\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{A remark on some nonlinear elliptic problems} { Lucio Boccardo} \begin{document} \setcounter{page}{47} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems,\newline Electronic Journal of Differential Equations, Conference 08, 2002, pp 47--52. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A remark on some nonlinear elliptic problems % \thanks{ {\em Mathematics Subject Classifications:} 35J60, 35J65, 35J70. \hfil\break\indent {\em Key words:} Dirichlet problem in $L^1$", uncontrolled growth. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002.} } \date{} \author{Lucio Boccardo} \maketitle \numberwithin{equation}{section} \begin{abstract} We shall prove an existence result of $W_0^{1,p}(\Omega)$ solutions for the boundary value problem $$\label{p} \begin{gathered} -\mathop{\rm div} a(x, u,\nabla u)=F \quad\mbox{in }\Omega\\ u=0\quad\mbox{on }\partial\Omega \end{gathered}$$ with right hand side in $W^{-1,p'}(\Omega)$. The features of the equation are that no restrictions on the growth of the function $a(x,s,\xi)$ with respect to $s$ are assumed and that $a(x,s,\xi)$ with respect to $\xi$ is monotone, but not strictly monotone. We overcome the difficulty of the uncontrolled growth of $a$ thanks to a suitable definition of solution (similar to the one introduced in \cite{B6} for the study of the Dirichlet problem in $L^1$) and the difficulty of the not strict monotonicity thanks to a technique (the $L^1$-version of Minty's Lemma) similar to the one used in \cite{BO}. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{defintion}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \def\elle#1{L^{#1}(\Omega)} \def\norm#1#2{\|#1\|_{#2}} \section{Introduction and assumptions} We deal with boundary value problems with differential operators $A$ defined as $$A(v)=-\mathop{\rm div}\,(a(x,v,\nabla v))$$ where $\Omega$ is a bounded domain of $\mathbb{R} ^N$, $N \geq 2$, $a: \Omega \times \mathbb{R} \times \mathbb{R} ^N \to \mathbb{R} ^N$ is a Carath\'eodory function (that is, measurable with respect to $x$ in $\Omega$ for every $(s,\xi)$ in $\mathbb{R} \times \mathbb{R}^N$, and continuous with respect to $(s,\xi)$ in $\mathbb{R} \times \mathbb{R}^N$ for almost every $x$ in $\Omega$). We assume that there exist a real positive constant $\alpha$, a continuous function $\beta(s)$ and a nonnegative function $k$ in $\elle{p'}$, where $1 < p$, such that for almost every $x$ in $\Omega$, for every $s$ in $\mathbb{R}$, for every $\xi$ and $\eta$ in $\mathbb{R}^N$ \begin{gather} a(x,s, \xi ) \cdot \xi \geq \alpha | \xi |^p\,, \label{coerc}\\ [a(x,s,\xi)-a(x,s, \eta)] \cdot [ \xi -\eta]\geq 0\,, \label{monot}\\ |a(x,s, \xi )|\leq \bigl(k(x) + [\beta( s)| \xi |]^{p-1}\bigr)\,. \label{cont} \end{gather} Thus, $A$ is not well defined on the whole $W_0^{1,p}(\Omega)$, but only in $W_0^{1,p}(\Omega)\cap\elle\infty$. Note that if we assume \begin{gather} [a(x,s,\xi)-a(x,s, \eta)] \cdot [ \xi -\eta]> 0,\quad \xi\neq \eta, \label{monot1} \\ |a(x,s, \xi )|\leq \bigl(k(x) + |s |^{p-1}+| \xi |^{p-1}\bigr) \label{cont1}, \end{gather} instead of \eqref{monot}, \eqref{cont}, $A$ turns out to be pseudomonotone, coercive and is hence surjective on $W_0^{1,p}(\Omega)$ (see \cite{Bre,Bro,HS,LL}). A model operator for our setting is $$-\sum_i \frac{\partial}{\partial x_i} \Big( (1+|v|^{\gamma_i}\chi_{_{E_i}})\frac{\partial v}{\partial x_i} \Big)$$ where $\gamma_i\geq 0$ and $\chi_{_{E_i}}$ is the characteristic function of the measurable subset $E_i\subset \Omega$. Concerning the right hand side of \eqref{p}, we assume that $$\label{F} F\in W^{-1,p'}(\Omega).