Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in $H^1\left({\mathbb{R}}^d\right)$ below the threshold for nonradial data when $1\le d\le 4$.