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\AtBeginDocument{{\noindent\small 
2004-Fez conference on Differential Equations and Mechanics \newline 
{\em Electronic Journal of Differential Equations}, 
Conference 11, 2004, pp. 11--22.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}} 
\setcounter{page}{11}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Solvability of quasilinear elliptic problems] 
{On the solvability of degenerated quasilinear elliptic problems}

\author[Y. Akdim, E. Azroul,  M. Rhoudaf\hfil EJDE/Conf/11 \hfilneg]
{Youssef Akdim,  Elhoussine Azroul,  Mohamed Rhoudaf}  % in alphabetical order

\address{D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P. 1796 Atlas  F\`es, Maroc}

\email[Y. Akdim]{akdimyoussef@yahoo.fr} 
\email[E. Azroul]{azroul\_elhoussine@yahoo.fr} 
\email[M. Rhoudaf]{rhoudaf\_mohamed@yahoo.fr}

\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{35J20, 35J25, 35J70} 
\keywords{Weighted Sobolev spaces; variational calculus, Hardy inequality}


\begin{abstract}
 In this article, we study the quasilinear elliptic problem
 \begin{gather*}
 Au = - \mathop{\rm div} (a(x,u,\nabla u)) = f(x,u,\nabla u)
\quad\mbox{in }  \mathcal{D}'(\Omega) \\
 u = 0   \quad\mbox{on }\partial\Omega\,,
 \end{gather*}
 where $A$ is a Leray-Lions operator from $W_0^{1,p}(\Omega,w)$ to
 its dual  $W^{-1,p'}(\Omega,w^*)$. We show that there exists a
solution
 in $W_0^{1,p}(\Omega,w)$ provided that
 $$
 |f(x,r, \xi)|\leq \sigma^{1/q} [ g(x)+|r|^\eta \sigma^{\eta/q}
 +   \sum_{i=1}^N w_i^{\delta/p}(x)|\xi_i|^\delta],
 $$
 where $g(x)$ is a positive function in  $L^{q'}(\Omega)$ and
$\sigma(x)$ is
 weight function and $0 \leq  \eta < \min  (p-1,q-1)$,
 $0 \leq \delta < (p-1)/q'$.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded open set in $\mathbb{R}^N$, $N \geq 2$,
and $p$ be a real number such that $1<p<\infty$.  Let $w=
\{w_i(x)$, $0\leq i\leq N\}$ be a vector  weight functions on
$\Omega$; i.e., each $w_i(x)$ is a measurable a.e. strictly
positive function  on $\Omega$, satisfying some integrability
conditions (see section2). Let us consider the problem
\begin{equation}
\begin{gathered}
Au=f(x,u,\nabla u) \quad \mbox{in } \mathcal{D}'(\Omega)\\
u = 0 \quad \mbox{on } \partial\Omega,
\end{gathered} \label{e1.1}
\end{equation}
where $A$ is a Leray-Lions operator $Au = - \mathop{\rm div}
(a(x,u,\nabla u))$ and  $f(x,r,\xi): \Omega \times \mathbb{R}
\times \mathbb{R}^N \to \mathbb{R}$ is a Carath\'eodory function.
Boccardo, Murat and Puel in \cite{bomupu} studied the problem
\eqref{e1.1} in the non weighted case, with $f$ satisfying the
condition
$$ |f(x,r,\xi)| \leq h(|r|)(1 + |\xi|^p),
$$
where $h$ is  increasing function from $\mathbb{R}^+$ into
$\mathbb{R}^+$. The existence result is proved assuming the
existence of the subsolution and supersolution in  $
W^{1,\infty}(\Omega)$, which play an important roll in their work.
Further in \cite{bomupu88} the author's studied the problem
\eqref{e1.1} with $f$ satisfies the hypotheses
\begin{gather}
|f(x,r, \xi)|\leq \beta[ g(x)+|r|^{p-1} + |\xi|^{p-1}], \label{e0.1}\\
f(x,r,\xi)r\geq\alpha |r|^{p}. \label{e0.2}
\end{gather}
Recently,  Tsang-Hai Kuo and Chiung-Chion Tsai \cite{tsch} proved
an existence result under the assumption
$$
 |f(x,r,\xi)| \leq c (1 + |r|^\delta + |\xi|^\eta).
$$
Our objective in this paper, is to study the problem \eqref{e1.1}
in weighted Sobolev spaces  where $f$ satisfying only  the growth
condition
$$
|f(x,r, \xi)|\leq\sigma^{1/q} [ g(x)+|r|^\eta
\sigma^{\frac{\eta}{q}} + \sum_{i=1}^N
w_i^{\delta/p}(x)|\xi_i|^\delta],
$$
where $g(x)$  is  a positive  function in $L^{q'}(\Omega)$,
$\sigma$ is  a weight function, and
$$ 0 \leq \eta < \min (p-1,q-1),\quad  0 \leq \delta < \frac{p-1}{q'}.
$$
Note that we obtain the existence result without assuming the
condition \eqref{e0.2}  and  without knowing a priori the
existence of subsolutions and  supersolutions. Let us point out
that this work can be see as a generalization of the work in
\cite{bomupu88} and \cite{tsch}.

