\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 63, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/63\hfil Existence of solutions] {Existence of solutions for some nonlinear elliptic equations} \author[A. Anane, O. Chakrone, M. Chehabi\hfil EJDE-2006/63\hfilneg] {Aomar Anane, Omar Chakrone, Mohammed Chehabi} % in alphabetical order \address{D\'epartement de Math\'ematiques et Informatique, Facult\'e des Sciences, Universit\'e Mohammed 1er, Oujda, Maroc} \email[Aomar Anane]{anane@sciences.univ-oujda.ac.ma} \email[Omar Chakrone]{chakrone@sciences.univ-oujda.ac.ma} \email[Mohammed Chehabi]{chehb\_md@yahoo.fr} \date{} \thanks{Submitted January 23, 2006. Published May 19, 2006.} \subjclass[2000]{35J15, 35J70, 35J85} \keywords{Boundary value problem; truncation; $L^1$; p-Laplacian; spectrum} \begin{abstract} In this paper, we study the existence of solutions to the following nonlinear elliptic problem in a bounded subset $\Omega$ of $\mathbb{R}^{N}$: \begin{gather*} -\Delta _{p}u = f(x,u,\nabla u)+\mu \quad \hbox{in } \Omega ,\\ u = 0 \quad \mbox{on }\partial \Omega , \end{gather*} where $\mu$ is a Radon measure on $\Omega$ which is zero on sets of $p$-capacity zero, $f:\Omega \times \mathbb{R}\times \mathbb{R} ^{N}\to \mathbb{R}$ is a Carath\'{e}odory function that satisfies certain conditions with respect to the one dimensional spectrum. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} We consider the quasilinear elliptic problem $$\label{eP} \begin{gathered} -\Delta _{p}u=f(x,u,\nabla u)+\mu \quad\text{in } \Omega , \\ u=0 \quad \text{on }\partial \Omega , \end{gathered}$$ where $\Omega$ is a bounded open set in $\mathbb{R}^{N}$, $N\geq 2$, $10, \end{equation*} They require also that for almost every$x\in \Omega $, for every$\xi $in$\mathbb{R}^{N}$, for every$s$in$\mathbb{R}$such that$|s|\geq \sigma $, \begin{equation*} g(x,s,\xi )\mathop{\rm sgn}(s)\geq \rho |\xi |^{p}, \end{equation*} where$\rho $and$\sigma $are two positive real numbers and$\mathop{\rm sgn}(s)$is the sign of$s$. Let$(\beta ,\alpha ,u)\in \mathbb{R}^{N}\mathbb{\times R\times} W_{0}^{1,p}(\Omega )\backslash \{0\}$. If$(\beta ,\alpha ,u)$is a solution of the problem \begin{gather*} -\Delta _{p}u=\alpha m(x)|u|^{p-2}u+\beta . |\nabla u|^{p-2}\nabla u \quad \text{in }\Omega , \\ u =0\quad\text{on } \partial \Omega , \end{gather*} where$10\}\neq 0\}$. In this case, the pair$(\beta ,\alpha )$is said to be a one dimensional eigenvalue and$u$the associated eigenfunction. We designate by$\sigma _{1}(-\Delta _{p},m)\subset \mathbb{R}^{N}\mathbb{\times R}$the set of one dimensional eigenvalues$(\beta ,\alpha )$with$\alpha \geq 0$. \begin{proposition} \label{prop1} (1)$\sigma _{1}(-\Delta _{p},m)$contains the union of the sequence of graphs of the functions$\Lambda _{n}:\mathbb{R}^{N}\to \mathbb{R}^{+}$,$n=1,2,\dots $, where$\Lambda _{n}(\beta )$is defined for every$\beta \in \mathbb{R}^{N}$by \begin{equation*} \frac{1}{\Lambda _{n}(\beta )} =\sup_{K\in A_{n}^{\beta }} \min_{u\in K} \int_{\Omega }e^{\beta .x}m(x)|u|^{p}dx. \end{equation*} with$A_{n}^{\beta }=\{K\subset S_{\beta }$,$K$compact symmetrical;$\gamma (K)\geq n\}$, $$S_{\beta }=\big\{ u\in W_{0}^{1,p}(\Omega ):\Big( \int_{\Omega }e^{\beta .x}m(x)|\nabla u|^{p}dx\Big) ^{1/p}=1\big\}$$ and$\gamma(K)$indicates the genus of$K$. (2)$\Lambda _{1}(.)