Dynamic of threshold solutions for energy-critical NLS

TL;DR

This paper investigates the behavior of solutions to the energy-critical focusing nonlinear Schrödinger equation at the threshold energy level, providing a classification and dynamical characterization of solutions at this critical point.

Contribution

It offers a detailed classification of solutions at the energy threshold E(W), extending previous work by characterizing the dynamics at this critical energy level.

Findings

Classified solutions at the energy threshold E(W).

Provided a dynamical characterization of the stationary solution W.

Extended understanding of solution behavior at critical energy levels.

Abstract

We consider the radial energy-critical non-linear focusing Schr\"odinger equation in dimension N=3,4,5. An explicit stationnary solution, W, of this equation is known. In a previous work by C. Carlos and F. Merle, the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u)=E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.