 \section{Preliminaries and Basic Assumptions}

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
\begin{gather}
w_i \in L^1_{\rm loc}(\Omega), \label{e2.1} \\
w_i^{-1/(p-1)} \in L^1_{\rm loc}(\Omega), \label{e2.2}
\end{gather}
for any $0\leq i\leq N$. We denote by $W^{1,p}(\Omega,w)$ the
space of
 real-valued functions $u \in L^p(\Omega,w_0)$ such that their
derivatives in the sense of distributions satisfies
$$
\frac{\partial u}{\partial x_i} \in L^p(\Omega, w_i)\quad
 \mbox{for } i=1,\dots, N.
$$
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{e2.3}
\end{equation}
The condition \eqref{e2.1} implies that $C_0^{\infty}(\Omega)$ is
a subspace 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 \eqref{e2.3}. Moreover, the condition
\eqref{e2.2} 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 $w^{*} = \{w_i^{*} = w_i^{1-p'}$, $i = 1, \dots , N
\}$ and $p'$ is the conjugate of $p$, i.e. $p' = \frac{p}{p-1}$.
For more details we refer the reader to \cite{drkuni}. We start by
stating the following assumptions:
\begin{itemize}
\item[(H1)] The expression
 \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 {e2.4}
\end{equation}
is a norm defined on $W_0^{1,p}(\Omega,w)$ and its equivalent to
the norm \eqref{e2.3}. And there exist a weight function $\sigma$
on $\Omega$ and a parameter $0 < q < \infty$, such that the Hardy
inequality
 \begin{equation}
 \Big({ \int_{\Omega}|u(x)|^q\sigma(x) \,dx}\Big)^{1/q}
 \leq c\Big({ \sum_{i=1}^N}{\int_{\Omega}|\frac{\partial u}{\partial
x_i}|^p w_i(x)
 \,dx}\Big)^{1/p},\label {e2.5}
 \end{equation}
holds for every $ u \in W_0^{1,p}(\Omega,w)$ with a constant $ c
>0$. Moreover, the imbedding
\begin{equation}
W_0^{1,p}(\Omega,w) \hookrightarrow \hookrightarrow
L^q(\Omega,\sigma), \label{e2.6}
\end{equation}
is compact.
\end{itemize}
Let $A$ be a nonlinear operator from $W_0^{1,p}(\Omega,w)$ into
its dual $W^{-1,p'}(\Omega,w^*)$ defined by
$$ A(u) = -{\mathop{\rm div}}(a(x,u,\nabla u)),
 $$
where $a(x,r,\xi) :\Omega \times\mathbb{R}\times \mathbb{R}^N \to
\mathbb{R}^N $ is a Carath\'eodory vector-valued function  that
satisfies  the following assumption:
\begin{itemize}
\item[(H2)] For $i=1,\dots ,N$,
\begin{gather}
|a_i(x,r, \xi)| \leq \beta w_i^{1/p}(x)[k(x)+
\sigma^{\frac{1}{p'}}|r|^{q/p'} + {
\sum_{j=1}^N}w_j^{\frac{1}{p'}} |\xi_j|^{p-1}]
\label{e2.7} \\
[a(x,r,\xi) - a(x,r,\eta)](\xi-\eta) > 0 \quad \mbox{for  all }
 \xi \neq \eta \in \mathbb{R}^N; \label{e2.8} \\
a(x,r,\xi) \xi \geq  \alpha { \sum_{i=1}^N}w_i |\xi_i|^p,
\label{e2.9}
\end{gather}
where $k(x)$ is a positive function in $ L^{p'}(\Omega)$ and
$\alpha $, $\beta$ are strictly positive constants.
\end{itemize}
Let $f(x,r, \xi)$ is a Carath\'eodory function satisfying the
following assumptions:
\begin{itemize}
\item[(H3)]
\begin{equation}
|f(x,r, \xi)|\leq \sigma^{1/q} [ g(x)+|r|^\eta
\sigma^{\frac{\eta}{q}} +  {
\sum_{i=1}^N}w_i^{\delta/p}(x)|\xi_i|^\delta], \label{e2.10}
\end{equation}
where $g(x)$ is a positive function in $L^{q'}(\Omega)$, and
\begin{equation}
0 \leq \eta <  \min(p-1,q-1),\quad  0 \leq \delta <
\frac{p-1}{q'}. \label{e2.11}
\end{equation}
\end{itemize}

\section{Main result}

Consider the  problem
\begin{equation}
\begin{gathered}
 - \mathop{\rm div}  \ a(x,u,\nabla u) = f(x,u,\nabla u) \quad
  \mbox{in } D'(\Omega) \\
  u = 0 \quad   \mbox{on } \partial \Omega \,.
\end{gathered}\label{e3.1}
\end{equation}

\begin{theorem} \label{thm3.1}
 Under hypotheses (H1)-(H3), there exist at least one solution to
\eqref{e3.1}.
\end{theorem}

We first give some definition and some lemmas that  will be used
in the proof of this theorem.


