Scattering for the Non-Radial Focusing Inhomogeneous Nonlinear Schrödinger–Choquard Equation

Abstract

In this paper, we study the long-time behavior of global solutions to the Schrödinger–Choquard equation

$${\rm{i}}{\partial _t}u + \Delta u = - ( {{I_\alpha } * {{\vert \cdot \vert}^b}{{\vert u \vert}^p}} ){\vert \cdot \vert^b}{\vert u \vert^{p - 2}}u.$$

Inspired by Murphy who gave a simple proof of scattering for the non-radial INLS, we find that the inhomogeneous term ∣x∣^b can replace the radial Sobolev embedding theorem, which allows us to prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.