Dynamics of radial threshold solutions for generalized energy-critical Hartree equation

TL;DR

This paper classifies all radial threshold solutions for the focusing energy-critical Hartree equation, using spectral theory, modulational analysis, and concentration compactness, with new results on nondegeneracy of nonlocal bubble solutions.

Contribution

It introduces a novel approach to prove nondegeneracy of nonlocal bubble solutions, enabling classification of all threshold solutions for the generalized energy-critical Hartree equation.

Findings

Classified all radial threshold solutions.

Established nondegeneracy of positive bubble solutions with nonlocal structure.

Extended classification techniques from NLS and NLW to Hartree equations.

Abstract

In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution}, later by D. Li and X. Zhang in \cite{LiZh:NLS, LiZh:NLW} in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in $\dot H^1(\R^N)$ (i.e. the spectral assumption in \cite{MiaoWX:dynamic gHartree}) by the nondegeneracy result of positive bubble solution in $L^{\infty}(\R^N)$ in \cite{LLTX:Nondegeneracy} and the Moser iteration method in \cite{DiMeVald:book}, which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions.