2004-Fez conference on Differential Equations and Mechanics. Electron. J. Diff. Eqns., Conference 11, 2004, pp. 11-21.

On the solvability of degenerated quasilinear elliptic problems

Youssef Akdim, Elhoussine Azroul, Mohamed Rhoudaf

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 $0 \leq \eta < \min (p-1,q-1)$, $0 \leq \delta &\lt; (p-1)/q'$.

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

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Youssef Akdim
Département de Mathématiques et Informatique,
Faculté des Sciences Dhar-Mahraz,
B.P. 1796 Atlas, Fès, Maroc.
e-mail: akdimyoussef@yahoo.fr
Elhoussine Azroul
Département de Mathématiques et Informatique,
Faculté des Sciences Dhar-Mahraz,
B.P. 1796 Atlas, Fès, Maroc.
e-mail: azroul_elhoussine@yahoo.fr
Mohamed Rhoudaf
Département de Mathématiques et Informatique,
Faculté des Sciences Dhar-Mahraz,
B.P. 1796 Atlas, Fès, Maroc.
e-mail: rhoudaf_mohamed@yahoo.fr

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