$$ The aim of this note is to prove existence of solutions for \eqref{p} under the weaker assumption \eqref{monot}, without using the almost everywhere convergence of the gradients of the approximate equations, since this is impossible to prove in our setting. The main tools of our proof are a version of Minty's Lemma, similar to that one used in \cite{BO} to study nonlinear boundary value problems in $L^1$, and a definition of solution, similar to the one introduced in \cite{B6} for the Dirichlet problem in $L^1$. Other existence results of finite energy solutions, similar to the one of Theorem \ref{exist}, can be found \cite{B2} (see also \cite{LePo, Po2,B1,B3} for existence results and \cite{Po1,Po3} for uniqueness results). \section{Existence} We recall that, for $k > 0$ and $s$ in $\mathbb{R}$, the truncating function is defined as $$T_k(s) = \min \big\{ k, \max \{ -k , s\}\big\}\,.$$ The composition of functions in $W_0^{1,p}(\Omega)$ with $T_k$ will play an important role in our approach to the existence of solutions of \eqref{p}. More precisely, we will use the following definition of solution, which is similar to the one introduced in \cite{B6} for the Dirichlet problem in $L^1$. \begin{defintion} \rm A function $u\in W_0^{1,p}(\Omega)$ is a $T$-solution of \eqref{p} if $$\label{sol} \begin{gathered} u\in W_0^{1,p}(\Omega),\quad \forall \varphi\in W_0^{1,p}(\Omega)\cap\elle\infty:\\ \int_\Omega a(x,u,\nabla u)\nabla T_k[u-\varphi] = \langle F, T_k[u -\varphi]\rangle \end{gathered}$$ \end{defintion} \begin{theorem}\label{exist} Under the assumptions \eqref{coerc}, \eqref{monot}, \eqref{cont}, \eqref{F} there exists a $T$-solution $u$ of \eqref{sol}. \end{theorem} \begin{remark} \rm Note that, even if $a(x,s, \xi )$ is unbounded with respect to $s$, the integral in \eqref{sol} is well defined, since $\nabla T_k[u -\varphi]$ is not zero on the subset $\{x\in\Omega: |u (x)-\varphi(x)|\leq k \}$, that is in subsets where $u$ is bounded. \end{remark} \begin{remark} \rm If in \eqref{sol} we take $\varphi=0$, we have $$\int_\Omega a(x,u,\nabla u)\nabla T_k(u) = \langle F, T_k(u)\rangle,$$ so that Lebesgue and Fatou Theorems imply that $\displaystyle a(x,u,\nabla u)\nabla u \in\elle 1$, but we are not able to prove that $\displaystyle a(x,u,\nabla u)\in\elle 1$, which would imply that $u$ is a solution in ${\cal D}'(\Omega)$, that is $$u\in W_0^{1,p}(\Omega): \int_\Omega a(x,u,\nabla u)\nabla \phi = \langle F, \phi\rangle , \quad\forall \phi\in{\cal D}(\Omega).$$ \end{remark} \paragraph{Proof of Theorem \ref{exist}.} Consider the approximate problems $$\label{pn} u_n\in W_0^{1,p}(\Omega):\, -\mathop{\rm div}(a(x, T_n(u_n),\nabla u_n)=F.$$ The solutions $u_n$ exist thanks to the Leray-Lions existence theorem (see \cite{LL}). Moreover, the use of $u_n$ as test function in \eqref{pn} and the assumption \eqref{coerc} imply that the sequence $\{u_n\}$ is bounded in $W_0^{1,p}(\Omega)$. Thus, there exists a function $u\in W_0^{1,p}(\Omega)$ and a subsequence $\{u_{n_j}\}$ such that $u_{n_j}$ converges weakly to $u$ in $W_0^{1,p}(\Omega)$ and almost everywhere. Now we use an idea of G.J. Minty (\cite{M}), in the framework of \cite{BO}: thanks to the monotonicity of $a(x,s,\xi)$ with respect to $\xi$, if $\varphi\in W_0^{1,p}(\Omega)\cap\elle\infty$ and $n>k+\norm{\varphi}{\elle\infty}$, we have that $$\int_\Omega a(x,u_{n_j}, \nabla \varphi)\nabla T_k[u_{n_j}-\varphi] \leq \langle F, T_k[u_{n_j}-\varphi]\rangle.