Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials

TL;DR

This paper proves the existence of positive stationary solutions for certain nonlinear Schrödinger equations with potentials that may decay rapidly or be compactly supported, showing solutions concentrate near local minima as a parameter tends to zero.

Contribution

It establishes existence and concentration of solutions without restrictions on the decay rate of the potential, including compactly supported potentials.

Findings

Solutions exist for small epsilon near local minima of V.

No restrictions on the decay rate of the potential V.

Solutions concentrate around the minimum as epsilon approaches zero.

Abstract

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.