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\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
\begin{equation} \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}
\end{equation}
   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
\begin{equation}\label{F}
F\in W^{-1,p'}(\Omega).
\end{equation}
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
\begin{equation}\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{equation}
\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
\begin{equation} \label{pn}
u_n\in  W_0^{1,p}(\Omega):\, -\mathop{\rm div}(a(x, T_n(u_n),\nabla u_n)=F.
\end{equation}
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.
This completes the proof of  Theorem \ref{exist}.
\hfill$\square$

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\noindent\textsc{Lucio Boccardo}\\
Dipartimento di Matematica, Universit\`a di Roma I,\\
Piazza A. Moro 2, 00185 Roma, Italia\\
e-mail: \texttt{boccardo@mat.uniroma1.it }
\end{document}