\noindent\textbf{Definition}\; Let $Y$ be a separable reflexive
Banach space, the operator $B$ from $Y$ to its dual $Y^*$ is
called of the calculus of  variations type, if $B$ is bounded and
is of the from,
 \begin{equation}B(u)=B(u,u), \label{e 3.2}
\end{equation}
where $(u,v) \to B(u,v)$ is an operator $Y \times Y$ into $Y^*$
satisfying the following properties:
\begin{equation}
\begin{aligned}
&\hbox{For $u \in Y$, the mapping $v \mapsto B(u,v)$ is bounded
and
hemicontinuous}\\
&\mbox{from $Y$ to $Y^*$ and $(B(u,u)-B(u,v),u-v) \geq 0$;}
\end{aligned} \label{e3.3}
\end{equation}
for $v \in Y$, the mapping $u \mapsto B(u,v)$ is bounded and
hemicontinuous from $Y$ to $Y^*$;
\begin{align}
&\begin{aligned} &\mbox{If $u_n \rightharpoonup u$  weakly in $Y$
and  if
$(B(u_n,u_n)-B(u_n,u),u_n-u) \to 0$,} \\
&\mbox{then $B(u_n,v) \rightharpoonup B(u,v)$ weakly  in $Y^*$,
for all $v \in Y$;}
\end{aligned} \label{e3.4}
\\
&\begin{aligned} &\mbox{If $u_n \rightharpoonup u$ weakly  in $Y$
and if
$B(u_n,v) \rightharpoonup \psi$ weakly in $Y^*$,} \\
&\mbox{then $(B(u_n,v),u_n) \to (\psi,u)$.}
\end{aligned} \label{e3.5}
\end{align}


\begin{lemma}[\cite{akazbe}] \label{lem3.1}
Let $g \in L^q(\Omega,\gamma)$, $g_n \in L^q(\Omega,\gamma)$, and
$\|g_n\|_{q,\gamma} \leq c$ $(1<q<\infty)$. If $g_n(x) \to g(x)$
a.e. in $\Omega$, then $g_n \rightharpoonup g$ weakly in
$L^q(\Omega,\gamma)$, where $\gamma$ is a weight function on
$\Omega$.
\end{lemma}

\begin{lemma} \label{lem3.2}
If $u_n \rightharpoonup u$ in $W_0^{1,p}(\Omega,w)$ and $v \in
W_0^{1,p}(\Omega,w)$, then $a_i(x,u_n, \nabla v) \to a_i(x,u,
\nabla v)$ in $L^{p'}(\Omega, w_i^*)$.
\end{lemma}

\begin{proof} From (H2), it follows that
\begin{equation}
\begin{aligned}
|a_i(x,u_{n},\nabla v)|^{p'} w_i^{\frac{-p'}{p}} &\leq \beta
[k(x)+|u_{n}|^{\frac{q}{p'}}\sigma^\frac{1}{p'} +
\sum_{j=1}^{N}|\frac{\partial v }{\partial
x_j}|^{p-1}w_j^{\frac{1}{p'}}]^{p'}\\
&\leq \gamma [k(x)^{p'}+|u_{n}|^{q}\sigma +\sum_{j=1}^{N}
|\frac{\partial v }{\partial x_j}|^{p}w_j],
\end{aligned} \label{e3.6}
\end{equation}
where $\beta$ and $\gamma$ are positive constants. Since $u_{n}
\rightharpoonup u$ weakly in $W_0^{1,p}(\Omega,w)$ and
$W_0^{1,p}(\Omega,w) \hookrightarrow \hookrightarrow
 L^q(\Omega,\sigma)$, it follows that $u_{n} \to u$ strongly in
 $L^q(\Omega,\sigma)$ and $u_{n} \to u$  a.e. in $\Omega$;
hence
\begin{equation}
|a_i(x,u_{n},\nabla v)|^{p'}w_i^* \to |a_i(x,u,\nabla
v)|^{p'}w_i^* \quad \mbox{ a.e. in } \Omega, \label{e3.7}
\end{equation}
and
$$
\gamma{\Big[k(x)^{p'}+|u_{n}|^{q}\sigma
+\sum_{j=1}^{N}|\frac{\partial v }{\partial x_j}|^{p}w_i \Big]}
\to \gamma{\Big[k(x)^{p'}+|u|^{q}\sigma
+\sum_{j=1}^{N}|\frac{\partial v }{\partial x_j}|^{p}w_j \Big]}
$$
 a.e. in $\Omega$.
Then, By Vitali's theorem,
\begin{equation}
 a_i(x,u_{n},\nabla v) \to a_i(x,u,\nabla v) \quad \mbox{strongly in }
 L^{p'}(\Omega,w_i^*),\mbox{ as }  n \to  +\infty.  \label{e3.8}
\end{equation}
\end{proof}

\begin{lemma}[\cite{akazbe}] \label{lem3.3}
Assume that (H1)--(H2) are satisfied, and let $(u_n)$ be a
sequence in $W_0^{1,p}(\Omega,w)$ 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, $u_n \to u$ in $W_0^{1,p}(\Omega,w)$.
\end{lemma}
 For $v \in W_0^{1,p}(\Omega,w)$, we associate the Nemytskii operator
$F$
 with respect to $f$,
$$
F(v, \nabla v)(x) = f(x,v, \nabla v) \mbox{ a.e.},  x \in  \Omega.
$$

\begin{lemma} \label{lem3.4}
 The mapping $v \mapsto F(v,\nabla v)$ is continuous from the space
 $W_0^{1,p}(\Omega,w)$  to $L^{q'}(\Omega, \sigma^{1-q'})$.
\end{lemma}

