Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential

Abstract

We study the strong instability of standing waves $e^{i\omega t}{\varphi }_{\omega }(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega \in \mathbb{R}$ is a frequency, and ${\varphi }_{\omega }\in H^1({\mathbb{R}}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if ${\partial }_{\lambda }^2{S}_{\omega }({\varphi }_{\omega }^{\lambda }){|}_{\lambda =1}\le 0$, then the standing wave is strongly unstable, where $S_{\omega }$ is the action, and ${\varphi }_{\omega }^{\lambda }(x):={\lambda }^{N/2}{\varphi }_{\omega }(\lambda x)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.

17 pages