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We study the strong instability of standing waves $e^{i\omegat}\phi_\omega(x)$ for nonlinear Schr\"{o}dinger equations with an$L^2$-supercritical nonlinearity and an attractive inverse power potential,where $\omega\in\mathbb{R}$ is a frequency, and $\phi_\omega\inH^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation.Recently, for nonlinear Schr\"odinger equations with a harmonic potential, Ohta(2018) proved that if$\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, then thestanding 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 scaling, which does not change the $L^2$-norm. In this paper, we provethe strong instability under the same assumption as the above-mentioned ininverse power potential case. Our proof is applicable to nonlinearSchr\"odinger equations with other potentials such as an attractive Dirac deltapotential.

Strong instability of standing waves for nonlinear Schr\"odinger equations with attractive inverse power potential