@article{Shi2022, 
author = {Chao Shi},
title = {Existence of stable standing waves for the nonlinear Schrödinger equation with mixed power-type and Choquard-type nonlinearities},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {3},
pages = {3802-3825},
keywords = {nonlinear Schrödinger equation, standing waves, orbital stability},
url = {https://www.sciopen.com/article/10.3934/math.2022211},
doi = {10.3934/math.2022211},
abstract = {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      ∂    t    ψ  +  Δ  ψ  +  λ      |    ψ            |        q    ψ  +      1                  |            x                        |                α                  (                  ∫                                            R                        N                                                            |                    ψ                                    |                        p                                                |                    x          −          y                                    |                        μ                                |                    y                                    |                        α                              d      y        )        |    ψ            |              p      −      2        ψ  =  0  ,where    N  ≥  3,    0  <  μ  <  N,    λ  >  0,    α  ≥  0,    2  α  +  μ  ≤      N  ,    0  <  q  <      4    N   and    2  −            2      α      +      μ        N    <  p  <            2      N      −      2      α      −      μ              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.}
}