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

Abstract

<abstract> In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities <p class="disp_formula">$ 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 ^{[<xref ref-type="bibr" rid="b14">14</xref>]} to the $ L^2 $-critical and $ L^2 $-supercritical nonlinearities. </abstract>