Existence and instability of standing waves with prescribed norm for a class of Schr\"odinger-Poisson equations

TL;DR

This paper investigates the existence and instability of standing waves with fixed $L^2$-norm for a class of Schr"odinger-Poisson equations in three dimensions, revealing conditions for existence, instability, and their relation to classical nonlinear Schr"odinger equations.

Contribution

It introduces a mountain pass approach to find critical points of the energy functional for Schr"odinger-Poisson equations with prescribed norm, and analyzes their stability and existence conditions.

Findings

Critical points exist for small charge c>0.

Standing waves are strongly unstable at the mountain pass level.

Threshold energy level determines global existence of solutions.

Abstract

In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schr\"odinger-Poisson-Slater equations in $\R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schr\"odinger-Poisson type equation \label{evolution1} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 % \text{in} \R^{3}, when $p \in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| \triangledown u|_{L^{2}(\mathbb{R}^3)}^2+1/4\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-\frac{1}{p}\int_{\mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= \{u \in H^1(\mathbb{R}^3) :|u|_{L^2(\R^3)}^2=c, c>0}.$$ For the values $p \in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0\in H^1(\R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $\gamma(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schr\"odinger-Poisson-Slater equation and the classical nonlinear Schr\"odinger equation.