\begin{proof}
By hypothesis (H3), we have
$$
|f(x,r, \xi)|\leq \sigma^{1/q} [ g(x)+|r|^\eta
\sigma^{\frac{\eta}{q}} +  {
\sum_{i=1}^N}w_i^{\delta/p}(x)|\xi_i|^\delta].
$$
Thanks to Young's inequality,
\begin{gather*}
 |r|^\eta \sigma^{\eta/q} \leq [\frac{\eta}{q-1} |r|^{q-1}
 \sigma^{(q-1)/q} + 1] \leq [|r|^{q-1} \sigma^{q'} +1],\\
w_i^{\sigma/p}|\xi_i|^\sigma \leq  [w_i^{1/q'}|\xi_i|^{p/q'} + 1],
\end{gather*}
which implies
 $$
 |f(x,r, \xi)| \leq \sigma^{1/q} [(N+2)+ g(x)+|r|^{q-1} \sigma^{1/q'}
 +  { \sum_{i=1}^N}w_i^{1/q'}|\xi_i|^{p/q'}]\,.
 $$
Then
$$
|f(x,r, \xi)|^{q'} \sigma^{-q'/q} \leq c_2[c_1  +
g(x)^{q'}+|r|^{(q-1)q'} \sigma +  { \sum_{i=1}^N}w_i|\xi_i|^p]\,.
$$
Since $f$ is a Carath\'eodory, and for all subset $E$ measurable,
such that $|E|< \eta $, we have
$$
{ \int_{E}|f(x,v, \nabla v)|^{q'} \sigma^{\frac{-q'}{q}} \,dx}
\leq c_2[c_3 + { \int_{E}|v|^q  \sigma \,dx} + { \int_{E} {
\sum_{i=1}^N}w_i|\frac{\partial v}{\partial x_i}|^p \,dx]}\,.
$$
Then by Vitali's theorem, we deduce the continuous of the operator
$F$. Moreover,
\begin{equation}
\Big({ \int_{\Omega }|f(x,v, \nabla v)|^{q'}
\sigma^{\frac{-q'}{q}} \,dx}\Big)^{1/q'}
  \leq c_2[ c + \\\|| v \||^{q/q'} + \|| v \||^{p/q'}].\label{e3.9}
\end{equation}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
{\bf Step (1)} We will show that the operator $B:
W_0^{1,p}(\Omega,w) \to W^{1,p'}(\Omega,w^*)$ defined by  $B(v)=
A(v)-f(x,v,\nabla v)$ is a calcul of  variational.

\noindent\textbf{Assertion 1.} Let
$$
B(u,v) = { -\sum_{i=1}^N}\frac{\partial a_i(x,u, \nabla
v)}{\partial x_i} - f(x,u, \nabla u).
$$
Then $B(v,v)= B(v)$  for all $v \in W_0^{1,p}(\Omega,w)$.

\noindent\textbf{Assertion 2.} We claim that the operator $v \to
B(u,v)$ is bounded for all $u \in W_0^{1,p}(\Omega,w)$. Let $\psi
\in W_0^{1,p}(\Omega,w)$, we have
 $$\langle B(u,v), \psi) \rangle
 = { \sum_{i=1}^N} { \int_{\Omega} a_i(x,u, \nabla v)
  \frac{\partial \psi}{\partial x_i}}
 - { \int_{\Omega} f(x,u, \nabla u) \psi \,dx}.
$$
 From H\"older's inequality, the growth condition \eqref{e2.7} and the
compact imbedding \eqref{e2.6}, we obtain
\begin{equation}
\begin{aligned}
&{ \sum_{i=1}^N} { \int_{\Omega} a_i(x,u, \nabla v)
\frac{\partial \psi}{\partial x_i}} \\
& \leq  { \sum_{i=1}^N} \Big({ \int_{\Omega} |a_i(x,u, \nabla
v)|^{p'} w_i^{\frac{-p'}{p}} \,dx} \Big)^{1/p'} \Big({
\int_{\Omega} |\frac{\partial \psi}{\partial x_i}|^pw_i \,dx}
\Big) ^{1/p} \\
& \leq  c_4 \|| \psi\|| { \sum_{i=1}^N} \Big({
\int_{\Omega}k(x)^{p'} + |u|^q \sigma + {
\sum_{j=1}^N}|\frac{\partial v}{\partial x_j}|^pw_j \,dx}
\Big)^{1/p'} \\
& \leq  c_5 \|| \psi\|| \big[c_6 + \|| u \||^{\frac{q}{p'}} + \||
v \||^{p-1} \big].
\end{aligned} \label{e3.11}
\end{equation}
Similarly, $$ { \int_{\Omega}f(x,u, \nabla u) \psi \,dx}  \leq
 \Big({ \int_{\Omega}|f(x,u, \nabla u)|^{q'}
\sigma^{\frac{-q'}{q}}\,dx}\Big)^{1/q'}
 \Big({ \int_{\Omega}|\psi|^q \sigma \,dx} \Big)^{1/q},
$$
by \eqref{e2.5} and \eqref{e3.9}, we have,
\begin{equation}
{ \int_{\Omega}f(x,u, \nabla u) \psi \,dx} \leq c \|| \psi\||
 \big[c_7 + \|| u \||^{q-1} + \|| u \||^{p/q'} \big]. \label{e3.12}
\end{equation}
Since $u$ and $v$ belong to $W_0^{1,p}(\Omega,w)$ and in view of
\eqref{e3.11} and \eqref{e3.12}, we deduce that $\langle B(u,v),
\psi \rangle$ is bounded in $W_0^{1,p}(\Omega,w)\times
W_0^{1,p}(\Omega,w) $.

