On $\big(p_{1}(x), p_{2}(x)\big)$-Laplace Equations

TL;DR

This paper studies the properties of the $(p_{1}(x), p_{2}(x))$-Laplace operator, establishing existence results for solutions to related differential equations with variable exponents.

Contribution

It introduces new analysis of the $(p_{1}(x), p_{2}(x))$-Laplace operator and proves existence of weak solutions using properties of the associated integral functional.

Findings

The integral functional admits an $(S_+)$ derivative inducing a homeomorphism.

Existence results for solutions to the $(p_{1}(x), p_{2}(x))$-Laplace equation are established.

Weak solutions exist under specified conditions in bounded smooth domains.

Abstract

In this paper, we investigate the $(p_{1}(x), p_{2}(x))$-Laplace operator, the properties of the corresponding integral functional and weak solutions to the related differential equations. We show that the integral functional admits a derivative of type $(S_+)$ which induces a homeomorphism between duality space pairs. As applications of the above results, we gave some existence results of the $(p_{1}(x), p_{2}(x))$-Laplace equation -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)=f(x,u) in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$ with Dirichlet boundary condition.