Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

Ge Dong,Xiaochun Fang,Pasquale Vetro

doi:10.1155/2021/9927898

Abstract

In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x , u , D u + a 0 x , u = f x , u , D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N , a 0 : Ω × ℝ ⟶ ℝ , and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering “≤,” which are called barrier solutions.

Highlights

Let Ω ⊂ RN be a bounded domain with Lipschitz boundary
In Ω coupled with Neumann or Dirichlet boundary condition in reflexive
Dong and Fang [2] studied the existence of weak solutions for the following equations:

Summary

Introduction

Let Ω ⊂ RN be a bounded domain with Lipschitz boundary. Fan [1] established a sub-supersolution method for the following differential equations of divergence form:. The purpose of this paper is to provide sufficient conditions which ensure that the solution set S of the following Dirichlet boundary problem We refer to some results in variable exponent Sobolev or Orlicz-Sobolev spaces [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]

Preliminaries

Existence Theorems