Strong instability of standing waves with negative energy for double power nonlinear Schr\"odinger equations

TL;DR

This paper proves the strong instability of certain standing waves with negative energy in double power nonlinear Schrödinger equations, extending previous results to include cases with non-positive second derivatives of the action functional.

Contribution

It improves prior instability results by establishing strong instability for standing waves with non-positive second derivatives of the action, covering more cases with negative energy.

Findings

Standing waves with $rac{ ext{d}^2}{ ext{d}\lambda^2}S_ ext{ω}( ext{φ}_ ext{ω}^ ext{λ})|_{ ext{λ}=1} extless=0 are strongly unstable.

The result applies to ground-state solutions with negative energy in double power nonlinear Schrödinger equations.

Extension of previous instability results to include cases with non-positive second derivatives of the action.

Abstract

We study the strong instability of ground-state standing waves $e^{i\omega t}\phi_\omega(x)$ for $N$-dimensional nonlinear Schr\"odinger equations with 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 $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the $L^2$-invariant scaling.