On symmetry and monotonicity of singular solutions to critical fractional Hartree equations
Abstract
This paper continues the study initiated in \cite{GuJevnikarYang2026b} of fractional Hartree equations with zero capacity singular sets. We are concerned here with the critical case, i.e. (−Δ)s u=(∫ℝN∖Γ u2μ,s*(y)/(|x−y|μ) d y)u2μ,s*−1 in ℝN∖Γ, where N≥2, s∈(0,1), μ∈(0,N) and 2μ,s* is the critical exponent associated to this class of problems. We prove symmetry and monotonicity with respect to the singular set Γ. The Kelvin transforms centered at regular points provide sufficient decay at infinity for both the transformed solutions and their associated Hartree coefficients, without imposing global integrability assumptions. Combined with a moving plane argument and a comparison estimate in critical Lebesgue spaces, this allows the moving plane method to cover the full interaction range.
Related
- Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity
- Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
- Asymptotic analysis and uniqueness of blowup solutions of non-quantized singular mean field equations
- A sufficient condition for the existence of solutions of singular linear-quadratic vector equations
- A Priori Estimates for Singular Fractional Stochastic Burgers Equations
BibTeX
@misc{gu2026symmetry,
title = {{On symmetry and monotonicity of singular solutions to critical fractional Hartree equations}},
author = {Guangze Gu and Aleks Jevnikar and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05525},
eprint = {2611.05525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05525},
}
RIS
TY - UNPB
TI - On symmetry and monotonicity of singular solutions to critical fractional Hartree equations
AU - Guangze Gu
AU - Aleks Jevnikar
AU - Zhipeng Yang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05525
UR - https://arxiv.org/abs/2611.05525
AB - This paper continues the study initiated in \cite{GuJevnikarYang2026b} of fractional Hartree equations with zero capacity singular sets. We are concerned here with the critical case, i.e. (−Δ)^s u=(∫_ℝ^N∖Γ u^2_μ,s^*(y)/(|x−y|^μ) d y)u^2_μ,s^*−1 in ℝ^N∖Γ, where N≥2, s∈(0,1), μ∈(0,N) and 2_μ,s^* is the critical exponent associated to this class of problems. We prove symmetry and monotonicity with respect to the singular set Γ. The Kelvin transforms centered at regular points provide sufficient decay at infinity for both the transformed solutions and their associated Hartree coefficients, without imposing global integrability assumptions. Combined with a moving plane argument and a comparison estimate in critical Lebesgue spaces, this allows the moving plane method to cover the full interaction range.
ER -