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:
−Δp(x)lu+δ|u|ζ(x)−2u=μg(x,u)+λf(x,u,∇u)in Ω,∂u∂η=0on ∂Ω,
where Δp(x)lu is the p(x)-Laplacian-like operator, Ω is a smooth bounded domain in RN, 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.
