Existence result for a Neumann boundary value problem governed by a class of p ( x )-Laplacian-like equation

Abstract

In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in  Ω , ∂ u ∂ η = 0 on  ∂ Ω , where Δ p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ, μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ), η is the outer unit normal to ∂ Ω and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.