\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Strongly nonlinear degenerated elliptic unilateral problems } { Y. Akdim, E. Azroul, \& A. Benkirane} \begin{document} \setcounter{page}{25} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 25--39. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Strongly nonlinear degenerated elliptic unilateral problems via convergence of truncations % \thanks{ {\em Mathematics Subject Classifications:} 35J15, 35J70, 35J85. \hfil\break\indent {\em Key words:} Weighted Sobolev spaces, Hardy inequality, variational ineqality, \hfil\break\indent strongly nonlinear degenerated elliptic operators, truncations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002. } } \date{} \author{Youssef Akdim, Elhoussine Azroul, \& Abdelmoujib Benkirane} \maketitle \begin{abstract} We prove an existence theorem for a strongly nonlinear degenerated elliptic inequalities involving nonlinear operators of the form $Au+g(x,u,\nabla u)$. Here $A$ is a Leray-Lions operator, $g(x,s,\xi)$ is a lower order term satisfying some natural growth with respect to $|\nabla u|$. There is no growth restrictions with respect to $|u|$, only a sign condition. Under the assumption that the second term belongs to $W^{-1,p'}(\Omega,w^*)$, we obtain the main result via strong convergence of truncations. \end{abstract} \numberwithin{equation}{section} \newtheorem{lem}{Lemma}[section] \newtheorem{thm}{Theorem}[section] \newtheorem{Def}{Definition}[section] \newtheorem{rem}{Remark}[section] \section{Introduction} Let $\Omega$ be a bounded open set of $\mathbb{R}^N$ and $p$ a real number such that $10$ independent of $u$. Moreover, the imbedding $$ X\hookrightarrow\hookrightarrow L^q(\Omega, \sigma), \eqno{(2.13)} $$ expressed by the inequality (2.12) is compact. Note that $(X,\||.|\|_X)$ is a uniformly convex (and thus reflexive) Banach space. \begin{rem} \label{rmk2.1} \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 $$ w_i^{-\nu}\in L^1(\Omega) $$ for all $i=1,\dots ,N$ (which is stronger than (2.2)). Then $$ \||u|\|_X=\Big(\sum_{i=1}^N\int_\Omega |\frac{\partial u(x)}{\partial x_i}|^p w_i(x)\,dx\Big)^{1/p} $$ is a norm defined on $W_0^{1,p}(\Omega,w)$ and is equivalent to (2.3). Moreover $$ W_0^{1,p}(\Omega,w)\hookrightarrow\hookrightarrow L^q(\Omega), $$ for all $1\leq q0,\ \mbox{ for all } \xi\not=\eta\in \mathbb{R}^N,\tag{2.15} \\ a(x,s, \xi).\xi\geq\alpha \sum_{i=1}^Nw_i|\xi_i|^{p},\tag{2.16} \end{gather} where $k(x)$ is a positive function in $L^{p'}(\Omega)$ and $\alpha,\beta$ are strictly positive constants. \paragraph{Assumption (H3)} Let $g(x,s,\xi)$ be a Carath\'eodory function satisfying the following assumptions: \begin{gather} g(x,s,\xi)s\geq 0\tag{2.17}\\ |g(x,s,\xi)|\leq b(|s|)\Big(\sum_{i=1}^Nw_i|\xi_i|^{p}+c(x)\Big), \tag{2.18} \end{gather} where $b:\mathbb{R}^+\to \mathbb{R}^+$ is a continuous increasing function and $c(x)$ is a positive function which lies in $L^1(\Omega)$. Now we recall some lemmas introduced in $\cite{akazbe}$ which will be used later. \begin{lem}[cf. \cite{akazbe}] \label{lem1} Let $g\in L^r(\Omega,\gamma)$ and let $ g_n\in L^r(\Omega,\gamma)$, with $\|g_n\|_{r,\gamma} \leq c \ \ \ (10,\\ \mbox{for all }v\in W_0^{1,p}(\Omega,w) \quad v\geq\psi\mbox{ a.e.