We claim that the operator $v \to B(u,v)$ is hemicontinuous for
all $u \in W_0^{1,p}(\Omega,w)$, i.e., the operator $\lambda \to
\langle B(u,v_1+\lambda v_2), \psi \rangle$ is continuous for all
$v_1, v_2, \psi \in W_0^{1,p}(\Omega,w)$. Since $a_i$ is a
Carath\'eodory function,
$$
a_i(x,u, \nabla (v_1+\lambda v_2)) \to  a_i(x,u, \nabla v_1) \quad
\mbox{a.e. in } \Omega \mbox{ as } \lambda \to 0.
$$
Further, we know from \eqref{e2.7} that $(a_i(x,u, \nabla (v_1 +
\lambda v_2))_\lambda$ is bounded in $L^{p'}(\Omega,w_i^*)$; thus,
by Lemma \ref{lem3.1}, we conclude
\begin{equation}
a_i(x,u, \nabla (v_1 + \lambda v_2)) \rightharpoonup   a_i(x,u,
\nabla v_1) \quad \mbox{weakly in } L^{p'}(\Omega,w_i^*),\mbox{ as
} \lambda \to 0\,. \label{e3.13}
\end{equation}
Hence,
\begin{equation}
\begin{aligned}
&{ \lim_{\lambda \to 0}}\langle B(u,v_1+\lambda v_2), \psi \rangle \\
& =  { \lim_{\lambda \to 0}}{ \sum_{i=1}^N} { \int_{\Omega}
 a_i(x,u, \nabla (v_1+ \lambda v_2)) \frac{\partial \psi}{\partial x_i}
\,dx}
 - { \int_{\Omega} f(x,u, \nabla u) \psi \,dx } \\
& =  { \sum_{i=1}^N} { \int_{\Omega} a_i(x,u, \nabla v_1)
 \frac{\partial \psi}{\partial x_i} \,dx}
 - { \int_{\Omega} f(x,u, \nabla u) \psi \,dx} \\
& =  \langle B(u,v_1), \psi \rangle  \quad\mbox{for all } v_1,
v_2, \psi \in W_0^{1,p}( \Omega,w).
\end{aligned} \label{e3.14}
\end{equation}
Similarly, we show that $u \to \langle B(u,v), \psi\rangle $ is
bounded and hemicontinuous for all  $v \in W_0^{1,p}(\Omega,w)$.
Indeed.  By \eqref{e3.9}, we have $f ((x,u_1+ \lambda u_2, \nabla
(u_1+ \lambda u_2)) )_ \lambda$ is bounded in $L^{q'}(\Omega,
\sigma^{1-q'})$ and as $f$ is a Carath\'eodory function then
$$
f(x,u_1+ \lambda u_2, \nabla (u_1+ \lambda u_2)) \to f(x,u_1,
\nabla u_1) \quad \mbox{ a.e.  in }\Omega.
$$
Hence, Lemma \ref{lem3.1} gives,
\begin{equation}
f(x,u_1+ \lambda u_2, \nabla (u_1+ \lambda u_2)) \rightharpoonup
f(x,u_1, \nabla u_2) \quad \mbox {weakly  in } L^{q'}(\Omega,
\sigma^{1-q'}) \mbox{ as } \lambda \to 0, \label{e3.15}
\end{equation}
On the other hand, as in \eqref{e3.13}, we have
\begin{equation}
a_i(x,u_1+ \lambda u_2, \nabla v) \rightharpoonup  a_i(x,u_1,
\nabla v) \quad \mbox{in } L^{p'}(\Omega,w_1^*),\mbox{ as }
\lambda \to 0. \label{e3.16}
\end{equation}
Combining \eqref{e3.15} and \eqref{e3.16},  we conclude that, $u
\to B(u,v)$ is bounded and hemicontinuous.

\noindent\textbf{Assertion 3.} From \eqref{e2.8}, we have,
$$\langle B(u,u)-B(u,v),u-v \rangle = { \sum_{i=1}^N} { \int_{\Omega}
\left(a_i(x,u, \nabla u)- a_i(x,u, \nabla v) \right)
(\frac{\partial u}{\partial x_i}-\frac{\partial v}{\partial x_i})}
\geq 0
$$

\noindent\textbf{Assertion 4.}
 Assume that $u_n \rightharpoonup  u$ weakly in $W_0^{1,p}(\Omega,w)$
and $\langle B(u_n,u_n)-B(u_n,u),u_n-u \rangle \to 0$, we claim
that $B(u_n,v) \rightharpoonup  B(u,v)$ weakly in
$W^{-1,p'}(\Omega,w^*)$.

