Stable standing waves for a class of nonlinear Schroedinger-Poisson equations

Abstract

We prove the existence of orbitally stable standing waves with prescribed \(L^2\)-norm for the following Schr\"odinger-Poisson type equation \label{intro} %{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when \(p\in \{8/3\}\cup (3,10/3)\). In the case \(3<p<10/3\) we prove the existence and stability only for sufficiently large \(L^2\)-norm. In case \(p=8/3\) our approach recovers the result of Sanchez and Soler \cite{SS} %concerning the existence and stability for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In a final section a further application to the Schr\"odinger equation involving the biharmonic operator is given.