}\\ u\in W_0^{1,p}(\Omega,w)\quad u\geq\psi\mbox{ a.e.}\\ g(x,u,\nabla u)\in L^1(\Omega). \end{gathered} \eqno(3.4) %tildeP $$ Conversely, if $u$ is a solution of (3.4) then $u$ is a solution of (3.3). \end{lem} The proof of this lemma is similar to the proof of \cite[Remark 2.2]{beel} for the non weighted case. \paragraph{Proof of theorem $\ref{thm1}$} {\bf Step (1) The approximate problem and a priori estimate.} Let $\Omega_\varepsilon$ be a sequence of compact subsets of $\Omega$ such that $\Omega_\varepsilon$ increases to $\Omega$ as $\varepsilon \to 0$. We consider the sequence of approximate problems, $$ \begin{gathered} \langle Au_\varepsilon,v-u_\varepsilon\rangle +\int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)(v-u_\varepsilon)\,dx\geq \langle f,v-u_\varepsilon\rangle \\ v\in W_0^{1,p}(\Omega,w)\quad v\geq\psi\mbox{ a.e.}\\ u_\varepsilon\in W_0^{1,p}(\Omega,w)\quad u_\varepsilon\geq\psi \mbox{ a.e.}\\ \end{gathered} \eqno{(3.5)} %Pe $$ where, $$g_\varepsilon(x,s,\xi)=\frac{g(x,s,\xi)}{1+\varepsilon|g(x,s,\xi)|} \chi_{\Omega_\varepsilon}(x),$$ and where $\chi_{\Omega_\varepsilon}$ is the characteristic function of $\Omega_\varepsilon$. Note that $g_\varepsilon(x,s,\xi)$ satisfies the following conditions, $$ g_\varepsilon(x,s,\xi)s\geq 0,\quad |g_\varepsilon(x,s,\xi)|\leq |g(x,s,\xi)|\quad \mbox{and}\quad | g_\varepsilon(x,s,\xi)|\leq \frac{1}{\varepsilon}. $$ We define the operator $G_\varepsilon:\ X\to X^*$ by, $$\langle G_\varepsilon u,v\rangle =\int_\Omega g_\varepsilon(x,u,\nabla u)v\,dx.$$ Thanks to H\"older's inequality we have for all $u\in X$ and $v \in X$, $$ \begin{aligned} |\int_\Omega g_\varepsilon(x,u,\nabla u)v\,dx| \leq&\Big(\int_\Omega |g_\varepsilon(x,u,\nabla u)|^{q'}\sigma^{-\frac{q'}{q}}\,dx\Big)^{1/q'} \Big(\int_\Omega |v|^{q}\sigma\,dx\Big)^{1/q}\\ \leq& \frac{1}{\varepsilon} \Big(\int_{\Omega_\varepsilon}\sigma^{1-q'} \,dx\Big)^{1/q'}\|v\|_{q,\sigma} \leq c_\varepsilon\||v|\|. \end{aligned} \eqno{(3.6)} $$ The last inequality is due to (2.11) and (2.13). \begin{lem}\label{lemexsoap} The operator $B_\varepsilon =A+G_\varepsilon$ from $X$ into its dual $X^*$ is pseudo-monotone. Moreover, $B_\varepsilon$ is coercive, in the sense that: There exists $v_0\in K_\psi$ such that $$ \frac{\langle B_\varepsilon v, v-v_0\rangle }{\||v|\|}\to +\infty \quad \mbox{as } \||v|\|\to \infty,\quad v\in K_\psi. $$ \end{lem} The proof of this lemma will be presented below. In view of lemma $\ref{lemexsoap}$, (3.5) has a solution by the classical result (cf. Theorem 8.1 and Theorem 8.2 chapter 2 $\cite{li}$). With $v=\psi^+$ as test function in (3.5), we deduce that $\int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)(u_\varepsilon-\psi^+)\geq 0$, then, $\langle Au_\varepsilon,u_\varepsilon\rangle \leq \langle f,u_\varepsilon-\psi^+\rangle +\langle Au_\varepsilon,\psi^+\rangle$, i.e., $$\int_\Omega