Strong Instability of Standing Waves for a Type of Hartree Equations

Abstract

In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities

$i \partial_{t} \psi+\Delta \psi+\left(|x|^{-2} *|\psi|^{2}\right) \psi+|\psi|^{p-1} \psi=0$

with 1<p<5. Using standard variational arguments, the existence of ground state solutions is obtained. And then we prove that when p≥3, the standing wave solution $e^{i \omega t} u_{\omega}(x)$ is strongly unstable for the frequency ω>0.

Keywords

Hartree equation / standing wave / variational arguments / strong instability / blowup