Ground states and concentration phenomena for the fractional Schrödinger equation

Abstract

We consider here solutions of the nonlinear fractional Schrödinger equation $${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $\epsilon$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $\epsilon$ is small.

arXiv comment22 Pages, Accepted for Publications in "Nonlinearity"