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

TL;DR

This paper establishes the existence and orbital stability of standing waves with prescribed L^2-norm for a class of nonlinear Schrödinger-Poisson equations, extending previous results and analyzing compactness of minimizing sequences.

Contribution

It proves new stability results for Schrödinger-Poisson equations with specific nonlinearities and provides a novel analysis of the compactness of minimizing sequences.

Findings

Existence of orbitally stable standing waves for certain p-values.

Stability results for large L^2-norm in specific cases.

Application to Schrödinger equations with biharmonic operators.

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.