We can write $\langle B(u_n,u_n)-B(u_n,u),u_n-u \rangle  \to 0$ as
$n \to \infty$,
\begin{align*}
&\Big\langle { \sum_{i=1}^N}- \big[\frac{\partial}{\partial x_i}
a_i(x,u_n, \nabla u_n) - \frac{\partial}{\partial x_i} a_i(x,u_n,
\nabla u) \big],u_n-u
\Big\rangle \\
&=   { \sum_{i=1}^N} { \int_{\Omega} \left[a_i(x,u_n, \nabla u_n)
- a_i(x,u_n, \nabla u) \right] \frac{\partial}{\partial
x_i}(u_n-u) \,dx} \to 0
\end{align*}
Then, by Lemma \ref{lem3.3}, we have $u_n \to u$ strongly in
$W_0^{1,p}(\Omega,w)$ and it follows from Lemma \ref{lem3.4} that
 \begin{equation}
 f(x,u_n, \nabla u_n) \to f(x,u, \nabla u) \quad \mbox{ in }
  L^{q'}(\Omega, \sigma^{1-q'}). \label{e3.17}
\end{equation}
Since $u_n \to u$ in $L^p(\Omega,w)$ and by \eqref{e2.7} and
$W_0^{1,p}(\Omega,w) \hookrightarrow \hookrightarrow L^q(\Omega,
\sigma)$, we can obtain from Lemma \ref{lem3.2} that
\begin{equation}
a_i(x,u_n, \nabla v) \to a_i(x,u, \nabla v)  \quad \mbox{in }
 L^{p'}(\Omega,w_i^*)\,. \label{e3.18}
 \end{equation}
This implies
 \begin{equation}
{\int_{\Omega} a_i(x,u_n, \nabla v)\frac{\partial \psi}{\partial
x_i} \,dx} \to { \int_{\Omega} a_i(x,u, \nabla v)\frac{\partial
\psi}{\partial x_i} \,dx}. \label{e3.19}
\end{equation}
On the other hand, by H\"olders inequality,
$$
{ \int_{\Omega} |f(x,u_n, \nabla u_n) \psi| \,dx} \leq  \Big( {
\int_{\Omega} |f(x,u_n, \nabla u_n)|^{q'} \sigma^{1-q'} \,dx}
\Big)^{1/q'} \Big({ \int_{\Omega} |\psi|^q \sigma \,dx}
\Big)^{1/q}.
$$
Thanks to \eqref{e3.17}, \eqref{e2.5}, and Lebegue's dominated
convergence theorem,  we obtain
\begin{equation}
{\int_{\Omega} f(x,u_n, \nabla u_n) \psi \,dx} \to { \int_{\Omega}
f(x,u, \nabla u) \psi \,dx}\,. \label{e3.20}\end{equation} Then,
we have
\begin{align*}
{ \lim_{n\to \infty}} \langle B(u_n,v), \psi \rangle &= {
\lim_{n\to \infty}}{ \sum_{i=1}^N} { \int_{\Omega} a_i(x,u_n,
\nabla v) \frac{\partial \psi}{\partial x_i} \,dx}
-{ \int_{\Omega} f(x,u_n, \nabla u_n) \psi \,dx}\\
&= { \sum_{i=1}^N} { \int_{\Omega} a_i(x,u, \nabla v)
 \frac{\partial \psi}{\partial x_i} \,dx}
 -{ \int_{\Omega} f(x,u, \nabla u) \psi \,dx}\\
&= \langle B(u,v), \psi \rangle , \quad \mbox {for all } \psi \in
W_0^{1,p}(\Omega,w).
\end{align*}


\noindent\textbf{ Assertion 5.} Assume $u_n\rightharpoonup u$
weakly   in $W_0^{1,p}(\Omega,w)$ and $B(u_n,v) \rightharpoonup
\psi$ weakly in  $W^{-1,p'}(\Omega,w)$. We claim that $\langle
B(u_n,v), u_n\rangle \to \langle \psi,u\rangle$. Thanks to $u_n
\rightharpoonup u $ in $ W_0^p(\Omega,w)$, we obtain by Lemma
\ref{lem3.2},
\begin{equation}
a_i(x,u_n, \nabla v) \to a_i(x,u, \nabla v) \quad \mbox{strongly
in } L^{p'}(\Omega,w_i^*)  \mbox{ as } n \to +\infty.
\label{e3.21}
\end{equation}
And so
\begin{equation}{
\int_{\Omega} a_i(x,u_n, \nabla v)\frac{\partial u_n}{\partial
x_i} \,dx} \to { \int_{\Omega} a_i(x,u, \nabla v)\frac{\partial
u}{\partial x_i} \,dx}. \label{e3.22}
\end{equation}
Hence together with
\begin{equation}
{ \sum_{i=1}^N} { \int_{\Omega} a_i(x,u_n, \nabla v)
\frac{\partial u}{\partial x_i} \,dx}-{ \int_{\Omega} f(x,u_n,
\nabla v)u \,dx} \to \langle \psi,u\rangle , \label{e3.23}
\end{equation} we have
\begin{align*}
\langle B(u_n,v), u_n) \rangle & =  { \sum_{i=1}^N} {
\int_{\Omega} a_i(x,u_n, \nabla v) \frac{\partial u_n}{\partial
x_i} \,dx}
-{ \int_{\Omega} f(x,u_n, \nabla u_n)u_n \,dx} \\
& = { \sum_{i=1}^N} { \int_{\Omega} a_i(x,u_n, \nabla v)(
\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial x_i})
\,dx} +{ \sum_{i=1}^N} { \int_{\Omega} a_i(x,u_n, \nabla v)
\frac{\partial u}{\partial x_i} \,dx} \\
&\quad -  { \int_{\Omega} f(x,u_n, \nabla u_n)u \,dx} -{
\int_{\Omega} f(x,u_n, \nabla u_n)(u_n-u) \,dx}.
\end{align*}
But in view of \eqref{e3.21} and \eqref{e3.22}, we obtain
 \begin{equation}
 { \sum_{i=1}^N} { \int_{\Omega} a_i(x,u_n, \nabla v)
 (\frac{\partial u_n}{\partial x_i}-\frac{\partial u}{\partial x_i})
\,dx} \to 0.
 \label{e3.24}\end{equation}
On the other hand, by H\"older's inequality,
\begin{align*}
&{ \int_{\Omega} |f(x,u_n, \nabla u_n)(u_n-u)| \,dx} \\
& \leq  \Big({ \int_{\Omega} |f(x,u_n, \nabla u_n)|^{q'}
\sigma^{1-q'} \,dx}
\Big)^{1/q'} \Big({ \int_{\Omega} |u_n-u|^q \sigma \,dx} \Big)^{1/q} \\
& \leq  c \|u_n-u \|_{L^q(\Omega,\sigma)} \to \quad \mbox{as } n
\to \infty
\end{align*}
i.e.,
\begin{equation}
{ \int_{\Omega} f(x,u_n, \nabla u_n)(u_n-u) \,dx} \to 0 \quad
\mbox{as }
 n \to \infty. \label{e3.25}
 \end{equation}
Thanks to \eqref{e3.23}, \eqref{e3.24} and \eqref{e3.25}, we
obtain
$$
\langle B(u_n,v), u_n \rangle \to \langle \psi,u \rangle .
$$

