Asymptotic behavior of positive large solutions of semilinear Dirichlet problems

Habib Maagli,Sonia Ben Othman,Rym Chemmam

doi:10.14232/ejqtde.2013.1.57

Abstract

Let be a smooth bounded domain in R n ; n 2: This paper deals with the existence and the asymptotic behavior of positive solutions of the following problems u = a(x)u ; > 1 and u = a(x)e u ; with the boundary conditionuj@ = +1: The weight functiona(x) is positive inC loc () ; 0 < < 1, and satises an appropriate assumption related to Karamata regular variation theory. Our arguments are based on the sub-supersolution method.

Highlights

Let Ω be a C2 bounded domain in Rn, n ≥ 2
A is a positive function in Clγoc (Ω), 0 < γ < 1, satisfying an appropriate condition related to Karamata regular variation theory and δ(x) = dist(x, ∂Ω)
We prove the existence of classical solutions for both problems (1.1) and (1.2) and we establish the asymptotic behavior of such solutions where the function a is required to be in a large class of functions related to Karamata regular variation theory

Summary

Introduction

Let Ω be a C2 bounded domain in Rn , n ≥ 2. We deal with existence and estimates of solutions to the following elliptic problems (1.1). Some results of existence and nonexistence of solutions to problems (1.1) and (1.2) are established when the weight a(x) is unbounded near ∂Ω (see [5], [18], [22], [23], [24]). We prove the existence of classical solutions for both problems (1.1) and (1.2) and we establish the asymptotic behavior of such solutions where the function a is required to be in a large class of functions related to Karamata regular variation theory. We obtain existence and asymptotic behavior of solutions of problems (1.1) and (1.2) when the weight a belongs to Clγoc(Ω) and satisfies a(x) ≈ δ (x)−2 L (δ (x)) , where L ∈ K and η 0.

The Karamata class K

Proofs of main results

Applications