Threshold scattering for the 2d radial cubic-quintic NLS

TL;DR

This paper proves scattering for the 2D radial cubic-quintic nonlinear Schrödinger equation at the critical threshold without requiring weighted Sobolev space assumptions, extending previous results to a broader class of initial data.

Contribution

It removes the weighted Sobolev space assumption and establishes scattering at the threshold for radial data in H^1 for the 2D cubic-quintic NLS.

Findings

Scattering established at the threshold for radial H^1 data.

Removed weighted Sobolev space assumption from previous results.

Extended scattering results to a broader class of initial data.

Abstract

We consider the cubic-quintic nonlinear Schr\"odinger equation in two space dimensions. For this model, X. Cheng established scattering for $H^1$ data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R. Carles and C. Sparber utilized the pseudoconformal energy estimate to obtain scattering at the sharp threshold for data belonging to a weighted Sobolev space. In this work, we remove the weighted assumption and establish scattering at the threshold for radial data in $H^1$.