Multiplicity, asymptotics and stability of standing waves for nonlinear Schr\"odinger equation with rotation

TL;DR

This paper investigates the existence, multiplicity, asymptotic behavior, and stability of standing waves with prescribed mass for a nonlinear Schrödinger equation with rotation, extending results to the physically relevant mass-supercritical regime.

Contribution

It extends previous results on the nonlinear Schrödinger equation to the mass-supercritical regime, establishing existence, stability, and asymptotic properties of standing waves with prescribed mass.

Findings

Existence of a ground state for small mass c>0

Mass collapse behavior as c approaches 0

Stability of the ground state standing wave

Abstract

In this article, we study the multiplicity, asymptotics and stability of standing waves with prescribed mass $c>0$ for nonlinear Schr\"odinger equation with rotation in the mass-supercritical regime arising in Bose-Einstein condensation. Under suitable restriction on the rotation frequency, by searching critical points of the corresponding energy functional on the mass-sphere, we obtain a local minimizer $u_c$ and a mountain pass solution $\hat{u}_c$. %under suitable assumptions on the related parameters. Furthermore, we show that $u_c$ is a ground state for small mass $c>0$ and describe a mass collapse behavior of the minimizers as $c\to 0$, while $\hat{u}_c$ is an excited state. Finally, we prove that the standing wave associated with $u_c$ is stable. Notice that the pioneering works \cite{aMsC,shYZ} imply that finite time blow-up of solutions to this model occurred in the mass-supercritical setting, therefore, we in the present paper obtain a new stability result. The main contribution of this paper is to extend the main results in \cite{JeSp,gYlW} concerning the same model from mass-subcritical and mass-critical regimes to mass-supercritical regime, where the physically most relevant case is covered.