Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents

Abstract

In this article, we study the following nonlinear Neumann boundary-value problem − div a(x, ∇u) + |u| p(x)−2 u = f in Ω, on ∂Ω, where Ω is a smooth bounded open domain in ℝ N , is the outer unit normal derivative on ∂Ω, div a(x, ∇u) a p(x)-Laplace type operator. We prove the existence and uniqueness of a weak solution for f ∈ L (p −)′(Ω), the existence and uniqueness of an entropy solution for L 1-data f independent of u and the existence of weak solutions for f dependent on u. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.