Isolated Singularities for Fractional Hartree Equations

Abstract

We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, \[ (-Δ)^s u = \left( \int_{\mathbb{R}^N\setminus\{0\}} \frac{u^p(y)}{|x-y|^μ}\,dy \right)u^q \quad \text{in } \mathbb{R}^N\setminus\{0\}. \] The puncture changes the passage from the differential equation to its integral form: a fractional fundamental-solution term may occur at the singular point. For non-removable blow-up singularities satisfying a fundamental-order upper bound and a weighted source condition, we derive the corresponding Riesz decomposition and retain its nonnegative singular term in an off-center Kelvin moving-spheres argument. In the range determined by two nonnegative Kelvin weights, this yields radial symmetry and strict radial monotonicity. We also construct and classify positive radial homogeneous singular solutions in the corresponding convergence regime. If a nonzero positive radial homogeneous scaling limit is independently known to exist and to solve the limiting equation, then its coefficient is uniquely determined.

33 pages, comments are welcome