Remarks on a multiplicity theorem for a perturbed Neumann problem

The aim of this article is to prove that, given two potential functionals Ψ1, Ψ2 on W1,2(Ω) which coincide on a set of the type {u ∈ W1,2(Ω): a u(x) b a.e. in Ω}, then under suitable summability conditions, certain local minima of Ψ1 are local minima for Ψ2 as well. An application of this result allows us to obtain a multiplicity theorem for a Neumann problem where we impose a less restrictive oscillating behavior on the nonlinearity than the one required in an analogous result recently established by B. Ricceri