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

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 $n \geq 3$.