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

We are going to study the standing waves for a generalized Choquard equation with potential: $$ -ipartial_t u-Delta u+V(x)u=(|x|^{-mu}ast|u|^p)|u|^{p-2}u, hbox{in} mathbb{R}timesmathbb{R}^3, $$ 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 .