Instability of standing waves for nonlinear Schrödinger equations with potentials

Abstract

We study the instability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with an attractive power nonlinearity $|u|^{p-1}u$ and a potential $V(x)$ in ${\mathbb{R}}^n$. Here, $\omega>0$ and $\phi_{\omega}(x)$ is a minimal action solution of the stationary problem. Under suitable assumptions on $V(x)$, we show that if $p>1+4/n$, $e^{i \omega t}\phi_{\omega}(x)$ is unstable for sufficiently large $\omega$. For example, our theorem covers a harmonic potential $V(x)=|x|^2$, to which the arguments in the previous papers [2], [14], and [19] are not directly applicable. As another application, we also prove a similar result for a nonlinear Schrödinger equation with a constant magnetic field.