Semiclassical stationary states for nonlinear Schrodinger equations with fast decaying potentials

We study the existence of positive solutions for a class of nonlinear Schrodinger equations of the type -epsilon(2)Delta u + Vu = u(p) in R(N), where N >= 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and N/N-2 < p < N+2/N-2 , then for small epsilon the problem admits positive solutions which concentrate as epsilon -> 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.