Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-\Delta)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^\mu}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fundamental solution term may appear at the singular point. Under weighted assumptions on the Hartree source, we derive a Riesz decomposition containing this singular term and incorporate it into a Kelvin moving-spheres argument to prove radial symmetry and monotonicity. The proof relies on a narrow-region principle based on a Hartree defect estimate, which controls the contribution of the reflected negative set to the Riesz interaction. We also construct explicit homogeneous singular solutions in a suitable regime and identify radial homogeneous blow-up profiles associated with the natural scaling.