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: $$-i{\mathrm{\partial }}_{t}u-\mathrm{\Delta }u+V(x)u=(\mathrm{\mid }x{\mathrm{\mid }}^{-\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,\text{in}\mathbb{R}\times {\mathbb{R}}^3,$$ where $V(x)$ is a real function, $0<\mu <3$, $2-\mu \mathrm{/}3<p<6-\mu$ and $\ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $\omega$ , the stability and instability of the standing waves $u=e^{i\omega t}\phi (x)$ are investigated .