Existence of stable standing waves for the nonlinear Schro spacing diaeresis dinger 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 Schro spacing diaeresis dinger type equation with mixed power-type and Choquard-type nonlinearities i partial derivative(t)psi + Delta psi + lambda vertical bar psi vertical bar(q)psi + 1/vertical bar x vertical bar(alpha) (integral(RN) vertical bar psi vertical bar(p)/vertical bar x - y vertical bar mu vertical bar y vertical bar(alpha)dy)vertical bar psi vertical bar(p-2)psi = 0, where N >= 3, 0 < mu < N, lambda>0, alpha >= 0, 2 alpha + mu <= N, 0 < q < 4/N and 2 - 2 alpha+mu/N < p < 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.