a(x,u_\varepsilon,\nabla u_\varepsilon)\nabla u_\varepsilon\,dx\leq \langle f,u_\varepsilon-\psi^+\rangle +\sum_{i=1}^N\int_\Omega a_i(x,u_\varepsilon,\nabla u_\varepsilon)\frac{\partial \psi^+}{\partial x_i}\,dx,$$ then, \begin{eqnarray*} \lefteqn{\alpha\sum_{i=1}^N\int_\Omega w_i|\frac{\partial u_\varepsilon}{\partial x_i}|^p\,dx}\\ &=&\alpha\||u_\varepsilon|\|^p\\ &\leq& \|f\|_{X^*}(\||u_\varepsilon|\|+\||\psi^+|\|)+\\ && +\sum_{i=1}^N\Big(\int_\Omega|a_i(x,u_\varepsilon,\nabla u_\varepsilon)|^{p'}w_i^{1-p'}\,dx\Big)^{\frac{1}{p'}} \Big(\int_\Omega |\frac{\partial \psi^+}{\partial x_i}|^pw_i\,dx \Big)^{1/p}\\ &\leq&\|f\|_{X^*}(\||u_\varepsilon|\|+\||\psi^+|\|)+\\ &&+c\sum_{i=1}^N\Big(\int_\Omega (k^{p'}+|u_\varepsilon|^q\sigma+\sum_{j=1}^{N}|\frac{\partial u_\varepsilon}{\partial x_j}|^pw_j)\,dx\Big)^{1/p'}\||\psi^+|\|. \end{eqnarray*} Using (2.13) the last inequality becomes, $$\alpha \||u_\varepsilon|\|^p\leq c_1 \||u_\varepsilon|\|+c_2\||u_\varepsilon|\|^{\frac{q}{p'}} +c_3\||u_\varepsilon|\|^{p-1}+c_4, $$ where $c_i$ are various positive constants. Then, thanks to (2.10) we can deduce that $u_\varepsilon$ remains bounded in $W_0^{1,p}(\Omega,w)$, i.e., $$\||u_\varepsilon|\|\leq \beta_0,\eqno{(3.7)} $$ where $\beta_0$ is some positive constant. Extracting a subsequence (still denoted by $u_\varepsilon$) we get $$ u_\varepsilon \rightharpoonup u\quad \mbox{weakly in $X$ and a.e. in } \Omega. $$ Note that $u\geq \psi$ a.e. \noindent{\bf Step (2) Strong convergence of $T_k(u_\varepsilon)$.} Thanks to (3.7) and (2.13) we can extract a subsequence still denoted by $u_\varepsilon$ such that $$ \begin{gathered} u_\varepsilon\rightharpoonup u \quad \mbox{weakly in } W_0^{1,p}(\Omega,w,)\\ u_\varepsilon\to u\quad\mbox{a.e.\ in }\Omega. \end{gathered} \eqno{(3.8)} $$ Let $k>0$ by lemma $\ref{lem4}$ we have $$ T_k(u_\varepsilon) \rightharpoonup T_k(u)\quad \mbox{weakly in }W_0^{1,p}(\Omega,w) \mbox{ as }\varepsilon\to 0.\eqno{(3.9)} $$ Our objective is to prove that $$T_k(u_\varepsilon)\to T_k(u)\quad \mbox{strongly in }W_0^{1,p}(\Omega,w) \mbox{ as }\varepsilon\to 0.\eqno{(3.10)} $$ Fix $k>\|\psi^+\|_\infty$, and use the notation $z_\varepsilon=T_k(u_\varepsilon)-T_k(u)$. We use, as a test function in (3.5), $$ v_\varepsilon=u_\varepsilon-\eta\varphi_\lambda(z_\varepsilon) \eqno{(3.11)} $$ where $\varphi_\lambda(s)=se^{\lambda s^2}$ and $\eta=e^{-4\lambda k^2}$. Then we can check that $v_\varepsilon$ is admisible test function. So that $$-\langle Au_\varepsilon,\eta \varphi_\lambda z_\varepsilon \rangle -\int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\eta\varphi_\lambda(z_\varepsilon)\,dx\geq -\langle f,\eta\varphi_\lambda(z_\varepsilon)\rangle $$ which implies that $$\langle Au_\varepsilon,\varphi_\lambda (z_\varepsilon) \rangle +\int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\varphi_\lambda(z_\varepsilon)\,dx\leq \langle f,\varphi_\lambda(z_\varepsilon)\rangle.\eqno{(3.12)}$$ Since $\varphi_\lambda(z_\varepsilon)$ is bounded in $X$ and converges a.e.