The uniqueness and concentration behavior of solutions for a nonlinear fractional Schr\"odinger system

TL;DR

This paper investigates the uniqueness and concentration behavior of solutions to a coupled nonlinear fractional Schrödinger system, revealing how solutions blow up and concentrate at specific points as interaction strengths reach critical levels.

Contribution

It proves the uniqueness of solutions using the implicit function theorem and analyzes their blow-up and concentration behavior under trapping potentials.

Findings

Solutions are unique under certain conditions.

Solutions blow up and concentrate at the flattest minimum point of potentials.

An optimal blow-up rate for solutions is established.

Abstract

The paper is concerned with a nonlinear system of two coupled fractional Schr\"odinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained minimization problem, the uniqueness of solutions to this system is proved via the implicit function theorem. Under a certain type of trapping potential, by establishing some delicate energy estimates, we present a detailed analysis on the concentration behavior of the solutions as the total strength of intraspecies and interspecies interactions tends to a critical value, where each component of the solutions blows up and concentrates at a flattest common minimum point of the associated trapping potentials. An optimal blow-up rate of solutions to the system is also given.