Comparison principles and symmetry for subquadratic fractional p-Laplacian equations
Abstract
This paper establishes a new comparison principle framework for the subquadratic fractional p-Laplacian, i.e.~1 < p < 2 under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for (−Δ)ps u = f(u) in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range s ∈ (0, 1/2) and 1/(1−s) < p < 2. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even C1,1 regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to (−Δ)ps u = f(u) under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.
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BibTeX
@misc{ye2026comparison,
title = {{Comparison principles and symmetry for subquadratic fractional $p$-Laplacian equations}},
author = {Dong Ye and Weimin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02626},
eprint = {2609.02626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02626},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Comparison principles and symmetry for subquadratic fractional p-Laplacian equations
AU - Dong Ye
AU - Weimin Zhang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02626
UR - https://arxiv.org/abs/2609.02626
AB - This paper establishes a new comparison principle framework for the subquadratic fractional p-Laplacian, i.e.~1 < p < 2 under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for (−Δ)_p^s u = f(u) in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range s ∈ (0, 1/2) and 1/(1−s) < p < 2. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even C^1,1 regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to (−Δ)_p^s u = f(u) under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.
ER -