Perturbation from Dirichlet problem involving oscillating nonlinearities

In this paper we prove that if the potential F(x, t) = f(0)(t) f (x, s) ds has a suitable oscillating behavior in any neighborhood of the origin (respectively +infinity), then under very mild conditions on the perturbation term g, for every k is an element of N there exists b(k) > 0 such that {-Delta u = f (x, u) + lambda g(x, u) in Omega, u = 0 on partial derivative Omega has at least k distinct weak solutions in W-0(1,2)(Omega), for every lambda is an element of R with vertical bar lambda vertical bar <= b(k). Moreover, information about the location of such solutions is also given. In fact, there exists a positive real number sigma > 0, which does not depend on lambda, such that the W-0(1,2)(Omega)-norm of each of those k solutions is not greater than sigma. (c) 2006 Elsevier Inc. All rights reserved.