Energy scattering for a 3D Hartree equation with inverse square potential

Abstract

This work studies the focusing inhomogeneous nonlinear equation of Hartree type

Here, the linear Schrödinger operator reads ${\mathcal{K}}_{\lambda }:=-\Delta +\frac{\lambda }{|x{|}^2}$ for some $\lambda >-\frac{1}{4}$. The Riesz potential is ${\mathcal{J}}_{\alpha }(x)=C_{\alpha }|x{|}^{-(3-\alpha )}$ for certain $0<\alpha <3$. The singular decaying term $|x{|}^{-\tau }$ for some $\tau >0$ gives an inhomogeneous non-linearity. One considers the inter-critical regime, namely, $1+\frac{2(1-\tau )+\alpha }{3}<p<1+2(1-\tau )+\alpha$. Moreover, one assumes that $p\ge 2$ in order to avoid a singular term $|u{|}^{p-2}$. Furthermore, one restricts $\lambda >0$ because there is no dispersive estimate $L^1\to L^{\infty }$ for $\lambda <0$. Contrarily to the homogeneous case $\tau =0$, for $\lambda >0$, there is a ground state which minimizes the associated Gagliardo–Nirenberg-type estimate. The purpose is to investigate the energy scattering of global solutions under the ground state threshold. One uses the method of Dodson–Murphy based on Tao’s scattering criteria and Morawetz estimates. The decay of the inhomogeneous term $|x{|}^{-\tau }$ avoids any spherically symmetric assumption.