On instability of radial standing waves for the nonlinear Schr\"odinger equation with inverse-square potential

TL;DR

This paper proves the strong instability of radial ground state standing waves for a focusing nonlinear Schrödinger equation with inverse-square potential in the supercritical case, extending previous stability results to this regime.

Contribution

It establishes the strong instability of radial ground state standing waves for the supercritical nonlinear Schrödinger equation with inverse-square potential, a case not covered in prior work.

Findings

Radial ground states are strongly unstable in the supercritical regime.

Extension of instability results to inverse-square potential case.

Builds on and generalizes previous stability analyses.

Abstract

We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schr\"odinger equation with inverse-square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^{\alpha} u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $u: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $c\ne 0$ satisfies $c<\lambda(d):=\left(\frac{d-2}{2}\right)^2$ and $\frac{4}{d} <\alpha<\frac{4}{d-2}$. This result extends a recent result of Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for the nonlinear Schr\"odinger equation with inverse-square potential}, \url{arXiv:1805.01245}] where the stability and instability of standing waves were shown in the $L^2$-subcritical and $L^2$-critical cases.