Concentration behaviour of normalized ground states of the mass critical fractional Schrödinger equations with ring-shaped potentials

Abstract

We consider $L^{2}$ -constraint minimizers of the mass critical fractional Schrödinger energy functional with a ring-shaped potential $V(x)=(|x|-M)^{2}$ , where $M>0$ and $x\in \mathbb {R}^{2}$ . By analysing some new estimates on the least energy of the mass critical fractional Schrödinger energy functional, we obtain the concentration behaviour of each minimizer of the mass critical fractional Schrödinger energy functional when $a\nearrow a^{\ast }=\|Q\|_{2}^{2s}$ , where $Q$ is the unique positive radial solution of $(-\Delta )^{s}u+su-|u|^{2s}u=0$ in $\mathbb {R}^{2}$ .