Strong Instability of Standing Waves for a Type of Hartree Equations (Journal of Partial Differential Equations, 2025)
In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities $$i\partial_tψ+Δψ+(|x|^{-2}*|ψ|^2)ψ+|ψ|^{p-1}ψ=0$$with $1 < p < 5.$ Using standard variational arguments, the existence of ground statesolutions 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 $ω>0.$
Keywords: Hartree equation; standing wave; variational arguments; strong instability; blowup.