Stability and instability of standing waves for a generalized Choquard equation with potential

Abstract

We are going to study the standing waves for a generalized Choquard equation with potential: $$ -i\partial_t u-\Delta u+V(x)u=(|x|^{-\mu}\ast|u|^p)|u|^{p-2}u, \ \ \hbox{in}\ \ \mathbb{R}\times\mathbb{R}^3, $$ where $V(x)$ is a real function, $0<\mu<3$, $2-\mu/3<p<6-\mu$ and $\ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $\omega$ , the stability and instability of the standing waves $u=e^{i \omega t}\varphi(x)$ are investigated .