On Elliptic Equations in Orlicz Spaces Involving Natural Growth Term and Measure Data

Mohammed Kbiri Alaoui

doi:10.1155/2012/615816

Abstract

The paper deals with the existence of solutions of elliptic equations in the framework of Orlicz spaces with right-hand side measure and natural growth term.

Highlights

We give the following definitions and results which will be used in our main result
The p-capacity Cp B, Ω of any set B ⊂ Ω with respect to Ω is defined in the following classical way
We say that u is a weak solution of the problem Pμ if u is measurable, Tk u ∈ W01LM Ω, 3.4 a ·, u, ∇u ∇v dx g u M |∇u| v dx v dμ ∀v ∈ D Ω

Summary

Introduction

Ω is a bounded domain of RN, with the segment property. a : Ω × R × RN → RN is a Caratheodory function i.e., measurable with respect to x in Ω for every s, ξ in R × RN, and continuous with respect to s, ξ in R × RN for almost every x in Ω such that there exist two N-functions P M and for all ξ, ξ∗ ∈ RN, ξ / ξ∗, the following hypotheses are true a x, s, ξ ξ ≥ αM. Conditions on which the function g have been considered in order to get a solution for all f in a given Lebesgue space. This is the way chosen in 1, 5, 6 under the hypothesis g ∈ L1. In 8 , the authors present some results concerning existence, nonexistence, multiplicity, and regularity of positive solutions for two elliptic quasilinear problems with Dirichlet data in a bounded domain. The second one, of unknown v, presents a source term of order 0 They gave and established a precise connection between problems in u and v. In the spirit of the work 5 , our purpose in this paper is to prove existence results in the setting of the Orlicz Sobolev space W1LM Ω when the operator does not satisfy the classical polynomial growth

Preliminaries

Useful Results

Existence Result

A Priori Estimates

General Case