Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials

Abstract

We study the stability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with critical power nonlinearity $|u|^{4/n}u$ and a potential $V(x)$ in $\mathbb R^n$. Here, $\omega\in \mathbb R$ and $\phi_{\omega}(x)$ is a ground state of the stationary problem. Under suitable assumptions on $V(x)$, we show that $e^{i \omega t}\phi_{\omega}(x)$ is stable for sufficiently large $\omega$. This result gives a different phenomenon from the case $V(x)\equiv 0$ where the strong instability was proved by M.I. Weinstein [25].