\ in $\Omega$ to zero and using (2.13), we have $\varphi_\lambda(z_\varepsilon) \rightharpoonup 0 $ weakly in $X$ as $\varepsilon\to 0$. Then $$\eta_1(\varepsilon)=\langle f,\varphi_\lambda(z_\varepsilon)\rangle\to 0,\eqno{(3.13)}$$ and since $g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\varphi_\lambda(z_\varepsilon) \geq 0$ in the subset $\{x\in \Omega : |u_\varepsilon(x)| \geq k\}$ hence (3.12) and (3.13) yield $$\langle Au_\varepsilon,\varphi_\lambda(z_\varepsilon)\rangle +\int_{\{|u_\varepsilon|\leq k\}}g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\varphi_\lambda(z_\varepsilon)\,dx \leq \eta_1(\varepsilon).\eqno{(3.14)} $$ We study each term in the left hand side of (3.14). We have, $$ \begin{aligned} \langle Au_\varepsilon, \varphi_\lambda(z_\varepsilon)\rangle =& \int_\Omega a(x,u_\varepsilon,\nabla u_\varepsilon)\nabla(T_k(u_\varepsilon)-T_k(u)) \varphi_\lambda'(z_\varepsilon)\,dx\\ =& \int_\Omega a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))\nabla(T_k(u_\varepsilon)-T_k(u))\varphi_\lambda'( z_\varepsilon)\, dx\\ &-\int_{\{|u_\varepsilon|>k\}} a(x,u_\varepsilon,\nabla u_\varepsilon)\nabla T_k(u)\varphi_\lambda'(z_\varepsilon)\,dx\\ =& \int_\Omega \left(a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))-a(x,T_k(u_\varepsilon),\nabla T_k(u))\right)\nabla(T_k(u_\varepsilon)\\ &-T_k(u))\varphi_\lambda'(z_\varepsilon)\,dx +\eta_2(\varepsilon), \end{aligned}\eqno{(3.15)} $$ where, $$ \begin{aligned} \eta_2(\varepsilon)=&\int_\Omega a(x,T_k(u_\varepsilon),\nabla T_k(u))\nabla(T_k(u_\varepsilon)-T_k(u))\varphi_\lambda'( z_\varepsilon)\, dx\\ &-\int_{\{|u_\varepsilon|>k\}}a(x,u_\varepsilon,\nabla u_\varepsilon)\nabla T_k(u)\varphi_\lambda'(z_\varepsilon)\,dx, \end{aligned} $$ which converges to 0 as $\varepsilon\to 0$. On the other hand, $$ \begin{aligned} |&\int_{\{|u_\varepsilon|\leq k\}} g_\varepsilon (x,u_\varepsilon,\nabla u_\varepsilon)\varphi_\lambda(z_\varepsilon)\,dx|\\ \leq& \int_{\{|u_\varepsilon|\leq k\}}b(k)[c(x)+\sum_{i=1}^N|\frac{\partial u_\varepsilon}{\partial x_i}|^pw_i]|\varphi_\lambda(z_\varepsilon)|\,dx\\ \leq& b(k)\int_{\{|u_\varepsilon|\leq k\}} c(x) |\varphi_\lambda(z_\varepsilon)|\; dx+\frac{b(k)}{\alpha}\int_{\{|u_\varepsilon|\leq k\}} a(x,u_\varepsilon,\nabla u_\varepsilon)\nabla u_\varepsilon |\varphi_\lambda(z_\varepsilon)|\,dx\\ =& \eta_3(\varepsilon) +\frac{b(k)}{\alpha}\int_\Omega a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))\nabla T_k(u_\varepsilon) |\varphi_\lambda(z_\varepsilon)|\,dx\\ =& \frac{b(k)}{\alpha}\int_\Omega \left(a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))-a(x,T_k(u_\varepsilon),\nabla T_k(u))\right)\nabla (T_k(u_\varepsilon)\\ &-T_k(u)) |\varphi_\lambda(z_\varepsilon)|\,dx +\eta_4(\varepsilon) \end{aligned}\eqno{(3.16)} $$ where $$\eta_3(\varepsilon)= b(k)\int_{\{|u_\varepsilon|\leq k\}} c(x) |\varphi_\lambda(z_\varepsilon)|\,dx\to 0\mbox{ as }\varepsilon \to 0$$ and \begin{eqnarray*} \eta_4(\varepsilon)&=&\eta_3(\varepsilon)+\frac{b(k)}{\alpha}\int_\Omega a(x,T_k(u_\varepsilon),\nabla T_k(u))\nabla (T_k(u_\varepsilon)-T_k(u)) |\varphi_\lambda(z_\varepsilon)|\,dx\\ & &+\frac{b(k)}{\alpha}\int_\Omega a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))\nabla T_k(u) |\varphi_\lambda(z_\varepsilon)|\; dx\to 0 \quad\mbox{as }\varepsilon \to 0. \end{eqnarray*} Note that, when $\lambda\geq\left(\frac{b(k)}{2\alpha}\right)^2$ we have $$\varphi_\lambda'(s)-\frac{b(k)}{\alpha}|\varphi(s)|\geq \frac{1}{2}.