Solutions for Neumann boundary value problems involving $(p_1(x),p_2(x))$-Laplace operators

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=λf(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}{\partialν}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partialν}=μg(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.

17 pages. arXiv admin note: substantial text overlap with arXiv:1205.1854

References in corpus (1)

• Existence for the steady problem of a mixture of two power-law fluids