Isolated Singularities for Fractional Hartree Equations

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, [Formula: see text] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental solution term may occur at the singular point. Under an upper bound at the scale of the fundamental solution and a weighted integrability condition on the source, we derive a Riesz decomposition with a nonnegative singular term. Under the same assumptions, if in addition the solution blows up at the puncture, [Formula: see text], and both Kelvin weights are nonnegative, an off-center moving spheres argument yields radial symmetry and strict radial decrease. We also classify positive radial homogeneous singular solutions under the corresponding convergence conditions. If a nonzero positive radial homogeneous scaling limit exists and solves the limiting equation, this classification uniquely determines its coefficient.