$$ Which combining with (3.14),(3.15) and (3.16) one obtains \begin{multline*} \int_\Omega\big(a(x,T_k(u_\varepsilon),\nabla T_k(u_\varepsilon))- a(x,T_k(u_\varepsilon),\nabla T_k(u))\big)\nabla (T_k(u_\varepsilon)-T_k(u)) \,dx\\ \leq \eta_5(\varepsilon)=2(\eta_1(\varepsilon)-\eta_2(\varepsilon) +\eta_4(\varepsilon))\to 0\quad \mbox{as }\varepsilon \to 0. \end{multline*} Finally lemma $\ref{lem5}$ implies (3.10) for any fixed $k\geq \|\psi\|_\infty$. \noindent {\bf Step (3) Passage to the limit.} In view of (3.10) we have for a subsequence, $$ \nabla u_\varepsilon\to \nabla u\quad\mbox{a.e.\ in } \Omega,\eqno{(3.17)} $$ which with (3.8) imply, $$ \begin{gathered} a(x,u_\varepsilon,\nabla u_\varepsilon)\to a(x,u,\nabla u) \quad\mbox{a.e.\ in }\Omega,\\ g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\to g(x,u,\nabla u)\mbox{a.e.\ in }\Omega,\\ g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)u_\varepsilon\to g(x,u,\nabla u)u\quad \mbox{a.e.\ in }\Omega. \end{gathered} \eqno{(3.18)} $$ On the other hand, thanks to (2.14) and (3.7) we have $a(x,u_\varepsilon,\nabla u_\varepsilon)$ is bounded in $\prod_{i=1}^NL^{p'}(\Omega,w_i^*)$ then by lemma \ref{lem1} we obtain $$ a(x,u_\varepsilon,\nabla u_\varepsilon)\rightharpoonup a(x,u,\nabla u)\quad \mbox{weakly in }\prod_{i=1}^NL^{p'}(\Omega,w_i^*).\eqno{(3.19)} $$ We shall prove that, $$ g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)\to g(x,u,\nabla u)\quad\mbox{ strongly in }\ L^1(\Omega).\eqno{(3.20)} $$ By (3.18), to apply Vitali's theorem it suffices to prove that $g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)$ is uniformly equi-integrable. Indeed, thanks to (2.17), (3.6) and (3.7) we obtain, $$0\leq \int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)u_\varepsilon\,dx\leq c_0,\eqno{(3.21)}$$ where $c_0$ is some positive constant. For any measurable subset $E$ of $\Omega$ and any $m>0$ we have, $$\int_E |g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)|\,dx=\int_{E\cap X_m^\varepsilon}|g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)|\,dx+\int_{E\cap Y_m^\varepsilon} |g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)|\,dx$$ where, $$ X_m^\varepsilon=\{x\in \Omega,\ |u_\varepsilon(x)|\leq m\},\quad Y_m^\varepsilon=\{x\in \Omega,\ |u_\varepsilon(x)|> m\}. $$ From these expressions, (2.18), and (3.21), we have $$ \begin{aligned} \int_E& |g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)|\,dx\\ =& \int_{E\cap X_m^\varepsilon}|g_\varepsilon(x,u_\varepsilon,\nabla T_m (u_\varepsilon))|\,dx+\int_{E\cap Y_m^\varepsilon} |g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)|\,dx\\ \leq& \int_{E\cap X_m^\varepsilon}|g_\varepsilon(x,u_\varepsilon,\nabla T_m (u_\varepsilon))|\,dx+\frac{1}{m}\int_\Omega g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)u_\varepsilon\,dx\\ \leq & b(m)\int_E (\sum_{i=1}^{N}w_i|\frac{\partial T_m(u_\varepsilon)}{\partial x_i}|^p+c(x))+\frac{c_0}{m}. \end{aligned}\eqno{(3.22)} $$ Since the sequence ($\nabla T_m(u_\varepsilon)$) strongly converges in $\prod_{i=1}^NL^p(\Omega,w_i),$ then (3.22) implies the equi-integrability of $g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)$. Moreover, since $g_\varepsilon(x,u_\varepsilon,\nabla u_\varepsilon)u_\varepsilon\geq 0$ a.e.\ in $\Omega$, then by (3.18), (3.21) and Fatou's lemma, we have $g(x,u,\nabla u)u\in L^1(\Omega)$. On the other hand, for $v\in L^\infty(\Omega)$, set $h=k+\|v\|_\infty$, then \begin{eqnarray*} |\frac{\partial T_k(v-u_\varepsilon)}{\partial x_i}|w_i^{1/p} &=& \chi_{\{|v-u_\varepsilon|\leq k\}}|\frac{\partial v}{\partial x_i}-\frac{\partial u_\varepsilon}{\partial x_i}|w_i^{1/p}\\ &\leq& \chi_{\{|u_\varepsilon|\leq h\}}|\frac{\partial v}{\partial x_i}-\frac{\partial u_\varepsilon}{\partial x_i}|w_i^{1/p}\\ &\leq& |\frac{\partial v}{\partial x_i}|w_i^{1/p}+|\frac{\partial T_h(u_\varepsilon)}{\partial x_i}|w_i^{1/p} \end{eqnarray*} which implies, using Vitali's theorem with (3.10) and (3.17) that $$ \nabla T_k(v-u_\varepsilon)\to \nabla T_k(v-u)\quad \mbox{strongly in } \prod_{i=1}^NL^p(\Omega,w_i)\eqno{(3.23)} $$ for any $v\in W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega)$. Thanks to lemma $\ref{lemeqso}$ and from (3.19), (3.20) and (3.23) we can pass to the limit in $$\langle Au_\varepsilon,T_k(v-u_\varepsilon)\rangle+\int_\Omega g_\varepsilon (x,u_\varepsilon,\nabla u_\varepsilon)T_k(v-u_\varepsilon)\geq\langle f,T_k(v-u_\varepsilon)\rangle$$ and we obtain, $$ \langle Au,T_k(v-u)\rangle+\int_\Omega g(x,u,\nabla u)T_k(v-u)\geq\langle f,T_k(v-u)\rangle \eqno{(3.24)} $$ for any $v\in W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega)$ and for all $k>0$. Taking for any $v\in W_0^{1,p}(\Omega,w)$ and $v\geq \psi$ the test function $T_m(v)$ which belongs to $W_0^{1,p}(\Omega,w)\cap L^\infty(\Omega) $ for $m\geq \|\psi^+\|_\infty$ and passing to the limit in (3.24) as $m\to \infty$, then $u$ is a solution of (3.4). Using again lemma \ref{lemeqso} we obtain the desired result, i.e., $u$ is a solution of (3.3). \paragraph{Proof of lemma $\ref{lemexsoap}$} By proposition 2.6 chapter 2 $\cite{li}$, it is sufficient to show that $B_\varepsilon$ is of the calculus of variations type in the sense of definition $\ref{defcava}$. Indeed put, $$ b_1(u,v,\tilde w)=\sum_{i=1}^N\int_\Omega a_i(x,u,\nabla v)\nabla \tilde w\,dx,\quad b_2(u,\tilde w)=\int_\Omega g_\varepsilon(x,u,\nabla u)\tilde w\,dx. $$ Then the mapping $\tilde w\mapsto b_1(u,v,\tilde w)+b_2(u,\tilde w)$ is continuous in $X$. Then $$ b_1(u,v,\tilde w)+b_2(u,\tilde w)=b(u,v,\tilde w)=\langle B_\varepsilon(u,v),\tilde w\rangle ,\quad B_\varepsilon(u,v)\in W^{-1,p'}(\Omega,w^*) $$ and we have $$B_\varepsilon(u,u)=B_\varepsilon u. $$ Using (2.14) and H\"older's inequality we can show that $A$ is bounded as in \cite{drkumu}, and thanks to (3.6) $B_\varepsilon$ is bounded. Then, it is sufficient to check (2.6)-(2.9). Next we show that (2.6) and (2.7) are true. By (2.15) we have, $$ (B_\varepsilon(u,u)-B_\varepsilon(u,v), u-v)=b_1(u,u,u-v)-b_1(u,v,u-v)\geq 0. $$ The operator $v\to B_\varepsilon(u,v)$ is bounded hemicontinuous. Indeed, we have $$ a_i(x,u,\nabla(v_1+\lambda v_2))\to a_i(x,u,\nabla v_1)\quad \mbox{ strongly in }L^{p'}(\Omega,w_i^*)\mbox{ as }\lambda\to 0.