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

In this paper, we investigate the existence of stable standing waves for the nonlinear Schr\"{o}dinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities \[ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi=0,~~(t,x)\in [0,T^\star)\times \mathbb{R}^N. \] By using concentration compactness principle, when one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper \cite {13} to the $L^2$-critical and $L^2$-supercritical nonlinearities.