Electronic Journal of Differential Equations,
Conference 08 (2002), pp. 47-52

Title: A remark on some nonlinear elliptic problems

Authors: Lucio Boccardo (Univ. a di Roma I, Roma, Italia)
 
Abstract: 
We shall prove an existence result of $W_0^{1,p}(\Omega)$
solutions for the
   boundary value problem
 $$\displylines{
   -\mathop{\rm div} a(x, u,\nabla u)=F \quad\mbox{in }\Omega\cr
   u=0\quad\mbox{on }\partial\Omega
 }$$
   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}.

Published October 21, 2002.
Math Subject Classifications: 35J60, 35J65, 35J70.
Key Words: Dirichlet problem in $L^1$"; uncontrolled growth.