| Abstract: |
| We consider strong instability of the standing wave $e^{i\omega t}\phi_\omega(x)$ for $N$-dimensional nonlinear Schr\odinger equations with $L^2$-supercritical nonlinearity and an attractive inverse power potential,
where $\omega$ is the frequency of the standing wave,
and $\phi_\omega$ is a ground state of the corresponding stationary equation.
Recently, Ohta proved that if $\partial_\lambda^2E(\phi_\omega^\lambda)|_{\lambda=1}\le0$,
then the standing wave for NLS with a harmonic potential is strongly unstable,
where $E$ is the energy,
and $\lambda\mapsto v^\lambda(x):=\lambda^{N/2}v(\lambda x)$ is the scaling, which does not change $L^2$-norm.
In this talk, we prove strong instability under the same assumption as the above-mentioned in inverse power potential case.
Our proof can be applicable to NLS with other potentials such as an attractive Dirac delta potential. |
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