Weak and entropy solutions for a class of nonlinear inhomogeneous Neumann boundary value problem with variable exponent

Abstract

We study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous Neumann boundary value problem involving the ð‘(ð‘¥)-Laplace of the form − div É‘(ð‘¥, ∇ð‘¢) + |ð‘¢| ^{ð‘(ð‘¥)−2} ð‘¢ = f in Ω, É‘(ð‘¥, ∇ð‘¢).η = 𜑠on ∂Ω, where Ω is a smooth bounded open domain in â„^N, N ≥ 3, ð‘ ∈ C(Ω) and ð‘(ð‘¥) > 1 for 𑥠∈ Ω. We prove the existence and uniqueness of a weak solution for data 𜑠∈ L^{(ð‘−) ”²} (∂Ω) and f ∈ L^{(ð‘−) ”²} (Ω), the existence and uniqueness of an entropy solution for L¹âˆ’data f and 𜑠independent of ð‘¢ and the existence of weak solutions for f dependent on ð‘¢ and ðœ‘ ∈ L^{(ð‘−) ”²} (Ω).