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

TL;DR

This paper analyzes the asymptotic behavior of solutions to a planar Hartree equation with isolated singularities, providing detailed descriptions near the singularity and extending results to more general cases and higher dimensions.

Contribution

It establishes a representation formula for singular solutions and extends asymptotic behavior results to equations with variable coefficients and higher dimensions.

Findings

Asymptotic behavior near the singularity characterized

Representation formula for singular solutions derived

Results extended to equations with non-negative coefficients and higher dimensions

Abstract

In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -\Delta u (x)= \left(\frac{1}{|x|^\alpha}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where $\alpha>0$, $\displaystyle \frac{1}{|x|^\alpha}*e^u\triangleq\int_{B_{1} \setminus \{0\}}\frac{e^u(y)}{|x-y|^\alpha}dy$, 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 $n \geq 3$.