Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case

TL;DR

This paper proves global well-posedness and scattering for radial solutions to a nonlinear Schrödinger equation with combined defocusing and focusing nonlinearities, using variational and concentration-compactness methods.

Contribution

It establishes the first comprehensive analysis of global behavior for such combined nonlinearities in the radial case, including profile decomposition and threshold characterization.

Findings

Global well-posedness below the threshold

Scattering results for radial solutions in dimensions up to 4

Profile decomposition in $H^1(Bbb R^d)$

Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1(\Bbb R^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1(\Bbb R^d)$ below the threshold for radial data when $d\leq4$.