Local asymptotics for singular solutions to critical Hartree equations

TL;DR

This paper analyzes the local behavior and symmetry of singular solutions to critical Hartree equations on punctured domains, extending classical results with new asymptotic and integral techniques.

Contribution

It introduces an asymptotic integral moving spheres method and establishes radial symmetry for solutions, extending classical theorems to Hartree equations.

Findings

Solutions behave like blow-up profiles near singularities

Established radial symmetry of positive singular solutions

Developed an asymptotic integral moving spheres technique

Abstract

We investigate the qualitative properties of a critical Hartree equation defined on punctured domains. Our study has two main objectives: analyzing the asymptotic behavior near isolated singularities and establishing radial symmetry of positive singular solutions. First, employing asymptotic analysis, we characterize the local behavior of solutions near the singularity. Specifically, we show that, within a punctured ball, solutions behave like the blow-up limit profile. This is achieved through classification results for entire bubble solutions, a standard blow-up procedure, and a removable singularity theorem, yielding sharp upper and lower bounds near the origin. To run the blow-up analysis, we develop an asymptotic integral version of the moving spheres technique, a technique of independent interest. Second, we establish the radial symmetry of blow-up limit solutions using an integral moving spheres method. On the technical level, we apply the integral dual method from Jin, Li, Xiong \cite{MR3694645, arxiv:1901.01678} to provide local asymptotic estimates within the punctured ball and to prove that solutions in the entire punctured space are radially symmetric with respect to the origin. Our results extend seminal theorems of Caffarelli, Gidas, and Spruck \cite{MR982351} to the setting of Hartree equations.