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*} -Δu (x)= \left(\frac{1}{|x|^α}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus{0} , \end{equation*} where $α&gt;0$, $$\displaystyle \frac{1}{|x|^{α}*e^{u\triangleq\int_{B_{1}}} \setminus {0}}\frac{e<sup>u(y)}{|x-y|<sup>α}dy$$, and the punctured ball $B_{1}\setminus{0}\subset \mathbb{R}<sup>2$. 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 $n \geq 3$.