeta_i+\omega_0
A_0(\eta)\quad\mbox{for } i\in I\\
b_i(x,\eta,\zeta_{I^c})=\omega|\zeta_i|^{p-1}\mathop{\rm sgn}\zeta_i+\omega_0
A_0(\eta)\quad \mbox{for }i\in I^c\mbox{ and }i\neq i_0\\
b_{i_0}(x,\eta,\zeta_{I^c})=\omega_0 A_0(\eta)\\
c_0(x,\eta,\zeta_I)=\sum_{j\in I}\omega^{1/q'}\bar
\omega^{1/q}|\zeta_j|^{\frac{p}{q'}}+\omega_0 B_0(\eta)\\
c_i(x,\eta,\zeta_I)=\sum_{j\in
I}\omega^{1/p+1/r}\bar\omega^{1/q}|\zeta_j|^{p/r}+\omega_0 B_1(\eta)\quad
\mbox{for } i\in I^c,
\end{gathered}
\eqno{(3.21)}
$$
with $1/p+1/r+1/q <1$.
The above functions define by $(3.21)$ satisfies the growth conditions
(H2')
if we suppose that
$$
\begin{gathered}
|\omega_0 A_0(\eta)|\leq \beta_1\omega^{1/p}
\bar\omega^{1/p'}|\eta|^{q/p'}\\
|\omega_0 B_0(\eta)|\leq \beta_2\bar\omega |\eta|^{q/q'}\\
|\omega_0 B_1(\eta)|\leq \beta_3\omega^{1/p}
\bar\omega^{1/q+1/r}|\eta|^{q/r},
\end{gathered}
\eqno{(3.22)}
$$
with $\beta_j \ j=1,2,3$ are some positive constants.
In particular, let us use the special weight functions $\omega_0,
\omega, \bar\omega$ expressed in terms of the distance to the boundary
$\partial \Omega$: denote $d(x)=\mathop{\rm dist}(x,\partial\Omega)$ and set
$$
\omega(x)=d^\lambda(x),\quad \omega_0(x)=d^{\lambda_0}(x),\quad
\bar\omega(x)=d^\mu(x).
$$
In this case the condition $(3.22)$ writes as
$$
\begin{gathered}
|A_0(\eta)|\leq \beta_1 d^{\lambda/p+\mu/p'-\lambda_0}|\eta|^{q/p}\\
|B_0(\eta)|\leq \beta_2 d^{\mu-\lambda_0}|\eta|^{q/q'}\\
|B_1(\eta)|\leq  \beta_3
d^{\lambda/p+\mu/q+\mu/r-\lambda_0}|\eta|^{q/r},
\end{gathered} \eqno{(3.23)}
$$
and the Hardy inequality reads
$$
\Big(\int_\Omega |u(x)|^q\;d^\mu(x)\,dx\Big)^{1/q}\leq
c\Big(\sum_{i\neq i_0}\int_\Omega |\frac{\partial u}{\partial
x_i}|^p\;d^\lambda(x)\,dx \Big)^{1/p},\eqno{(3.24)}
$$
and the corresponding imbedding (1.7) is compact for $1\leq p\leq
q<\infty$
(resp. $1\leq q<p<\infty,$), if and only if $\lambda\neq p-1$,
$\frac{N}{q}-\frac{N}{p}+1\geq 0$,
$\frac{\mu}{q}-\frac{\lambda}{p}+\frac{N}{q}-\frac{N}{p}+1>0$,
(resp. $\lambda\in \mathbb{R}$,
$\frac{\mu}{q}-\frac{\lambda}{p}+\frac{1}{q}-\frac{1}{p}>0$)
(see \cite{op-ku}).
Moreover, the two monotonicity conditions (2.4) and (2.5)
are satisfied:
\begin{multline*}
\sum_{i\in I}\left(b_i(x,\eta,\zeta_I)-b_i(x,\eta,\bar\zeta_I)\right)
\left(\zeta_i-\bar\zeta_i\right)\\
=\omega(x)\sum_{i\in I}\left(|\zeta_i|^{p-1}\mathop{\rm sgn}\zeta_i-|\bar
\zeta_i|^{p-1}\mathop{\rm sgn}\bar\zeta_i\right)
\left(\zeta_i-\bar\zeta_i\right)>0
\end{multline*}
for almost all $x\in \Omega$ and for all $\zeta,\bar\zeta\in \mathbb{R}^N$
with
$\zeta_I\neq \bar\zeta_I$, since $\omega>0\ a.e.$ in $\Omega$;
and
\begin{multline*}
\sum_{i\in I^c}\left(b_i(x,\eta,\zeta_{I^c})-b_i(x,\eta,\bar\zeta_{I^c})\right)
\left(\zeta_i-\bar\zeta_i\right)\\
=\omega(x)\sum_{\stackrel{i\in I^c}{i\neq i_0}}
\left(|\zeta_i|^{p-1}sgn\zeta_i-|\bar\zeta_i|^{p-1}sgn\bar\zeta_i\right)
\left(\zeta_i-\bar\zeta_i\right)\geq 0
\end{multline*}
for almost all $x\in \Omega$ and for all $\zeta,\bar\zeta\in \mathbb{R}^N$.
This last inequality can not be strict, since for $\zeta_{I^c}\neq
\bar\zeta_{I^c}$ with $\zeta_{i_0}\neq \bar\zeta_{i_0}$ but $
\zeta_i=\bar\zeta_i$ for all $i\in I^c$ and $i\neq i_0$, the
corresponding expression is zero.
Finally, the hypotheses of theorem 3.1 are verify, then the mapping $T$
defined as $(3.1)$ corresponding to $(3.21)$ is pseudo-monotone.

