Scattering threshold for the focusing coupled Schrödinger system revisited

This note investigates the coupled Schrödinger system iu˙j+Δuj=-∑k=1majk|uk|p|uj|p-2uj.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\dot{u}_j +\Delta u_j= -\left( \sum \nolimits _{k=1}^{m}a_{jk}|u_k|^p\right) |u_j|^{p-2}u_j. \end{aligned}$$\end{document}Indeed, beyond the mass-energy threshold given in Saanouni (Appl Anal, 2020. https://doi.org/10.1080/00036811.2020.1808201), a scattering versus finite time blow-up dichotomy is obtained in the mass super-critical and energy sub-critical regime. Moreover, one extends the previous work [18] to the non-radial case.