$is the first eigensurface of the spectrum of$\sigma _{1}(-\Delta _{p},m)$in the sense \begin{equation*} \sigma _{1}(-\Delta _{p},m)\subset \{ (\beta ,\alpha )\in \mathbb{R}^{N} \mathbb{\times R}\text{; }\Lambda _{1}(\beta )\leq \alpha \} \end{equation*} \end{proposition} The proof of the above proposition can be found in \cite{Anan}. When$\mu =h\in W^{-1,p'}(\Omega )$, Anane, Chakrone and Gossez have proved in \cite{Anan} the existence of a solution to \eqref{eP}, in the sense \begin{equation*} \int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\langle h,v\rangle \end{equation*} for every$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$. This is done under the hypotheses of non-resonance with respect to the spectrum of one dimensional$\sigma_{1}(-\Delta _{p},1)$: There exists ($\beta ,\alpha )\in \mathbb{R}^{N}\times \mathbb{R}$with$\alpha <\Lambda _{1}(\beta ,-\Delta _{p},1)$where$\Lambda_{1}(.,-\Delta_{p},1)$is the first eigensurface of the spectrum of one dimensional$\sigma _{1}(-\Delta _{p},1)$, such that for all$\delta >0$there exists$a_{\delta }\in L^{p'}(\Omega)$such that $$\label{eP1} f(x,s,\xi )s\leq \alpha |s|^{p}+\beta |\xi |^{p-2}\xi s +\delta (|s|^{p-1}+|\xi |^{p-1}+a_{\delta }(x))|s|$$ for almost every$x\in \Omega$and for all$(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$; and for all$k>0$there exist$\phi _{k}\in L^{1}(\Omega )$and$ b_{k}\in \mathbb{R}$such that $$\label{eP2} \max_{|s|\leq k} |f(x,s,\xi )|\leq b_{k}|\xi |^{p}+\phi _{k}(x)$$ for almost every$x\in \Omega $and for all$\xi \in \mathbb{R}^{N}$. \begin{remark} \rm \begin{enumerate} \item If$f(x,u,\nabla u)=\alpha m(x)|u|^{p-2}u +\beta .|\nabla u|^{p-2}\nabla u$, then \eqref{eP} has a solution for every$\mu \in W^{-1,p'}(\Omega )$, in the usual sense \begin{equation*} \int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx +\langle h,v\rangle_{W^{-1,p'}(\Omega ),W_{0}^{1,p}(\Omega )} \end{equation*} for every$v\in W_{0}^{1,p}(\Omega )$, if and only if$(\beta,\alpha )\notin \sigma _{1}(-\Delta _{p},m)$. \item If$\mu \notin W^{-1,p'}(\Omega )$, problem \eqref{eP} does not have always a solution. Indeed in the case$10$there exists$a_{\delta }\in L^{p'}(\Omega )$such that $$\label{eP3} f(x,s,\xi )s\leq -\rho |\xi |^{p}|s|+\alpha | s|^{p}+\beta |\xi |^{p-2}\xi s+\delta (|s| ^{p-1}+|\xi |^{p-1}+a_{\delta }(x))|s|$$ for almost every$x\in \Omega$and for all$(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$, where ($\beta ,\alpha )\in \mathbb{R}^{N}\times \mathbb{R}$satisfies the same conditions as in \eqref{eP1} and$\rho $is a positive real number. In the case$\delta =1$, there exists$a_{1}\in L^{p'}(\Omega )$such that $$\label{eP4} f(x,s,\xi )\mathop{\rm sgn}(s)\leq -\rho |\xi |^{p}+\alpha '|s| ^{p-1}+\beta '|\xi |^{p-1}+a_{1}(x)$$ for almost every$x\in \Omega$and for all$(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$, where$\alpha '=\alpha +1$and$\beta '=|\beta |+1$. \begin{remark} \rm \begin{enumerate} \item The conditions of the sign given in \cite{Boc} imply \eqref{eP3} in the case$\alpha =0$and$\beta =0$. \item The hypothesis \eqref{eP2} and \eqref{eP3} are satisfied for example if \begin{equation*} f(x,s,\xi )=-\rho |\xi |^{p}\mathop{\rm sgn}(s)+\alpha |s| ^{p-2}s+\beta |\xi |^{p-2}\xi +g(x,s,\xi )+l(x,s,\xi ) \end{equation*} where$g$and$l$satisfy \begin{gather*} g(x,s,\xi )s\leq 0, \\ |g(x,s,\xi )|\leq b(|s|)(|x|^{p}+c(x)), \\ sl(x,s,\xi )\leq