On $(p_1(x),p_2(x))$-Laplace Equations

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.

15pages