2025/05/25 by João Henrique Andrade, Tao Feng, Andrade, João Henrique +5 · 1 citation
Computer Science · Mathematics · #35B09 #35B40 #35J30 #35J60 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Ball (mathematics) #FOS: Mathematics #Gravitational singularity #Infinity #Limit (mathematics) #SPHERES #Singular perturbation #Singularity #Spectral Theory in Mathematical Physics #Symmetry (geometry) #Symmetry in biology
paper · pdf · doi:10.48550/arxiv.2505.19021
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \citeMR3694645, 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 \citeMR982351 to the setting of Hartree equations.