Existence of stable standing waves for the nonlinear Schrödinger equation with mixed power-type and Choquard-type nonlinearities

The aim of this paper is to study the existence of stable standing waves for the following nonlinear Schrödinger type equation with mixed power-type and Choquard-type nonlinearities

$$i{\mathrm{\partial }}_t\psi +\mathrm{\Delta }\psi +\lambda |\psi {|}^q\psi +\frac{1}{|x{|}^{\alpha }}\left({\int }_{{\mathbb{R}}^N}\frac{|\psi {|}^p}{|x-y{|}^{\mu }|y{|}^{\alpha }}dy\right)|\psi {|}^{p-2}\psi =0,$$

where $N\ge 3$, $0<\mu <N$, $\lambda >0$, $\alpha \ge 0$, $2\alpha +\mu \le N$, $0<q<\frac{4}{N}$ and $2-\frac{2\alpha +\mu }{N}<p<\frac{2N-2\alpha -\mu }{N-2}$. We firstly obtain the best constant of a generalized Gagliardo-Nirenberg inequality, and then we prove the existence and orbital stability of standing waves in the $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical cases by the concentration compactness principle in a systematic way.