The asymptotic behavior of solutions to a class of inhomogeneous problems: an Orlicz–Sobolev space approach

Andrei Grecu,Denisa Stancu-Dumitru

doi:10.14232/ejqtde.2021.1.38

Abstract

The asymptotic behavior of the sequence {vn} of nonnegative solutions for a class of inhomogeneous problems settled in Orlicz–Sobolev spaces with prescribed Dirichlet data on the boundary of domain Ω is analysed. We show that {vn} converges uniformly in Ω as n → ∞, to the distance function to the boundary of the domain.

Highlights

Let Ω ⊂ RN (N ≥ 2) be a bounded domain with smooth boundary ∂Ω
We say that vn is a weak solution of problem (1.1) if vn ∈ W01,Φn (Ω) and the following relation holds true φn(|∇vn |∇vn|
The main goal of this section is to prove the existence of weak solutions of problem (1.1) for each positive integer n

Summary

Introduction

Let Ω ⊂ RN (N ≥ 2) be a bounded domain with smooth boundary ∂Ω. We consider the family of problems− div φn(|∇v|) |∇v| ∇v = λev in Ω, v = 0 on ∂Ω, (1.1)where for each positive integer n, the mappings φn : R → R are odd, increasing homeomorphisms of class C1 satisfying Lieberman-type condition N − < φ−n. We say that vn is a weak solution of problem (1.1) if vn ∈ W01,Φn (Ω) and the following relation holds true φn(|∇vn |∇vn|

Objectives

Results

Conclusion