Solutions for Neumann boundary value problems involving $\big(p_{1}(x), p_{2}(x)\big)$-Laplace operators

TL;DR

This paper investigates the existence and multiplicity of solutions for nonlinear Neumann boundary value problems involving variable exponent p-Laplace operators, using variational methods in variable exponent Sobolev spaces.

Contribution

It introduces new results on solution existence and multiplicity for variable exponent p-Laplace equations with Neumann boundary conditions, expanding the understanding of such nonlinear problems.

Findings

Existence of solutions under certain conditions.

Multiple solutions established via variational methods.

Application of variable exponent Sobolev space theory.

Abstract

In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$ with Neumann boundary condition given by |\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partial\nu}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partial\nu}=\mu g(x,u) on $\partial\Omega$. Under appropriate conditions on the source and boundary nonlinearities, we obtain a number of results on existence and multiplicity of solutions by variational methods in the framework of variable exponent Lebesgue and Sobolev spaces.