A quasilinear Neumann problem involving the p(x)-Laplacian

We study the following Neumann problem: {−Δp(x)u+α(x)|u|p(x)−2u=α(x)f(u)+λg(x,u)in Ω∂u∂ν=0on ∂Ω and we prove that, under suitable assumptions on the functions α, f, p and g, the Ricceri two-local-minima theorem, together with the Palais-Smale property, ensures the existence of at least three solutions of it. This work could be considered a possible extension of some results by Cammaroto, Chinnì and Di Bella who handled the case where p(x) is constant.