\textbf{Step 2.} We claim that the operator $B$ satisfies the
coercivity condition
\begin{equation}
\lim_{\||v\||{\to + \infty}} \frac{\langle B(v),v \rangle
}{\||v\||} = \infty . \label{e3.10}
\end{equation}
Since
$$ \langle Bv,v \rangle  = { \sum_{i=1}^N} {\int_{\Omega} a_i(x,v,
\nabla v) \frac{\partial v}{\partial x_i}\,dx}-{ \int_{\Omega}
f(x,v, \nabla v)v \,dx}.
$$
Then, using \eqref{e2.9}, we have
\begin{equation}
\langle Bv,v \rangle \geq \alpha { \sum_{i=1}^N}w_i|
\frac{\partial v}{\partial x_i}|^p -{ \int_{\Omega} f(x,v, \nabla
v)v \,dx}. \label{e3.27}
\end{equation}
Moreover,
\begin{equation}
\begin{aligned}
&{ \int_{\Omega} f(x,v, \nabla v)v \,dx}\\
&\leq { \int_{\Omega} \sigma^{1/q} g(x)v \,dx} +{ \int_{\Omega}
|v|^{\eta+1}\sigma^{(\eta+1)/q} \,dx} +{ \int_{\Omega}{
\sum_{i=1}^N}w_i^{\delta/p}| \frac{\partial v}{\partial
x_i}|^\delta \sigma^{1/q} |v| \,dx}.
\end{aligned}\label{e3.28}
\end{equation}
Thanks to  H\"older's inequality and \eqref{e2.5}, we have
\begin{equation}
{ \int_{\Omega} \sigma^{1/q} g(x)v \,dx}
 \leq  \Big({ \int_{\Omega}|g(x)|^{q'} \,dx} \Big)^{1/q'}
 \Big({ \int_{\Omega}|v|^q \sigma \,dx} \Big)^{1/q}
 \leq  c \||v \||.\label{e3.29}
\end{equation}
On the other hand, by H\"older's inequality,
$$
{ \sum_{i=1}^N}w_i^{\delta/p}|\frac{\partial v}{\partial
x_i}|^\delta \sigma^{1/q} |v| \leq c{ \sum_{i=1}^N} \Big({
\int_{\Omega}w_i^\frac{\delta q'}{p}|\frac{\partial v}{\partial
x_i}|^{\delta q'} \,dx} \Big)^{1/q'} \Big({ \int_{\Omega}|v|^q
\sigma  \,dx} \Big)^{1/q}.
$$
In view of \eqref{e2.5}, we have
\begin{equation}
{ \sum_{i=1}^N}w_i^{\delta/p}|\frac{\partial v}{\partial
x_i}|^\delta \sigma^{1/q} |v| \leq c{ \sum_{i=1}^N} \Big({
\int_{\Omega}w_i^\frac{\delta q'}{p}| \frac{\partial v}{\partial
x_i}|^{\delta q'} \,dx} \Big)^{1/q'} \||v\||.
 \label{e3.30}
\end{equation}
Since  $0 \leq \frac{\delta q'}{p}<1$, hence by H\"older's
inequality, we deduce
\begin{equation}
\Big({ \int_{\Omega}w_i^{\delta/ q'}{p}|\frac{\partial v}{\partial
x_i} |^{\delta q'} \,dx} \Big)^{1/q'} \leq \Big({ \int_{\Omega}w_i
|\frac{\partial v}{\partial x_i}|^p \,dx} \Big )^{\delta/p},
\label{e3.31}
\end{equation}
remark that,
\begin{equation}
(a+b)^r \geq c(a^r+b^r) \quad \mbox{if }  0 \leq r<1.
\label{e3.32}
\end{equation}
Combining  \eqref{e3.30}, \eqref{e3.31} and \eqref{e3.32}, we
conclude that
\begin{equation}
 { \sum_{i=1}^N}w_i^{\delta/p}|\frac{\partial v}{\partial x_i}|^\delta
 \sigma^{1/q} |v|
 \leq  c \||v\|| \Big({ \sum_{i=1}^N}{ \int_{\Omega}w_i|
 \frac{\partial v}{\partial x_i}|^p \,dx} \Big)^{\delta/p}
 \leq  c\||v\|| \; \||v\||^\delta. \label{e3.33}
\end{equation}
Further, $ 0 \leq \frac{\eta+1}{q} < 1 $, then by  H\"older's
inequality and \eqref{e2.6}, we deduce
\begin{equation}
{ \int_{\Omega}|v|^{\eta+1}\sigma^{(\eta+1)/q} \,dx} \leq c
\||v\||^{\eta+1}. \label{e3.34}\end{equation} Then from
\eqref{e3.27}, \eqref{e3.29}, \eqref{e3.33} and \eqref{e3.34}, we
deduce that
$$
\langle Bv,v \rangle  \geq \alpha \||v\||^{p-1}-c_1 -c_2
\||v\||^\eta-c_3 \||v\||^{\delta-1}
$$
and since $p-1>\eta$ and $p>\delta$, we conclude that
$\frac{\langle Bv,v \rangle }{\|v\|} \to + \infty$. Finally, the
proof of Theorem is complete thanks to the classical Theorem in
\cite{li}.
\end{proof}

