Ground states and concentration phenomena for the fractional Schr\"odinger equation

Research paper by Mouhamed Moustapha Fall, Fethi Mahmoudi, Enrico Valdinoci

Indexed on: 24 Apr '15Published on: 24 Apr '15Published in: Mathematics - Analysis of PDEs

Abstract

We consider here solutions of the nonlinear fractional Schr\"odinger 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.