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 = ( ∫ℝN∖{0} up(y)/(|x−y|μ) dy )uq in ℝN∖{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. 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, p,q≥1, 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.
Versions
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BibTeX
@misc{gu2026isolated,
title = {{Isolated Singularities for Fractional Hartree Equations}},
author = {Guangze Gu and Aleks Jevnikar and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.10990},
eprint = {2606.10990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10990},
}
RIS
TY - UNPB
TI - Isolated Singularities for Fractional Hartree Equations
AU - Guangze Gu
AU - Aleks Jevnikar
AU - Zhipeng Yang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2606.10990
UR - https://arxiv.org/abs/2606.10990
AB - We study isolated singularities of positive solutions to a fractional Hartree equation with Riesz interaction, (−Δ)^s u = ( ∫_ℝ^N∖{0} u^p(y)/(|x−y|^μ) dy )u^q in ℝ^N∖{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. 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, p,q≥1, 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.
ER -