An example of stable excited state on nonlinear Schr\"odinger equation with nonlocal nonlinearity

In this article, we consider nonlinear Schr"odinger equation with nonlocal nonlinearity which is a generalized model of the Schr"odinger-Poisson system (Schr"odinger-Newton equations) in low dimensions. We first prove the global well-posedness in wider space than in previous result and show the stability of standing waves including excites states. It turns out that an example of stable excite states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.