On the instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential

Abstract

We study the strong instability of ground state standing waves for the focusing $L^2$-supercritical NLS with inverse-square potential $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+c|x{|}^{-2}u=-|u{|}^{\alpha }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$ where $d\ge 3$, $u:\mathbb{R}\times {\mathbb{R}}^d\to \mathbb{C}$, $c\ne 0$ satisfies $c<\lambda \left(d\right):={\left(\left(d-2\right)/2\right)}^2$ and $4/d<\alpha <4/\left(d-2\right)$. This result extends a recent result of Bensouilah et al. [On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential. J. Math. Phys. 59 (2018), 101505] where the stability and instability of standing waves were shown in the $L^2$-subcritical and $L^2$-critical cases.

Keywords:

• Nonlinear Schrödinger equation
• inverse-square potential
• ground states
• instability

AMS SUBJECT CLASSIFICATIONS:

• 35B35
• 35Q55

Acknowledgments

The author would like to express his deep gratitude to his wife-Uyen Cong for her encouragement and support. The authors would like to thank the reviewers for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the Labex CEMPI [grant number ANR-11-LABX-0007-01].