C(|s|^{q-1}+|x|^{q-1}+d(x))|s| \end{gather*} with$b$continuous,$c(x)\in L^{1}(\Omega )$,$q0$, \begin{equation*} T_{k}(s)=\begin{cases} k\mathop{\rm sgn}(s) & \text{if }|s|>k, \\ s & \text{if } |s|\leq k, \end{cases} \end{equation*} and$G_{k}(s)=s-T_{k}(s)$. \begin{lemma} \label{lem1} Let$g\in L^{\infty }(\Omega )$and$F\in (L^{p'}(\Omega ))^{N}$. Under the hypotheses \eqref{eP2} and \eqref{eP3}, the problem $$\label{eP6} \begin{gathered} -\Delta _{p}u=f(x,u,\nabla u)+g- \mathop{\rm div}F \quad \text{in }\Omega, \\ u=0 \quad\text{on }\partial \Omega , \end{gathered}$$ admits a solution$u\in W_{0}^{1,p}(\Omega )$in the sense that$f(x,u,\nabla u)$and$f(x,u,\nabla u)u$are in$L^{1}(\Omega )$, and that $$\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }gv +\int_{\Omega }F\nabla v$$ for every$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty}(\Omega )$and for$v=u$. \end{lemma} \begin{proof} Letting$l=g-\mathop{\rm div}F$, we have$l\in W^{-1,p'}(\Omega )$. Then \eqref{eP3} implies \eqref{eP1}, and Lemma \ref{lem1} is a particular case of a result in \cite{Anan}. \end{proof} \begin{lemma} \label{lem2}$\mathcal{M}_{0}^{p}(\Omega )=L^{1}(\Omega )+W^{-1,p'}(\Omega )$for every$1N$, then$L^{1}(\Omega )\subset W^{-1,p'}(\Omega )$; therefore,$\mathcal{M}_{0}^{p}(\Omega )=W^{-1,p'}(\Omega )$. Then the existence of a solution of \eqref{eP5} is a consequence of \cite[Theorem 7.1]{Anan}. That is why, we assume that$11\}}\varphi _{1}f(x,u_{n},\nabla u_{n})dx. \end{align*} By \eqref{eP2}, we have \begin{align*} |\int_{\{|u_{n}|\leq 1\}}\varphi _{1}f(x,u_{n},\nabla u_{n})dx| & \leq \int_{\{|u_{n}|\leq 1\}}| \varphi _{1}||f(x,u_{n},\nabla u_{n})|dx \\ & \leq \int_{\{|u_{n}|\leq 1\}}|\varphi_{1}| [ b_{1}|\nabla u_{n}|^{p}+\phi _{1}(x)]dx \\ & \leq b_{1}\int_{\{|u_{n}|\leq 1\}}| \varphi _{1}||\nabla u_{n}|^{p}dx+\varphi (1)\| \phi_{1}\| _{L^{1}} \\ & \leq b_{1}\int_{\Omega }|\varphi _{1}||\nabla (T_{1}(u_{n}))|^{p}dx+\varphi (1)\| \phi _{1}\| _{L^{1}}. \end{align*} On the other hand, on $\{|u_{n}|>1\}$, $T_{1}(u_{n})=\mathop{\rm sgn}(u_{n})$, so $\varphi (T_{1}(u_{n}))=\mathop{\rm sgn}(u_{n})$ $e^{\theta }$ and by \eqref{eP4}, we get \begin{align*} &\int_{\{|u_{n}|>1\}}\varphi _{1}f(x,u_{n},\nabla u_{n})dx\\ & = \int_{\{|u_{n}|>1\}}e^{\theta }f(x,u_{n},\nabla u_{n})\mathop{\rm sgn}(u_{n})dx\\ & \leq e^{\theta }\int_{\{|u_{n}|>1\}}[-\rho | \nabla u_{n}|^{p}+\alpha '|u_{n}|^{p-1} + \beta '|\nabla u_{n}|^{p-1}+a_{1}(x)]dx. \end{align*} Adding the above inequalities, by \eqref{eP10}, we obtain \begin{aligned} & \int_{\Omega }[\varphi _{1}'-b_{1}|\varphi _{1}|]|\nabla (T_{1}(u_{n}))|^{p}dx+\rho e^{\theta }\int_{\{|u_{n}|>1\}}|\nabla u_{n}|^{p}dx \\ &\leq \| F\| _{L^{p'}}\varphi '(1)\| T_{1}(u_{n})\| _{1,p} +\varphi (1)\| \widetilde{g}\| _{L^{1}}+\varphi (1)\| \phi _{1}\| _{L^{1}} \\ &\quad +e^{\theta }\int_{\{|u_{n}|>1\}}[\alpha '| u_{n}|^{p-1}+\beta '|\nabla u_{n}|^{p-1}+a_{1}(x)]dx. \end{aligned} \label{eP11} Using H\"{o}lder's inequality, we have \begin{gather*} \int_{\{|u_{n}|>1\}}|\nabla u_{n}|^{p-1}dx \leq \| u_{n}\| _{1,p}^{p-1}(\mathop{\rm meas}(\Omega ))^{1/p}, \\ \int_{\{|u_{n}|>1\}}|u_{n}|^{p-1}dx \leq \|u_{n}\| _{p}^{p-1}(\mathop{\rm meas}(\Omega ))^{1/p}. \end{gather*} By Poincar\'{e}'s inequality, there exists $c>0$ such that \begin{equation*} \| u_{n}\| _{p}\leq c\| \nabla u_{n}\| _{p}. \end{equation*} So \begin{equation*} \int_{\{|u_{n}|>1\}}|u_{n}|^{p-1}dx \leq