A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let.. R-n be a bounded domain, with a boundary of class C-2, and let f,g : R. R be two continuous functions, alpha is an element of L-infinity(Omega), with ess inf(Omega)a > 0, beta is an element of L-p(Omega), with p > n. If [graphics] and if the set of all global minima of the function xi ->xi(2)/2 - integral(xi)(0) f(t)dt has at least k >= 2 connected components, then, for each lambda > 0 small enough, the Neumann problem [graphics] admits at least k + 1 strong solutions in W-2,W-p(Omega).