Dynamics of subcritical threshold solutions for the 4d energy-critical NLS

TL;DR

This paper extends the understanding of the 4d energy-critical nonlinear Schrödinger equation by analyzing the dynamics of solutions near the ground state energy, including non-radial cases, revealing scattering and convergence behaviors.

Contribution

It generalizes previous radial results to non-radial solutions, providing a comprehensive description of solution dynamics at the ground state energy level.

Findings

Radial solutions either scatter or converge to the ground state.

Non-radial solutions exhibit similar dichotomous behavior.

Extension of previous results to broader symmetry classes.

Abstract

We study dynamics of the 4$d$ energy-critical nonlinear Schr\"odinger equation at the ground state energy. Previously, Duyckaerts and Merle [Geom. Funct. Anal. (2009)] proved that any radial solution with kinetic energy less than that of the ground state either scatters in both time directions or coincides (modulo symmetries) with a heteroclinic orbit, which scatters in one time direction and converges to the ground state in the other. We extend this result to the non-radial setting.