Asymptotic Behaviour of Some Equations in Orlicz Spaces

Volume 4, Issue 5, 2003

Article 98

ASYMPTOTIC BEHAVIOUR OF SOME EQUATIONS IN ORLICZ SPACES

D. MESKINE AND A. ELMAHI

DÉPARTEMENT DE MATHÉMATIQUES ET INFORMATIQUE
FACULTÉ DES SCIENCES DHAR-MAHRAZ
B.P 1796 ATLAS-FÈS, FÈS MAROC.
E-Mail: meskinedriss@hotmail.com

C.P.R DE FÈS, B.P 49, FÈS, MAROC.

Received 26 March, 2003; Accepted 05 August, 2003.
Communicated by: A. Fiorenza

ABSTRACT. In this paper, we prove an existence and uniqueness result for solutions of some bilateral problems of the form

$\displaystyle \left\{ \begin{array}{l} \langle Au, v-u \rangle \geq \langle f, v-u \rangle , \ \forall v\in K \\ [10pt] u\in K \end{array} \right.$

where

is a standard Leray-Lions operator defined on $W_{0}^{1}L_{M}(\Omega)$ , with

an N-function which satisfies the $\Delta_2$ -condition, and where

is a convex subset of $W_{0}^{1}L_{M}(\Omega)$ with obstacles depending on some Carathéodory function

. We consider first, the case $f\in W^{-1}E_{\overline M}(\Omega)$ and secondly where $f\in L^{1}(\Omega)$ . Our method deals with the study of the limit of the sequence of solutions

of some approximate problem with nonlinearity term of the form $\vert g(x,u_n)\vert^{n-1}g(x,u_n)\times M(\vert\nabla u_n\vert)$ .

Key words:
Strongly nonlinear elliptic equations, Natural growth, Truncations, Variational inequalities, Bilateral problems.

2000 Mathematics Subject Classification:
35J25, 35J60.

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