Blow-up rate and local uniqueness for fractional Schr\"odinger equations with nearly critical growth

TL;DR

This paper investigates the blow-up behavior and concentration phenomena of ground states for a fractional Schrödinger equation with nearly critical growth, providing precise blow-up rates and local uniqueness results.

Contribution

It establishes the exact blow-up rate of ground states as the parameter approaches zero and proves local uniqueness of concentrating solutions for radial potentials.

Findings

Ground states blow up at a rate proportional to psilon^{-rac{N-2s}{4s}}.

Concentration points are localized precisely.

Local uniqueness of concentrating ground states for radial potentials is proved.

Abstract

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schr\"odinger equation $ (-\Delta)^s u+V(x)u= u^{2_s^*-1-\varepsilon} \ \ \text{in}\ \ \mathbb{R}^N, $ where $\varepsilon>0$, $s\in (0,1)$, $2^*_s:=\frac{2N}{N-2s}$, $N>4s$. We show that the ground state $u_{\varepsilon}$ blows up and precisely with the following rate $\|u_{\varepsilon}\|_{L^\infty(\mathbb{R}^N)}\sim \varepsilon^{-\frac{N-2s}{4s}}$, as $\epsilon\rightarrow 0^+$. 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.