Scattering threshold for radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in $d\geq 5$

Abstract

We extend the scattering result for the radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in $d\leq 4$ given by Cheng et al. to the case $d\geq 5$. The main ingredient is a suitable long time perturbation theory which is applicable for $d\geq 5$. The paper will therefore give a full characterization on the scattering threshold for the radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in all dimensions $d\geq 3$.