c^{p-1}\| u_{n}\| _{1,p}^{p-1}(\mathop{\rm meas} (\Omega ))^{1/p}. \end{equation*} Replacing this in (\ref{eP11}) and using that $\varphi _{1}'-b_{1}|\varphi _{1}|\geq \frac{1}{2}$, we obtain \begin{equation*} \frac{1}{2}\int_{\Omega }|\nabla (T_{1}(u_{n}))| ^{p}dx+\rho e^{\theta }\int_{\{|u_{n}|>1\}}|\nabla u_{n}|^{p}dx\leq c_{1}\| u_{n}\| _{1,p}+c_{2}\| u_{n}\| _{1,p}^{p-1}+c_{3}, \end{equation*} where $c_{1}=\| F\| _{L^{p'}}\varphi '(1)$, $c_{2}=e^{\theta }[\alpha 'c^{p-1}+\beta '](\mathop{\rm meas}(\Omega ))^{ \frac{1}{p}}$ and $c_{3}=\varphi (1)\| \widetilde {g}\| _{L^{1}}+\varphi (1)\| \phi _{1}\| _{L^{1}}+e^{\theta }\| a_{1}(x)\| _{L^{1}}$. Set $c_{4}=\min (\frac{1}{2},\rho e^{\theta })$, we have \begin{equation*} c_{4}\| u_{n}\| _{1,p}^{p}\leq c_{1}\| u_{n}\| _{1,p}+c_{2}\| u_{n}\|_{1,p}^{p-1}+c_{3}, \end{equation*} since $p>1$, $(u_{n})_{n}$ is a bounded sequence in $W_{0}^{1,p}(\Omega )$. \end{proof} For a subsequence, still denoted by $(u_{n})_{n}$, we have $$\begin{gathered} u_{n}\rightharpoonup u \quad \text{weakly in }W_{0}^{1,p}(\Omega ), \\ u_{n}\to u \quad \text{strongly in }\ L^{p}(\Omega ), \\ u_{n}(x)\to u(x)\quad \text{for almost every }x\in \Omega . \end{gathered} \label{eP12}$$ \begin{lemma} \label{lem5} For every $k>0$, the sequence $(T_{k}(u_{n}))_{n}$ converges strongly to $T_{k}(u)$ in $W_{0}^{1,p}(\Omega )$. \end{lemma} \begin{proof} Let $k>0$. Consider $\varphi (s)=se^{\theta s^{2}}$ with $\theta =\frac{b^{2}}{4a^{2}}$, $a=1$ and $b=a_{k}$ ($a_{k }\geq 0$ is given by \eqref{eP2}. Setting \begin{gather*} a(\xi )=|\xi |^{p-2}\xi,\quad \forall \xi \in \mathbb{R}^{N}, \varphi _{n}=\varphi (T_{k}(u_{n})-T_{k}(u)), \quad \varphi_{n}'=\varphi '(T_{k}(u_{n})-T_{k}(u)). \end{gather*} By \eqref{eP12}, the continuity of $\varphi$ and $\varphi'$, and the dominated convergence theorem, we have $$\begin{gathered} \varphi _{n}\rightharpoonup 0 \quad\text{and}\quad \varphi _{n}'\rightharpoonup 1 \quad \text{weak-\ast in L^{\infty }(\Omega ) and a. e. x\in \Omega }, \\ \varphi _{n}\to 0\quad\text{and}\quad \varphi _{n}'\to 1 \quad\text{in L^{q}(\Omega ) for every q\geq 1}. \end{gathered} \label{eP13}$$ We will denote by $\varepsilon _{n}$\ any quantity which converges to zero as $n$ tends to infinity. Let $v=\varphi _{n}$, be a test function in (\ref{eP9}). Then \begin{aligned} &\int_{\Omega }a(\nabla u_{n})\nabla (T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx \\ & = \int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx +\int_{\Omega }g_{n}\varphi _{n}dx +\int_{\Omega }F\nabla (T_{k}(u_{n})-T_{k}(u))\varphi_{n}' \\ &:= A+B+C+D \end{aligned} \label{eP14} For the third term on the right-hand side: Since $\varphi _{n}\rightharpoonup 0$ weak-$\ast$ in $L^{\infty}(\Omega )$ and $g_{n}\to g\$in $L^{1}(\Omega )$, we have $\int_{\Omega }g_{n}\varphi _{n}dx\to 0$ so that $$C=\varepsilon _{n}. \label{eP15}$$ For the forth term on the right-hand side: It is clear that $F\varphi _{n}'\to F$ in $(L^{p'}(\Omega ))^{N}$ and $T_{k}(u_{n})\rightharpoonup T_{k}(u)\$ weakly in $W_{0}^{1,p}(\Omega )$, so that $$D=\varepsilon _{n}. \label{eP16}$$ For the second term on the right-hand side: \begin{align*} &\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx\\ &=\int_{\{|u_{n}|>k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx + \int_{\{|u_{n}|\leq k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx := B_1+B_2. \end{align*} On