$$ The weak convergence of the sequence $\{ u_{n_j} \}$ in $W_0^{1,p}(\Omega)$ and the remark that $\nabla T_k[u_{n_j}-\varphi]$ is not zero on the subset $\{x\in\Omega: |u_{n_j}(x)-\varphi(x)|\leq k \}$ (subset of $\{x\in\Omega: |u_{n_j}(x)|\leq k +\norm{\varphi}{\elle\infty}\}$) allow to pass to the limit in the previous inequality, so that $$\int_\Omega a(x,u, \nabla \varphi) \nabla T_k[u-\varphi] \leq \langle F, T_k[u-\varphi]\rangle.$$ Let $h$ and $k$ be positive real numbers, let $t$ belong to $(-1,1)$, and let $\psi$ be a function in $W_0^{1,p}(\Omega) \cap \elle\infty$. Choose $\varphi =T_h(u) + t\,T_k[u-\psi]$ in the previous inequality. Setting $G_k(s) = s - T_k(s)$, we obtain \begin{eqnarray*} I &=& \int_\Omega a(x,u,\nabla T_h(u) + t \nabla T_k[u-\psi] ) \nabla T_k(G_h(u) - t\,T_k[u-\psi] ) \\ &\leq& \langle F, T_k(G_h(u) - t , T_k[u-\psi]) \rangle = J\,. \end{eqnarray*} We then have \begin{eqnarray*} I &=& \int_{\{| G_h(u) - t\,T_k[u-\psi] | \leq k \}}{a(x,u,\nabla T_h(u) + t\,\nabla T_k[u-\psi]) \nabla G_h(u)} \\ && - t\,\int_{\{| G_h(u) - t\,T_k[u-\psi] | \leq k \}}{a(x,u,\nabla T_h(u) + t\,\nabla T_k[u-\psi] ) \nabla T_k[u-\psi] } \\ &=& H + L\,. \end{eqnarray*} Choosing $h \geq k + \norm{\psi}{\elle\infty}$, we have $|T_k[u-\psi]| = k$ on the set $\{|u|\geq h\}$; on the same set, we have $\nabla T_h(u) = 0$. Since $\nabla G_h(u)$ is different from zero only on $\{|u| \geq h\}$, we obtain $$H = \int_{\{|G_h(u) - t\,T_k[u-\psi] | \leq k \}}{a(x,u,0) \nabla G_h(u)} = 0\,,$$ being $a(x,s,0) = 0$ as a consequence of \eqref{coerc}. Since $\nabla T_k[u-\psi]$ is different from zero only on the set $\{x\in\Omega:|u(x)-\psi(x)| \leq k\}$, and on this set $|u| \leq k + \norm{\psi}{\elle\infty} \leq h$, then \begin{eqnarray*} \lefteqn{\{ | G_h(u) - t\,T_k[u-\psi] | \leq k \} \cap \{|u-\psi| \leq k\}}\\ &=&\{ | - t\,T_k[u-\psi] | \leq k \} \cap \{|u-\psi| \leq k\} \\ &=& \{|u-\psi| \leq k\}\,, \end{eqnarray*} where the last passage is due to the fact that $|t| < 1$. Hence, $$L = -t \, \int_\Omega {a(x,u,\nabla u + t\,T_k[u-\psi] ) \nabla T_k[u-\psi] }\,,$$ and so, for $h \geq k + \norm{\psi}{\elle\infty}$, we have $$I = - t \, \int_\Omega {a(x,u,\nabla u + t\,T_k[u-\psi] ) \nabla T_k[u-\psi] }\,.$$ On the other hand, we have, since $|t| < 1$, $T_k(s)$ is odd and $u\in W_0^{1,p}(\Omega)$, $$\lim_{h \to +\infty} \langle F, T_k(G_h(u) - t\,T_k[u-\psi]) \rangle = - t \langle F, T_k[u-\psi] \rangle$$ We thus have proved that $$-t \, \int_\Omega {a(x,u,\nabla u + t\,T_k[u-\psi] ) \nabla T_k[u-\psi] } \leq - t \langle F,\, T_k[u-\psi] \rangle \,,$$ for every $\psi \in W_0^{1,p}(\Omega) \cap \elle\infty$, and for every $k > 0$. Choosing $t > 0$, dividing by $t$, and then letting $t$ tend to zero, we obtain $$\int_\Omega {a(x,u,\nabla u) \nabla T_k[u-\psi] } \geq \langle F,\, T_k[u-\psi] \rangle \,,$$ while the reverse inequality is obtained choosing $t < 0$, dividing by $-t$, and then letting $t$ tend to zero. 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