\section{Examples}
Let us consider the Carath\'eodory functions
$$ a_i(x,r,\xi) = w_i|\xi_i|^{p-1}\mathop{\rm sgn}(\xi_i)
$$
Where $w_i(x) (i=1,\dots ,N) $ are a given weight functions
strictly positive almost everywhere in $\Omega$. We shall assume
that the weight function satisfies
 $w_i(x) = w(x)$, $x \in \Omega$ for $i=0,\dots ,N$.
It is easy to show that the $a_i(x,s,\xi)$ are Carath\'eodory
function satisfying the growth condition \eqref{e2.7} and the
coercivity \eqref{e2.9}. On the other side, the monotonicity
condition \eqref{e2.8}
 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)\\
&=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) > 0
\end{align*}
 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
$\Omega$. We consider the Hardy inequality \eqref{e2.5} in the
form
$$
\Big({ \int_{\Omega}|u(x)|^q \sigma(x) \,dx}\Big)^{1/q} \leq
c\Big({\int_{\Omega}|\nabla u(x)|^p w(x)\,dx}\Big)^{1/p},
$$
where $\sigma $ and $ q $ are defined in \eqref{e2.5}. In
particular, let us use a special weight functions $w$ and $\sigma$
expressed in terms of the distance to the bounded $\partial
\Omega$. Denote $d(x)=\mathop{\rm dist}(x,\partial \Omega)$ and
set
$$
w(x)=d^\lambda(x), \quad \sigma(x)=d^\mu(x).
$$
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:
\begin{itemize}
\item[(i)] For, $1<p \leq q< \infty$,
\begin{equation}
\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.
\label{e5.1}
\end{equation}
\item[(ii)] For $1 \leq q <p< \infty$,
\begin{equation}
\lambda<p-1, \quad
\frac{\mu}{q}-\frac{\lambda}{p}+\frac{1}{q}-\frac{1}{p}+1 >0.
\label{e5.2}
\end{equation}
\end{itemize}

\noindent\textbf{Remarks.}\; 1. Condition \eqref{e5.1} or
Condition \eqref{e5.2} is sufficient for the compact imbedding
\eqref{e2.6} to hold; see for example \cite[example 1]{drkumu},
\cite[example 1.5]{drkuni}, and  \cite[Theorems 19.17, 19.22]{Ku}.

Let us consider the Carath\'eodory function
$$
f(x,r,\xi)= d^{\frac{\mu}{q}}(x) \Big(d^{\frac{\mu
\delta}{q}}(x)|r|^\eta + { \sum_{i=1}^N} d^{\frac{\lambda
\delta}{p}}(x)|\xi_i|^\delta + g(x) \Big),
$$
with $g \in L^{q'}(\Omega)$, $\sigma(x)$ is weight function and $0
\leq  \eta < \min  (p-1,q-1)$, $0 \leq \delta < \frac{p-1}{q'}$.
Because of its definition, $ f(x,r,\xi)$ satisfies the growth
condition \eqref{e2.10}. Also the hypotheses of Theorem
\ref{thm3.1} are satisfied. Therefore,
 the problem
\begin{align*}
&\sum_{i=1}^N \int_\Omega \Big(d^\lambda (x)|\frac{\partial
u}{\partial x_i}|^{p-2}
  \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial
x_i}\Big)\,dx\\
&= \int_\Omega d^{\mu/q}(x) \Big(d^{\mu \delta/q}(x)|u|^\eta + {
\sum_{i=1}^N} d^{\lambda \delta/p} (x)|\xi_i|^\delta + g(x) \Big)v
\,dx\,,
\end{align*}
for all $v\in W_0^{1,p}(\Omega,w)$, has at last one  solution.


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\end {document}