\eqno{(3.25)} $$ On the other hand, $\left(g_\varepsilon(x,u_1+\lambda u_2,\nabla(u_1+\lambda u_2))\right)_\lambda$ is bounded in $L^{q'}(\Omega,\sigma^{1-q'})$ and $g_\varepsilon(x,u_1+\lambda u_2,\nabla(u_1+\lambda u_2))\to g_\varepsilon(x,u_1,\nabla u_1)\ \ \ a.e.$ in $\Omega$, hence lemma \ref{lem1} gives $$\begin{gathered} g_\varepsilon(x,u_1+\lambda u_2,\nabla(u_1+\lambda u_2))\rightharpoonup g_\varepsilon(x,u_1,\nabla u_1)\\ \mbox{ weakly in }L^{q'}(\Omega,\sigma^{1-q'})\mbox{ as }\lambda \to 0. \end{gathered}\eqno{(3.26)} $$ Using (3.25) and (3.26) we can write $$b(u,v_1+\lambda v_2,\tilde w)\to b(u,v_1,\tilde w)\ \ \mbox{ as }\lambda \to 0 \ \ \forall u,v_i,\tilde w\in X.$$ Similarly we can prove (2.7). \noindent Proof of assertion (2.8). Assume that $u_n\rightharpoonup u$ weakly in $X$ and $(B(u_n,u_n)-B(u_n,u),u_n-u)\to 0$. We have, \begin{multline*} (B(u_n,u_n)-B(u_n,u),u_n-u)\\ = \sum_{i=1}^N\int_\Omega \left(a_i(x,u_n,\nabla u_n)-a_i(x,u_n,\nabla u)\right)\nabla(u_n-u)\,dx\to 0, \end{multline*} then, by lemma $\ref{lem5}$, $u_n\to u$ strongly in $X$, which gives $$ b(u_n,v,\tilde w)\to b(u,v,\tilde w)\quad \forall \tilde w\in X, $$ i.e., $B_\varepsilon(u_n,v)\rightharpoonup B_\varepsilon(u,v) \quad \mbox{weakly in }X^*$. It remains to prove (2.9). Assume that $$u_n\rightharpoonup u\quad\mbox{ weakly in }X\eqno{(3.27)} $$ and that $$B(u_n,v)\rightharpoonup \psi\quad\mbox{weakly in }X^*.\eqno{(3.28)} $$ Thanks to (2.13), (2.14) and (3.27) we obtain, $$a_i(x,u_n,\nabla v)\to a_i(x,u,\nabla v)\quad \mbox{in }L^{p'}(\Omega,w_i^*)\mbox{ as } n\to \infty, $$ then, $$b_1(u_n,v,u_n)\to b_1(u,v,u).\eqno{(3.29)} $$ On the other hand, by H\"older's inequality, \begin{align*} |b_2(u_n,u_n-u)|\leq &\left(\int_\Omega|g_\varepsilon(x,u_n,\nabla u_n)|^{q'}\sigma^{\frac{-q'}{q}}\; dx\right)^{1/q'}\left(\int_\Omega|u_n-u|^q\sigma\,dx \right)^{1/q}\\ \leq &\frac{1}{\varepsilon}\left(\int_{\Omega_\varepsilon}\sigma^{\frac{-q'}{q}}\; dx\right)^{1/q'}\|u_n-u\|_{L^q(\Omega,\sigma)}\to 0\quad \mbox{as }n\to \infty, \end{align*} i.e., $$b_2(u_n,u_n-u)\to 0\mbox{ as }n\to \infty,\eqno{(3.30)} $$ but in view of (3.28) and (3.29) we obtain $$b_2(u_n,u)=(B_\varepsilon(u_n,v),u)-b_1(u_n,v,u)\to (\psi,u)-b_1(u,v,u) $$ and from (3.30) we have $b_2(u_n,u_n)\to (\psi,u)-b_1(u,v,u)$. Then, $$(B_\varepsilon(u_n,v),u_n)=b_1(u_n,v,u_n)+b_2(u_n,u_n)\to (\psi,u). $$ Now show that $B_\varepsilon$ is coercive. Let $v_0\in K_\psi$. From H\"older's inequality, the growth condition (2.14) and the compact imbedding (2.13) we have \begin{eqnarray*} \langle Av,v_0\rangle&=&\sum_{i=1}^N \int_\Omega