Global well-posedness, blow-up and stability of standing waves for supercritical NLS with rotation

TL;DR

This paper investigates the global existence, blow-up, and stability of standing waves in a supercritical nonlinear Schrödinger equation with rotation, providing new conditions and constructions for stable solutions.

Contribution

It establishes conditions for global solutions and blow-up, proves instability of certain standing waves, and constructs stable standing wave solutions in a supercritical setting.

Findings

Conditions for global existence and blow-up are established.

Strong instability of certain standing waves is proven.

Orbitally stable standing waves are constructed via minimization.

Abstract

We consider the focusing mass supercritical nonlinear Schr\"odinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_{\Omega}$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are $L^{2}$-supercritical.