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Local asymptotics for singular solutions to critical Hartree equations

João Henrique Andrade, Tao Feng, Paolo Piccione, Minbo Yang

TL;DR

This work analyzes local behavior near isolated singularities for critical Hartree equations on punctured domains. It leverages a blend of blow-up analysis, integral representations, Kelvin transforms, and an integral moving spheres framework to obtain radial symmetry of blow-up limits and a precise local asymptotic description, namely $u(x)=(1+o(1))u_\infty(|x|)$ as $x\to 0$, where $u_\infty$ solves the blow-up limit. A dual integral formulation and careful handling of double convolution kernels enable sharp upper bounds and a robust asymptotic theory for singular solutions. The results extend the classical symmetry and Liouville-type classifications of the local Yamabe-type problems to the nonlocal Hartree setting, providing new tools for understanding concentration phenomena in nonlocal critical equations.

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.

Local asymptotics for singular solutions to critical Hartree equations

TL;DR

This work analyzes local behavior near isolated singularities for critical Hartree equations on punctured domains. It leverages a blend of blow-up analysis, integral representations, Kelvin transforms, and an integral moving spheres framework to obtain radial symmetry of blow-up limits and a precise local asymptotic description, namely as , where solves the blow-up limit. A dual integral formulation and careful handling of double convolution kernels enable sharp upper bounds and a robust asymptotic theory for singular solutions. The results extend the classical symmetry and Liouville-type classifications of the local Yamabe-type problems to the nonlocal Hartree setting, providing new tools for understanding concentration phenomena in nonlocal critical equations.

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.

Paper Structure

This paper contains 10 sections, 17 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kelvin transform
  4. Symmetry of Delaunay-type solutions
  5. Integral representation
  6. Radial symmetry
  7. Local asymptotic behavior
  8. Integral representation
  9. Upper bound estimate
  10. Asymptotic radial symmetry

Key Result

Theorem A

Let $n\geqslant 3$ and $\alpha \in(0,n)$. If $1<q_1,q_2<\infty$ satisfy $\frac{1}{q_1}+\frac{1}{q_2}+\frac{n-\alpha}{n}=2$, then there exists $H=H(n, q_1, q_2, \alpha)>0$ such that, for all $f_1 \in L^{q_1}(\mathbb{R}^n)$ and $f_2 \in L^{q_2}(\mathbb{R}^n)$, it holds Moreover, when $q_1=q_2=\frac{2n}{n+\alpha}$, the best constant $H(n,\alpha,q_1,q_2)=H_n(\alpha)$ is given by Furthermore, the equ

Theorems & Definitions (35)

  • Theorem A
  • Theorem B
  • Theorem 1.1: Symmetry
  • Theorem 1.2: Asymptotics
  • Conjecture 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 25 more