Nonlinear bound states with prescribed angular momentum in the mass supercritical regime

TL;DR

This paper investigates the existence, stability, and regularity of nonlinear bound states with angular momentum in supercritical regimes, revealing new solutions relevant to rotating Bose-Einstein condensates.

Contribution

It introduces new non-radially symmetric solutions with prescribed angular momentum in the supercritical regime, extending previous subcritical results.

Findings

Existence of two non-radially symmetric solutions, one stable and one unstable.

Conditions for regularity and stability of local minimizers.

Identification of mountain pass solutions as strongly unstable.

Abstract

In this paper, we consider the existence, orbital stability/instability and regularity of bound state solutions to nonlinear Schr\"odinger equations with super-quadratic confinement in two and three spatial dimensions for the mass supercritical case. Such solutions, which are given by time-dependent rotations of a non-radially symmetric spatial profile, correspond to critical points of the underlying energy function restricted on the double constraints consisting of the mass and the angular momentum. The study exhibits new pictures for rotating Bose-Einstein condensates within the framework of Gross-Pitaevskii theory. It is proved that there exist two non-radially symmetric solutions, one of which is local minimizer and the other is mountain pass type critical point of the underlying energy function restricted on the constraints. Moreover, we derive conditions that guarantee that local minimizers are regular, the set of those is orbitally stable and mountain pass type solutions are strongly unstable. The results extend and complement the recent ones in \cite{NSS}, where the consideration is undertaken in the mass subcritical case.