the set $\{|u_{n}|>k\}$, $\varphi _{n}$ has the same sign as $u_{n}$, so by \eqref{eP4}, \begin{align*} &f(x,u_{n},\nabla u_{n})\varphi _{n} \\ &\leq -\rho |\nabla u_{n}|^{p}|\varphi _{n}|+\alpha '| u_{n}|^{p-1}|\varphi _{n}|+\beta '|\nabla u_{n}|^{p-1}|\varphi_{n}|+a_{1}(x)|\varphi _{n}|\\ &\leq [\alpha '|u_{n}|^{p-1}+\beta '|\nabla u_{n}|^{p-1}+a_{1}(x)]|\varphi _{n}|. \end{align*} By Lemma \ref{lem4} and \eqref{eP13}, we have $B_{1}\leq \varepsilon _{n}$, so that \begin{equation*} \int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx \leq \int_{\{|u_{n}|\leq k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx+\varepsilon _{n}. \end{equation*} By \eqref{eP2}, we have \begin{align*} |\int_{\{|u_{n}|\leq k\}}f(x,u_{n},\nabla u_{n})\varphi _{n}dx| & \leq \int_{\{|u_{n}|\leq k\}}| f(x,u_{n},\nabla u_{n})||\varphi _{n}|dx \\ & \leq \int_{\{|u_{n}|\leq k\}}[b_{k}| \nabla u_{n}|^{p}+\phi _{k}(x)]|\varphi _{n}|dx \\ & \leq b_{k}\int_{\Omega }|\nabla T_{k}(u_{n})| ^{p}| \varphi _{n}|dx+\int_{\Omega }\phi _{k}(x)|\varphi _{n}|dx, \end{align*} and \begin{align*} \int_{\Omega }|\nabla T_{k}(u_{n})|^{p}|\varphi _{n}|dx &= \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla T_{k}(u_{n})| \varphi_{n}|dx \\ & = \int_{\Omega }(a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u)))(\nabla T_{k}(u_{n})-\nabla T_{k}(u))|\varphi _{n}|dx \\ & \quad + \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla T_{k}(u)| \varphi_{n}|dx \\ & \quad + \int_{\Omega }a(\nabla T_{k}(u))(\nabla T_{k}(u_{n})-\nabla T_{k}(u))|\varphi _{n}|dx. \end{align*} By \eqref{eP13}, since $(T_{k}(u_{n}))_{n}$ is bounded in $W_{0}^{1,p}(\Omega )$, we have \begin{aligned} &\int_{\Omega }f(x,u_{n},\nabla u_{n})\varphi _{n}dx\\ &\leq \varepsilon _{n}+b_{k}\int_{\Omega }\big(a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u)) \big)\big(\nabla T_{k}(u_{n})-\nabla T_{k}(u)\big)| \varphi _{n}|dx. \end{aligned} \label{eP17} For the firs term on the right-hand side (A): We verify easily that $a(\nabla T_{k}(u_{n}))+a(\nabla G_{k}(u_{n}))=a(\nabla u_{n})$, so that \begin{align*} &\int_{\Omega }a(\nabla u_{n})\nabla (T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx \\ & = \int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla (T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx +\int_{\Omega }a(\nabla G_{k}(u_{n}))\nabla (T_{k}(u_{n})\\ &\quad -T_{k}(u))\varphi _{n}'dx := A_1+A_2. \end{align*} We have $\nabla (T_{k}(u_{n}))=0$ if $\nabla (G_{k}(u_{n}))\neq 0$, so \begin{align*} A_{2} &= -\int_{\Omega }a(\nabla G_{k }(u_{n}))\nabla (T_{k}(u))\varphi _{n}'dx \\ & = -\int_{\Omega }a(\nabla G_{k }(u_{n}))\nabla (T_{k}(u))\chi_{\{|u_{n}|\geq k\}}\varphi _{n}'dx. \end{align*} Since $\nabla T_{k}(u)=0$ on the set $\{|u|\geq k\}$, $\nabla T_{k}(u)\chi _{\{|u_{n}|\geq k\}}\to 0$ for almost every $x\in \Omega$, so, by Lebesgue theorem $A_{2}=\varepsilon _{n}$. For $(A_1)$, we have \begin{align*} &\int_{\Omega }a(\nabla T_{k}(u_{n}))\nabla (T_{k}(u_{n})- T_{k}(u))\varphi _{n}'dx\\ &=\int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))]\nabla (T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx\\ &\quad +\int_{\Omega }a(\nabla T_{k}(u))\nabla (T_{k}(u_{n})-T_{k}(u))\varphi _{n}'dx := A_{1.1}+A_{1.2} \end{align*} By (\ref{eP13}) , since $T_{k}(u_{n})\rightharpoonup T_{k}(u)$ weakly in $W_{0}^{1,p}(\Omega )$, we have $A_{1.2}=\varepsilon _{n}$. Thus $$A=\int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u)) ]\nabla \big(T_{k}(u_{n})-T_{k}(u)\big)\varphi _{n}'dx+\varepsilon _{n}. \label{eP18}$$ By \eqref{eP15}, \eqref{eP16}, \eqref{eP17}, \eqref{eP18} and from \eqref{eP14}, we obtain \begin{equation*} \int_{\Omega }[a\big(\nabla T_{k}(u_{n})\big)-a\big( \nabla T_{k}(u)\big)]\nabla \big(T_{k}(u_{n})-T_{k}(u) \big) [\varphi _{n}'-b_{k}|\varphi _{n}|] dx\leq \varepsilon _{n}. \end{equation*} Since $\varphi _{n}'-b_{k}|\varphi _{n}|\geq \frac{1}{2}$ with $a=1$ and $b=b_{k})$ and \begin{gather*} [a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))]\nabla \big(T_{k}(u_{n}) -T_{k}(u)\big)\geq 0, \\ \int_{\Omega }[a(\nabla T_{k}(u_{n}))-a(\nabla T_{k}(u))] \nabla \big(T_{k}(u_{n})-T_{k}(u)\big)dx=\varepsilon _{n}; \end{gather*} therefore, \begin{equation*} \langle -\Delta _{p}(T_{k}(u_{n}))+\Delta _{p}(T_{k}(u)),T_{k}(u_{n})-T_{k}(u)\rangle \to 0. \end{equation*} Since $T_{k}(u_{n})\rightharpoonup T_{k}(u)$ weakly in $W_{0}^{1,p}(\Omega )$, \begin{gather*} \langle -\Delta _{p}(T_{k}(u)),T_{k}(u_{n})-T_{k}(u)\rangle \to 0,\\ \langle -\Delta _{p}(T_{k}(u_{n})),T_{k}(u_{n})-T_{k}(u)\rangle \to 0. \end{gather*} Since $-\Delta _{p}$ belongs to the class $(S^{+})$ (see \cite{Mu}), $T_{k}(u_{n})\to T_{k}(u)$ strongly in $W_{0}^{1,p}(\Omega )$. \end{proof} \begin{lemma} \label{lem6} The following to limit hold: $$\begin{gathered} \lim_{k\to +\infty } [\sup_{n\in \mathbb{N}} \int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}dx] =0, \\ \lim_{k\to +\infty }[\sup_{n\in \mathbb{N}} \int_{\{|u_{n}|\geq k\}}|f(x,u_{n},\nabla u_{n})|dx]=0. \end{gathered} \label{eP19}$$ \end{lemma} \begin{proof} For the first limit, we define $\psi : \mathbb{R}\to \mathbb{R}^{+}$ by $\psi (-s)=-\psi (s)$ for all $s\in \mathbb{R}$ and $\psi (s)=\begin{cases} 0 & \text{if }0\leq s\leq k-1, \\ s-(k-1) & \text{if } k-1\leq s\leq k, \\ 1 & \text{if } s\geq k, \end{cases}$ where $k>1$, so that $\psi$ is continuous, bounded in $\mathbb{R}$ and $\psi (u_{n})\in W_{0}^{1,p}(\Omega )$. We choose $v=\psi (u_{n})$, as a test function in \eqref{eP9} we have \begin{align*} &\int_{\Omega }|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi (u_{n})dx\\ &=\int_{\Omega }f(x,u_{n},\nabla u_{n})\psi (u_{n})dx+\int_{\Omega }g_{n}\psi (u_{n})dx+\int_{\Omega }F\nabla \psi (u_{n})dx. \end{align*} Using Young's inequality, we obtain \begin{align*} \int_{\Omega }|\nabla \psi (u_{n})|^{p}dx & \leq \int_{\Omega }f(x,u_{n},\nabla u_{n})\psi (u_{n})dx +\int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx \\ &\quad + c\int_{\{ k-1<|u_{n}|k-1\}}f(x,u_{n},\nabla u_{n})\psi (u_{n})dx \\ & \leq \int_{\{|u_{n}|>k-1\}}[-\rho |\nabla u_{n}|^{p}|\psi (u_{n})|+\alpha '| u_{n}|^{p-1}|\psi (u_{n})| \\ &\quad + \beta '|\nabla u_{n}|^{p-1}|\psi (u_{n})|+a_{1}(x)|\psi (u_{n})|]dx. \end{align*} From (\ref{eP20}), we have \begin{aligned} &\rho \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}| \psi (u_{n})|dx \\ &\leq \int_{\{|u_{n}|\geq k-1\}}|g_{n}|dx+c\int_{\{ k-1<|u_{n}| k-1\}}|u_{n}|^{p-1}|\psi (u_{n})|dx \\ &\quad + \beta '\int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p-1}|\psi (u_{n})|dx + \int_{\{|u_{n}|>k-1\}}a_{1}(x)|\psi (u_{n})|dx. \end{aligned} \label{eP21} Since $u_{n}\to u$ in $L^{p}(\Omega )$, there exists $v\in L^{p}(\Omega )$ such that $|u_{n}|\leq |v|$. Since $|g_{n}|\leq |\widetilde {g}|$, $|\widetilde {g}|\in L^{1}(\Omega )$ and $|\psi (s)|\leq 1$, we have \begin{align*} &\rho \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}| \psi (u_{n})|dx\\ &\leq \int_{\Omega }[|\widetilde {g}|+c|F|^{p'}+\alpha '|v|^{p-1} +a_{1}(x)]\chi_{\{|v|\geq k-1\}}dx + \beta '\int_{\{|v|>k-1\}}|\nabla u_{n}|^{p-1}dx \\ &\leq \int_{\Omega }r(x)\chi _{\{|v|\geq k-1\}}dx +\beta '\| u_{n}\|_{1,p}^{p-1}(\int_{\Omega }\chi _{\{| v|\geq k-1\}}dx)^{1/p}, \end{align*} where $r(x)=|\widetilde {g}|+c|F|^{p'}+\alpha '|v|^{p-1}+a_{1}(x)$. We have $r\in L^{1}(\Omega )$ and $(u_{n})_{n}$ is bounded in $W_{0}^{1,p}(\Omega )$, so that \begin{equation*} \lim_{k\to +\infty }[\sup_{n\in \mathbb{N}} \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx ]=0. \end{equation*} Since \begin{align*} \int_{\{|u_{n}|\geq k\}} |\nabla u_{n}|^{p}dx &=\int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx \\ &\leq \int_{\{|u_{n}|>k-1\}}|\nabla u_{n}|^{p}|\psi (u_{n})|dx, \end{align*} it follows that \begin{equation*} \lim_{k\to +\infty }[\sup_{n\in \mathbb{N}} \int_{\{|u_{n}|\geq k\}}|\nabla u_{n}|^{p}dx]=0. \end{equation*} For the second limit, we let $l:\Omega \times \mathbb{R\times R}^{N}\to \mathbb{R}$ defined by \begin{equation*} l(x,s,\xi )=f(x,s,\xi )-\alpha |s|^{p-1}\mathop{\rm sgn}(s)-\beta | \xi |^{p-2}\xi -\big(|s|^{p-1}+|\xi |^{p-1} +a_{1}(x)\big)\mathop{\rm sgn}(s). \end{equation*} From \eqref{eP3}, we get $l(x,s,\xi )s\leq -\rho |\xi |^{p}|s|$ for almost every $x\in \Omega$, and for all $(\xi ,s)\in \mathbb{R}^{N}\times \mathbb{R}$. By \eqref{eP20} and using that $\psi (s)$ has the same sign as $s$ and that it is zero if $|s|\leq k-1$, we have \begin{align*} 0 & \leq \int_{\{|u_{n}|\geq \ k-1\}}|g_{n}|dx \\ & \quad + c\int_{\{ k-1<|u_{n}|0$be fixed. Let now$E$be a measurable subset of$\Omega $, we have \begin{equation*} \int_{E}|\nabla u_{n}|^{p}dx=\int_{E\cap \{|u_{n}| \leq k\}}|\nabla u_{n}|^{p}dx+\int_{E\cap \{|u_{n}| >k\}}|\nabla u_{n}|^{p}dx. \end{equation*} By lemma \ref{lem6} there exists$k>0$such that for all$n\in \mathbb{N}$, $\int_{\{|u_{n}|>k\}}|\nabla u_{n}|^{p}dx\leq \frac{ \varepsilon }{2}.$ For$k$fixed, we have $\int_{E\cap \{|u_{n}|\leq k\}}|\nabla u_{n}|^{p}dx \leq \int_{E}|\nabla T_{k}(u_{n})|^{p}dx.$ Since$T_{k}(u_{n})$converges strongly to$T_{k}(u)$in$W_{0}^{1,p}(\Omega )$, there exists$\gamma >0$such that \begin{equation*} \mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N}\text{ \ }\int_{E}|\nabla T_{k}(u_{n})|^{p}dx\leq \frac{\varepsilon }{2}, \end{equation*} so that \begin{equation*} \forall n\in \mathbb{N}\ \ \int_{E\cap \{|u_{n}|\leq k\}}|\nabla u_{n}|^{p}dx\leq \frac{\varepsilon }{2}. \end{equation*} Then, there exists$\gamma >0$such that \begin{equation*} \mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N}\int_{E}| \nabla u_{n}|^{p}dx\leq \varepsilon . \end{equation*} Therefore, the sequence$\{|\nabla u_{n}|^{p}\}$is equi-integrable in$L^{1}(\Omega )$. By Lemma \ref{lem5} we have$\nabla u_{n}\to \nabla u$for almost every$x\in \Omega $, so,$|\nabla u_{n}|^{p}\to |\nabla u|^{p}$strongly in$L^{1}(\Omega )$, thus the sequence$(u_{n})_{n}$converges strongly to$u$in$W_{0}^{1,p}(\Omega )$. \end{proof} \begin{lemma} The sequence$(f(x,u_{n},\nabla u_{n}))_{n}$converges to$f(x,u,\nabla u)$in$L^{1}(\Omega )$. \end{lemma} \begin{proof} We begin by proving that the sequence$\{|f(x,u_{n},\nabla u_{n})|\}$is equi-integrable