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We study the existence of stationnary positive solutions for a class ofnonlinear Schroedinger equations with a nonnegative continuous potential V.Amongst other results, we prove that if V has a positive local minimum, and ifthe exponent of the nonlinearity satisfies N/(N-2)<p<(N+2)/(N-2), then forsmall epsilon the problem admits positive solutions which concentrate asepsilon goes to 0 around the local minimum point of V. The novelty is that norestriction is imposed on the rate of decay of V. In particular, we cover thecase where V is compactly supported.

Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials