Scattering for non-radial 3D NLS with combined nonlinearities: the interaction Morawetz approach

TL;DR

This paper introduces a novel proof technique for scattering in non-radial 3D NLS with combined nonlinearities, utilizing interaction Morawetz estimates and new bounds to handle energy-critical cases.

Contribution

It provides a new proof of scattering below the ground state energy for intercritical NLS with combined focusing and defocusing nonlinearities, including a rate of blow-up for symmetric solutions.

Findings

Established scattering below the ground state energy level.

Developed a new bound for the Pohozaev functional of localized functions.

Determined the blow-up rate for symmetric solutions.

Abstract

We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schr\"odinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order nonlinearity with a defocusing perturbation. Our strategy combines interaction Morawetz estimates \`a la Dodson-Murphy and a new crucial bound for the Pohozaev functional of localized functions, which is essential to overcome the lack of a monotonicity condition. Furthermore, we give the rate of blow-up for symmetric solutions.