a_i(x,v,\nabla v)\frac{\partial v_0}{\partial x_i}\,dx\\ &\leq& \sum_{i=1}^N \Big(\int_\Omega |a_i(x,v,\nabla v)|^{p'}w_i^{\frac{-p'}{p}}\,dx\Big)^{\frac{1}{p'}} \Big(\int_\Omega |\frac{\partial v_0}{\partial x_i}|^pw_i\,dx\Big)^{1/p}\\ &\leq & c_1\||v_0|\|\Big( \int_\Omega k(x)^{p'}+|v|^q\sigma+\sum_{j=1}^N |\frac{\partial v}{\partial x_j}|^pw_j \,dx\Big)^{\frac{1}{p'}}\\ &\leq & c_2(c_3+\||v|\|^{\frac{q}{p'}}+\||v|\|^{p-1}), \end{eqnarray*} where $c_i$ are various constants. Thanks to (2.16), we obtain $$ \frac{\langle Av,v\rangle}{\||v|\|}-\frac{\langle Av,v_0\rangle}{\||v|\|} \geq \alpha \||v|\|^{p-1}-\||v|\|^{p-2}- \||v|\|^{\frac{q}{p'}-1}-\frac{c}{\||v|\|}. $$ In view of (2.10) we have $p-1>\frac{q}{p'}-1$. Then, $$\frac{\langle Av,v-v_0\rangle}{\||v|\|}\to \infty \quad \mbox{as }\||v|\|\to \infty. $$ Since $\langle G_\varepsilon v,v \rangle \geq 0$ and $\langle G_\varepsilon v,v_0 \rangle$ is bounded, we have $$ \frac{\langle B_\varepsilon v,v-v_0\rangle}{\||v|\|} \geq \frac{\langle Av,v-v_0\rangle}{\||v|\|}-\frac{\langle G_\varepsilon v,v_0\rangle}{\||v|\|}\to \infty \quad\mbox{as }\||v|\|\to \infty. $$ \begin{rem} \label{rmk3.2} \rm Assumption (2.10) appears to be necessary to prove the boundedness of $(u_\varepsilon)_\varepsilon$ in $W_0^{1,p}(\Omega,w)$ and the coercivity of the operator $B_\varepsilon$. While Assumption (2.11) is necessary to prove the boundedness of $G_\varepsilon$ in $W_0^{1,p}(\Omega,w)$. Thus, when $g\equiv 0$, we don't need to assume (2.11). \end{rem} \begin{thebibliography}{99} \frenchspacing \bibitem{akazbe} Y. A{\sc kdim}, E. A{\sc zroul} and A. B{\sc enkirane}, {\em Existence of solutions for quasilinear degenerated elliptic equations, }{Electronic J. Diff. Eqns., vol. {\bf 2001} N 71 (2001) 1-19.} \bibitem{bebomu} A. B{\sc ensoussan}, L. B{\sc occardo} and F. M{\sc urat}, {\em On a non linear partial differential equation having natural growth terms and unbounded solution,}{ Ann. Inst. Henri Poincar\'e {\bf 5} $N^°4$ (1988), 347-364.} \bibitem{beel} A. B{\sc enkirane} and A. E{\sc lmahi}, {\em Strongly nonlinear elliptic unilateral problems having natural growth terms and $L^1$ data, }{Rendiconti di Matematica, Serie VII vol. {\bf 18}, (1998), 289-303.} \bibitem{drkumu} 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{drkuni} 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{drni} P. D{\sc rabek} and F. N{\sc icolosi}, {\em Existence of Bounded Solutions for Some Degenerated Quasilinear Elliptic Equations,}{ Annali di Mathematica pura ed applicata (IV), Vol. CLXV (1993), pp. 217-238}. \bibitem{li} J. L. L{\sc ions}, {\em Quelques m\'ethodes de r\'esolution des probl\`emes aux limites non lin\'eaires, }{Dunod, Paris (1969).} \end{thebibliography} \noindent\textsc{Youssef Akdim} (e-mail: y.akdim1@caramail.com)\\ \textsc{Elhoussine Azroul} (e-mail: elazroul@caramail.com)\\ \textsc{Abdelmoujib Benkirane} (e-mail: abenkirane@fsdmfes.ac.m)\\[2pt] D\'epartement de Math\'ematiques et Informatique\\ Facult\'e des Sciences Dhar-Mahraz\\ B.P 1796 Atlas F\`es, Maroc \end{document}