Dynamics for the focusing, energy-critical nonlinear Hartree equation

TL;DR

This paper investigates the dynamics of radial solutions to the focusing energy-critical nonlinear Hartree equation at the threshold energy level, establishing regularity and uniqueness of solutions to a related nonlocal elliptic equation.

Contribution

It introduces a novel approach using the moving plane method to prove regularity and uniqueness of solutions to a nonlocal elliptic equation, advancing understanding of threshold dynamics.

Findings

Proves regularity and uniqueness of solutions to a nonlocal elliptic equation.

Analyzes the spectral properties of the linearized operator at threshold energy.

Provides insights into the behavior of solutions at the energy-critical level.

Abstract

In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when $E(u_0)<E(W)$, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy. Our arguments closely follow those in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution, DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in $L^{\frac{2d}{d-2}}(\R^d)$ is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution.