Stability of standing waves for nonlinear Schrödinger equations with potentials

Abstract

We study the stability 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, $\omega\in \mathbb R$ 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 $p <1+4/n$ and sufficiently large $\omega$, or for $1 <p <2^*-1$ and $\omega$ close to $-\lambda_1$, where $\lambda_1$ is the lowest eigenvalue of the operator $-\Delta+V(x)$. We give an improvement of previous results such as Rose and Weinstein [19], or Grillakis, Shatah and Strauss [11], for unbounded potentials $V(x)$ which cannot be treated by the standard perturbation argument.