Symmetry and monotonicity of singular solutions for the Hartree equation

In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-\mathrm{\Delta }u\text{\hspace{0.17em}⁣}=\left({\int }_{{\mathbb{R}}^N\setminus \mathrm{\Gamma }}\frac{F(u(y))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}dy\right)f(u(x)),x\in {\mathbb{R}}^N\setminus \mathrm{\Gamma },$$ where $F$ is the primitive of $f$ and $\mathrm{\Gamma }$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties.