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Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity
Abstract
We study positive weak singular solutions of the subcritical fractional Hartree equation (−Δ)s u = ( ∫ℝN∖Γ up(y)/(|x−y|μ)d y )uq in ℝN∖Γ, where 0<s<1, N>2s, 0<μ<N, and Γ⊂{x1=0} is compact with zero (s,2)-capacity. For solutions with a nonremovable singular point on Γ, we prove hyperplane symmetry and strict transverse monotonicity under global Lebesgue integrability conditions. For N≥2, we also prove these conclusions under upper bounds on p and q, using only the integrability required for weak solutions. We also establish local regularity and construct solutions with logarithmic singularities along a line segment.
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BibTeX
@misc{gu2026symmetry,
title = {{Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity}},
author = {Guangze Gu and Aleks Jevnikar and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24684},
eprint = {2610.24684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24684},
}
RIS
TY - UNPB
TI - Symmetry and Monotonicity for Subcritical Fractional Hartree Equations with Singular Sets of Zero Capacity
AU - Guangze Gu
AU - Aleks Jevnikar
AU - Zhipeng Yang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24684
UR - https://arxiv.org/abs/2610.24684
AB - We study positive weak singular solutions of the subcritical fractional Hartree equation (−Δ)^s u = ( ∫_ℝ^N∖Γ u^p(y)/(|x−y|^μ)d y )u^q in ℝ^N∖Γ, where 0<s<1, N>2s, 0<μ<N, and Γ⊂{x_1=0} is compact with zero (s,2)-capacity. For solutions with a nonremovable singular point on Γ, we prove hyperplane symmetry and strict transverse monotonicity under global Lebesgue integrability conditions. For N≥2, we also prove these conclusions under upper bounds on p and q, using only the integrability required for weak solutions. We also establish local regularity and construct solutions with logarithmic singularities along a line segment.
ER -