Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations

Abstract

We study the strong instability of ground-state standing waves $e^{i\omega t}{\varphi }_{\omega }(x)$ for $N$-dimensional nonlinear Schrödinger equations with focusing double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if ${\partial }_{\lambda }^2{S}_{\omega }({\varphi }_{\omega }^{\lambda }{)|}_{\lambda =1}\le 0$, 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 $L^2$-invariant scaling.