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 $$i\partial_t\psi+\Delta\psi+(|x|^{-2}*|\psi|^2)\psi+|\psi|^{p-1}\psi=0$$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 $e^{ iωt}u_ω(x)$ is strongly unstable for the frequency $ω>0.$