Blow-up rate and local uniqueness for fractional Schrödinger equations with nearly critical growth

Abstract

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation $(-\Delta {)}^{s}u+V(x)u=u^{2_s^{\ast }-1-\epsilon }\text{in}{\mathbb{R}}^N,$ where $\epsilon >0$, $s\in (0,1)$, $2_s^{\ast }:=\frac{2N}{N-2s}$, $N>4s$. We show that the ground state $u_{\epsilon }$ blows up and precisely with the following rate $\Vert u_{\epsilon }{\Vert }_{L^{\infty }({\mathbb{R}}^N)}\sim {\epsilon }^{-\frac{N-2s}{4s}}$, as $\epsilon \to 0^{+}$. We also localize the concentration points and, in the case of radial potentials $V$, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.