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

TL;DR

This paper studies solutions to the nonlinear fractional Schrödinger equation, showing how solutions concentrate at critical points of the potential and establishing uniqueness and concentration behavior under specific conditions.

Contribution

It proves that solutions concentrate at critical points of the potential and establishes uniqueness of ground states for small epsilon under radial symmetry.

Findings

Solutions concentrate at critical points of V.

Ground states concentrate at the global minimum of V as epsilon approaches zero.

Uniqueness of minimizers is shown for radial potentials when epsilon is small.

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.