Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials

We study the existence of stationnary positive solutions for a class of nonlinear Schroedinger equations with a nonnegative continuous potential V. Amongst other results, we prove that if V has a positive local minimum, and if the exponent of the nonlinearity satisfies N/(N-2)<p<(N+2)/(N-2), then for small epsilon the problem admits positive solutions which concentrate as epsilon goes to 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.