Multiplicity, asymptotics and stability of standing waves for nonlinear Schrödinger equation with rotation

Abstract

In this article, we study the multiplicity, asymptotics and stability of standing waves with prescribed mass c>0 for nonlinear Schrödinger 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 uc and a mountain pass solution uˆc. Furthermore, we show that uc is a ground state for small mass c>0 and describe a mass collapse behavior of the minimizers as c→0, while uˆc is an excited state. Finally, we prove that the standing wave associated with uc is stable. Notice that the pioneering works [2,8] 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 [4,17] concerning the same model from mass-subcritical and mass-critical regimes to mass-supercritical regime, where the physically most relevant case is covered.