Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity

We study positive weak singular solutions of the subcritical fractional Hartree equation $$(-\mathrm{\Delta }{)}^{s}u=\left({\int }_{{\mathbb{R}}^N\setminus \mathrm{\Gamma }}\frac{u^p(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}dy\right)u^q\text{in }{\mathbb{R}}^N\setminus \mathrm{\Gamma },$$ where $0<s<1$, $N>2s$, $0<\mu <N$, and $\mathrm{\Gamma }\subset \{x_1=0\}$ is compact with zero $(s,2)$-capacity. For solutions with a nonremovable singular point on $\mathrm{\Gamma }$, we prove hyperplane symmetry and strict transverse monotonicity under global Lebesgue integrability conditions. For $N\ge 2$, we also prove these conclusions under upper bounds on $p$ and $q$, using only the integrability required for weak solutions. We also establish local regularity and construct solutions with logarithmic singularities along a line segment.