2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems
Electron. J. Diff. Eqns., Conf. 08, 2002, pp. 47-52.

A remark on some nonlinear elliptic problems

Lucio Boccardo

Abstract:
We shall prove an existence result of $W_0^{1,p}(\Omega)$ solutions for the
$$\displaylines{ -\mathop{\rm div} a(x, u,\nabla u)=F \quad\hbox{in }\Omega\cr\ u=0\quad\hbox{on }\partial\Omega }$$
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 [1] 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 [5].

Published October 21, 2002.
Subject lassfications: 35J60, 35J65, 35J70.
Key words: Dirichlet problem in $L^1$, uncontrolled growth.

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Lucio Boccardo
Dipartimento di Matematica
Universita di Roma I
Piazza A. Moro 2, 00185 Roma, Italia
e-mail: boccardo@mat.uniroma1.it

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