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.