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

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.