2004-Fez conference on Differential Equations and Mechanics.
Electronic Journal of Differential Equations,
Conference 11, 2004, pp. 11-21.

Title: On the solvability of degenerated quasilinear elliptic problems

Authors: Youssef Akdim (Faculte des Sciences Dhar-Mahraz, Maroc)
   Elhoussine Azroul (Faculte des Sciences Dhar-Mahraz, Maroc)
   Mohamed Rhoudaf (Faculte des Sciences Dhar-Mahraz, Maroc)
 
Abstract: 
 In this article, we study the quasilinear elliptic problem
 $$\displaylines{
 Au = - \mathop{\rm div} (a(x,u,\nabla u)) = f(x,u,\nabla u)
 \quad\hbox{in }  \mathcal{D}'(\Omega) \cr
 u = 0   \quad\hbox{on }\partial\Omega\,,
 }$$
 where $A$ is a Leray-Lions operator from $W_0^{1,p}(\Omega,w)$ to
 its dual  $W^{-1,p'}(\Omega,w^*)$. We show that there exists a
solution
 in $W_0^{1,p}(\Omega,w)$ provided that
 $$
 |f(x,r, \xi)|\leq \sigma^{1/q} [ g(x)+|r|^\eta \sigma^{\eta/q}
 +   \sum_{i=1}^N w_i^{\delta/p}(x)|\xi_i|^\delta],
 $$
 where $g(x)$ is a positive function in  $L^{q'}(\Omega)$ and
$\sigma(x)$ is
 weight function and $0 \leq  \eta < \min  (p-1,q-1)$,
 $0 \leq \delta < (p-1)/q'$.

Published October 15, 2004/
Math Subject Classifications: 35J20, 35J25, 35J70.
Key Words: Weighted Sobolev spaces; variational calculus, Hardy inequality.