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 \begin{document}$ i\partial_t \psi+\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, $\end{document} where $ N\geq3 $, $ 0 < \mu < N $, $ \lambda > 0 $, $ \alpha\geq0 $, $ 2\alpha+\mu\leq{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.