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

Abstract

We study positive weak singular solutions of the subcritical fractional Hartree equation \[ (-Δ)^s u = \left( \int_{\mathbb{R}^N\setminusΓ} \frac{u^p(y)}{|x-y|^μ}d y \right)u^q \quad\text{in }\mathbb{R}^N\setminusΓ, \] where \(0 2s\), \(0<μ<N\), and \(Γ\subset\{x_1=0\}\) is compact with zero \((s,2)\)-capacity. For solutions with a nonremovable singular point on $Γ$, we prove hyperplane symmetry and strict transverse monotonicity under global Lebesgue integrability conditions. For $N\ge2$, 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.