Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

Adil Siddique,Zia Bashir,Wojciech Sałabun,Andrii Shekhovtsov

doi:10.3390/fractalfract6030128

Abstract

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: Gϵ∗=sup∫ΩG(u)ϵq(x)dx:∥∇u∥Lp(.)≤ϵ,u=0 on ∂Ω, with the help of Γ-convergence, where G:R→R is upper semicontinuous, non zero in the L1 sense, 0≤G(u)≤c|u|q(.), Ω is a bounded open subset of Rn, n≥3, 1<p(.)<n, and p(.)≤q(.)≤p∗(.). For special choices of G, we can study Bernoulli’s free-boundary and plasma problems in variable exponent Lebesgue spaces.

Highlights

The aim of this paper is to study the asymptotic behavior of low energy extremals
We establish the Γ-convergence of low energy extremals for a class of general functions in settings of variable exponent Lebesgue spaces
This work will help to determine low energy solutions. They form a spike near concentration points in the domain

Summary

Introduction

The aim of this paper is to study the asymptotic behavior of low energy extremals. with Γ-convergence, when e → 0. We refer to the section for the details of variable exponent Sobolev space. Bonder and Silva [12] and Yongqiang [13] proved the concentration/compactness principle for variable exponent Lebesgue spaces, independently, to deal with the following nonlinear elliptic PDE with critical growth at infinity:. Bashir et al [18] extended the work of Bonder and Silva [12] and proved generalized concentration/compactness principles for variable exponent Lebesgue spaces. They initiated the work on Problem (1) and proved the existence of low energy extremals.

Preliminaries

Proposed Theorems

Proofs of Theorems

Conclusions

Methods