in$L^{1}(\Omega )$. Let$\varepsilon >0$be fixed. Let now$E$be a measurable subset of$\Omega , we have \begin{align*} &\int_{E}|f(x,u_{n},\nabla u_{n})|dx\\ &=\int_{E\cap \{|u_{n}|\leq k\}}|f(x,u_{n},\nabla u_{n})|dx +\int_{E\cap\{|u_{n}|>k\}}|f(x,u_{n},\nabla u_{n})|dx. \end{align*} By Lemma \ref{lem6}, there existsk>0$such that \begin{equation*} \forall n\in \mathbb{N},\; \int_{E\cap \{|u_{n}|>k\}}| f(x,u_{n},\nabla u_{n})|dx\leq \frac{\varepsilon }{2}. \end{equation*} When$k$is fixed, by \eqref{eP2} we have \begin{equation*} \int_{E\cap \{|u_{n}|\leq k\}}|f(x,u_{n},\nabla u_{n})| dx\leq \int_{E}[b_{k}|\nabla T_{k}(u_{n})|^{p}+\phi _{k}(x) ]dx. \end{equation*} Since$\phi _{k}\in L^{1}(\Omega )$and$T_{k}(u_{n})\to T_{k}(u) $strongly in$W_{0}^{1,p}(\Omega )$, there exists$\gamma >0$such that \begin{equation*} \mathop{\rm meas}(E)<\gamma \Rightarrow \forall n\in \mathbb{N} \; \int_{E}[b_{k}|\nabla T_{k}(u_{n})|^{p}+\phi _{k}(x)]dx \leq \frac{\varepsilon }{2}, \end{equation*} so that \begin{equation*} \forall n\in \mathbb{N}\; \int_{E\cap \{|u_{n}|\leq k\}}|f(x,u_{n},\nabla u_{n})|dx\leq \frac{\varepsilon }{2}. \end{equation*} Therefore, the sequence$\{|f(x,u_{n},\nabla u_{n})|\}_{n}$is equi-integrable in$L^{1}(\Omega )$. Since$f:\Omega \times \mathbb{R} \times \mathbb{R}^{N}\to \mathbb{R}$is a Carath\'{e}odory function, we have$f(x,u_{n},\nabla u_{n})\to \ f(x,u,\nabla u)$for almost every$x\in \Omega $. so$f(x,u_{n},\nabla u_{n})\to \ f(x,u,\nabla u)$strongly in$L^{1}(\Omega )$. \end{proof} Going back to the the proof of Theorem 1.1, by \eqref{eP9} we have that for every$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty}(\Omega )$, \begin{equation*} \int_{\Omega }|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla v\,dx=\int_{\Omega }f(x,u_{n},\nabla u_{n})v\,dx+\int_{\Omega }g_{n}v+\int_{\Omega }F\nabla v. \end{equation*} As$n$approaches infinity, we get that for every$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, \begin{equation*} \int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx =\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }gv+\int_{\Omega}F\nabla v. \end{equation*} Thus the problem \begin{gather*} -\Delta _{p}u = f(x,u,\nabla u)+\mu \quad\text{in } \Omega , \\ u = 0 \quad\text{on }\partial \Omega \end{gather*} admits a solution$u\in W_{0}^{1,p}(\Omega )$in the sense that$f(x,u,\nabla u)\in L^{1}(\Omega )$, and for every$v\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega )\$, \begin{equation*} \int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega }f(x,u,\nabla u)v\,dx+\int_{\Omega }v\,d\mu. \end{equation*} \end{proof} \begin{thebibliography}{9} \bibitem{Anan} A. Anane, O. Chakrone, J. P. Gossez; \emph{Spectre d'ordre sup\'{e}rieur et probl\`{e}mes aux limites quasi-lin\'{e}aires}, Bollettino U.M.I. (8) 4-B (2001), 483-519. \bibitem{Mu} J. Berkovits, V. Mustonen; \emph{Nonlinear mapping of monotone type (classification and degree theory)}, 1988, Math. Univer. Oulu, Linnanmaa, Oulu, Finland. \bibitem{Boc} L. Boccardo, T. Gallouet, L. Orsina; \emph{Existence and nonexistence of solutions for some nonlinear elliptic equations}, J. Anal. Math. 75. (1997), 203-223. \bibitem{Bocc} L. Boccardo, T. Gallouet, L. Orsina; \emph{Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data}, Ann. Inst. H. Poincar\'{e}, Anal. Non Lin\'{e}aire, 5 (1996), 539-551. \end{thebibliography} \end{document}