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Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities

Tao Feng, Minbo Yang, Xianmei Zhou

TL;DR

The paper addresses the asymptotic behavior near isolated singularities for the planar Hartree equation with a nonlocal exponential nonlinearity, and extends the results to general nonnegative coefficients and to higher-order Hartree equations in $n\ge 3$. It develops a representation formula that decomposes singular solutions into a logarithmic part, a regular polyharmonic/harmonic component, and a logarithmic potential $v$ driven by the nonlocal term. Under finite total curvature and small-$r$ decay assumptions, the authors prove that the singularity is captured by a logarithmic term with coefficient $b$ and that the remainder is Hölder continuous, with precise thresholds on $b$ depending on $\alpha$ and the dimension. The results unify the treatment of even and odd dimensions and extend to a general coefficient $K(x)$, offering a PDE-based framework for nonlocal Hartree-type problems and a rigorous classification of local singularities.

Abstract

In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -Δu (x)= \left(\frac{1}{|x|^α}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where $α>0$, $\displaystyle \frac{1}{|x|^α}*e^u\triangleq\int_{B_{1} \setminus \{0\}}\frac{e^u(y)}{|x-y|^α}dy$, and the punctured ball $B_{1}\setminus\{0\}\subset \mathbb{R}^2$. Under the finite total curvature condition, by establishing a representation formula for singular solutions, we obtain the asymptotic behavior of the solutions near the origin. We also extend this asymptotic behavior results to the case with a general non-negative coefficient $K(x)$, and to the higher-order Hartree-type equations in any dimension $n \geq 3$.

Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities

TL;DR

The paper addresses the asymptotic behavior near isolated singularities for the planar Hartree equation with a nonlocal exponential nonlinearity, and extends the results to general nonnegative coefficients and to higher-order Hartree equations in . It develops a representation formula that decomposes singular solutions into a logarithmic part, a regular polyharmonic/harmonic component, and a logarithmic potential driven by the nonlocal term. Under finite total curvature and small- decay assumptions, the authors prove that the singularity is captured by a logarithmic term with coefficient and that the remainder is Hölder continuous, with precise thresholds on depending on and the dimension. The results unify the treatment of even and odd dimensions and extend to a general coefficient , offering a PDE-based framework for nonlocal Hartree-type problems and a rigorous classification of local singularities.

Abstract

In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -Δu (x)= \left(\frac{1}{|x|^α}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where , , and the punctured ball . Under the finite total curvature condition, by establishing a representation formula for singular solutions, we obtain the asymptotic behavior of the solutions near the origin. We also extend this asymptotic behavior results to the case with a general non-negative coefficient , and to the higher-order Hartree-type equations in any dimension .
Paper Structure (6 sections, 8 theorems, 107 equations)

This paper contains 6 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction and main results
  2. Asymptotic behavior at isolated singularities in even dimension case
  3. Representation formula
  4. Two-dimensional case
  5. Even-dimensional case ($n \geq 4$)
  6. Asymptotic behavior at isolated singularities in odd dimension case

Key Result

Theorem 1.1

Suppose that $u\in C^{2}\left(B_{1}\setminus\{0\}\right)$ is a solution of second-order Hartree in d2 and $\alpha>0$. If $u$ satisfies second-order assumption, then there exists a constant $b>-2$ such that where $h \in C^{\infty}(B_{1})$ is a solution of $-\Delta h =0$ in $B_{1}$ and $v$ is defined by Moreover, if $\alpha \in \left(0,2\right)$ and $b > \frac{\alpha-4}{2}$, then there exists a H$

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5: Asymptotic behavior
  • Remark 1.6: Valid range of $b$
  • Remark 1.7: General coefficient $K(x)$
  • Proposition 2.1: Representation Formula
  • proof
  • Lemma 2.2
  • ...and 11 more