\section{Specific case}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ satisfying the cone
condition.
In this section we assume in addition that the collection of weight
functions $\omega=\{\omega_i(x) \; i=0,\dots,N\}$ satisfy
$\omega_0(x)=1$ and the integrability condition:
There exists $\nu\in ]\frac{N}{p},\infty[\cap
[\frac{1}{p-1},\infty[$  such that
$$
\omega_i^{-\nu}\in L^1(\Omega) \quad \forall i=1,\dots,N.\eqno{(4.1)}
$$
Note that $(4.1)$ is stronger than $(2.2)$.

\begin{remark}[\cite{dr-ku-ni1}] \label{rm4.1} \rm
\begin{enumerate}
\item Assumptions (2.1) and (4.1) imply that,
$$
\||u|\|_X=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial
u(x)}{\partial x_i}|^p\omega_i(x)\,dx\Big)^{1/p}
$$
is a norm defined on $W_0^{1,p}(\Omega,\omega)$ and it's equivalent to
(2.3), and that
$$ W_0^{1,p}(\Omega,\omega)\hookrightarrow\hookrightarrow L^q(\Omega)
\eqno{(4.2)}
$$
for all $1\leq q<p_1^*$ if $p\nu<N(\nu+1)$ and $q\geq 1$ is arbitrary
if $p\nu \geq N(\nu+1)$ where $p_1=\frac{p\nu}{\nu+1}$ and
$p_1^*=\frac{Np_1}{N-p_1}=\frac{Np\nu}{N(\nu+1)-p\nu}$ is the Sobolev
conjugate of $p_1$.

\item Hypotheses (H1)  holds for all $q$ such that $1<q<p_1^*$ and
$\bar\omega\equiv 1$.
\end{enumerate}
\end{remark}
In the sequel, we replace (4.1) by the hypothesis
\begin{enumerate}
\item[(\~H1)]
If $\frac{2N}{N+1}<p<N$  there exists
$ \nu \in ]\frac{N}{p},\infty[\cap]\frac{1}{(p-1)-\frac{p}{p^*}},\infty[$
such that
$\omega_i^{-\nu}\in L^1(\Omega)$, for all $i=1,\dots,N$.
If $p=N$ there exists $ \nu \in ]1,\infty[$ such that
$\omega_i^{-\nu}\in L^1(\Omega)$ for all $i=1,\dots,N$.
If $p>N$ there exist
$\nu \in ]\frac{N}{p-N},\infty[\cap[\frac{1}{(p-1)},\infty[$
such that $\omega_i^{-\nu}\in L^1(\Omega)$ for all $i=1,\dots,N$.
\end{enumerate}

\begin{remark} \label{rm4.2}\rm
\begin{enumerate}
\item Hypothesis (\~H1) guarantees the existence of
$r$ satisfying $\frac{1}{r}+\frac{1}{p}+\frac{1}{p_1^*}<1$, where $p_1^*$
is the Sobolev conjugate of $p_1$ in the case $\frac{2N}{N+1}<p\leq N$ and
where
$p_1^*=\infty$ in the case $p>N$ (since $p_1>N$ due to
$\nu>\frac{N}{p-N}$).
\item If $1<p\leq\frac{2N}{N+1}$ we can't find a real $r>1$ such
that
$\frac{1}{r}+\frac{1}{p}+\frac{1}{p_1^*}<1$, since
$\frac{1}{p}+\frac{1}{p_1^*}>\frac{1}{p}+\frac{1}{p^*}\geq 1$.
\item Note that (\~H1) is stronger than (4.1), then the
compact imbedding (4.2) is satisfied whenever (\~H1) is
assumed.
\end{enumerate}
\end{remark}

\begin{theorem} \label{thm4.1}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$. And
assume that  (2.1), (\~H1),  (H2'), (2.4) and
(2.5) are satisfied.
Then the operator $T$ defined in $(3.1)$ is pseudo-monotone in
$X=W_0^{1,p}(\Omega,\omega)$.
Moreover, assume the degenerate ellipticity condition
$$
\sum_{i=0}^N a_i(x,\xi)\xi_i\geq
c_0\sum_{i=1}^N\omega_i(x)|\xi_i|^p
$$
for a.e.\ $x\in \Omega$, some $c_0>0$ and all $\xi\in \mathbb{R}^{N+1}$.
Then for any $f\in X^*$ the Dirichlet associated problem
$$
\langle Tu,v\rangle =\langle f,v \rangle \quad \mbox{for all }  v\in X,
$$
has at least one  solution $u\in X$.
\end{theorem}

\begin{thebibliography}{00} \frenchspacing

\bibitem{az} E. A{\sc zroul}, {\em Sur certains probl\`emes
elliptiques
(non lin\'eaires d\'eg\'en\'er\'es ou singuliers) et calcul des
variations,}
{Th\`ese de Doctorat d'Etat, Facult\'e des Sciences  Dhar-Mahraz,
F\`es,
Maroc, janvier 2000.}

\bibitem{dr-ku-mu} 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{dr-ku-ni1} P. D{\sc rabek}, A. K{\sc ufner} and F.
N{\sc
icolosi}, {\em Non linear elliptic equations, singular and degenerate
cases,}{ University of West Bohemia, (1996).}

\bibitem{dr-ku-ni 2} P. D{\sc rabek}, A. K{\sc ufner} and F.
N{\sc
icolosi}, {\em On the solvability of degenerated quasilinear elliptic
equations of higher order,}{ Journal of differential equations, {\bf
109}
(1994), 325-347.}

\bibitem{go-mu} J.P. G{\sc ossez} and V. M{\sc ustonen}, {\em
Pseudo-monotoncity and the Leray-Lions condition, Differential Integral
Equations}{ {\bf 6} (1993), 37-45.}

\bibitem{la} R. L{\sc andes}, {\em On Galerkin's method in the
existence
theory of quasilinear elliptic equations,}{ J. Functional Analysis {\bf
39}
(1980), 123-148.}

\bibitem{le-li} J. L{\sc eray} and J.L. L{\sc ions}, {\em Quelques
r\'esultats de Vi$\tilde s$ik sur des probl\`emes elliptiques non
lin\'eaires par les m\'ethodes de Minty-Browder,}{ Bul. Soc. Math.
France
{\bf 93} (1965), 97-107.}

\bibitem{op-ku} B. O{\sc pic} and A. K{\sc ufner}, {\em Hardy-type
inequalities, Pitman Research Notes in Mathematics Series {\bf
219}(Longman Scientific and Technical, Harlow, 1990).}

\end{thebibliography}

\noindent\textsc{Youssef Akdim} (e-mail y.akdim1@caramail.com)\\
\textsc{Elhoussine Azroul} (e-mail elazroul@caramail.com)\\
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P 1796 Atlas F\`es, Maroc

\end{document}