Scattering for the critical 2-D NLS with exponential growth

TL;DR

This paper proves scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth in the radial setting, using a combination of a priori estimates and Sobolev embedding characterizations.

Contribution

It establishes the first radial $H^1$-scattering result for the critical 2-D NLS with exponential nonlinearity, leveraging advanced compactness and embedding techniques.

Findings

Proves $H^1$-scattering in the radial case for the critical 2-D NLS.

Utilizes a priori estimates and Sobolev embedding characterizations.

Highlights the importance of radial symmetry and boundedness away from the origin.

Abstract

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schr\"odinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.