The direct moving plane method for weak solutions of the fractional p-Laplacian
Abstract
In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional p-Laplacian in the singular range 1<p≤2. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fractional p-Laplacian equations in a bounded domain. We also consider the nonlocal quasilinear Lane--Emden equation (−Δ)spu=uq in ℝn. In the Sobolev critical case, we establish radial symmetry, monotonicity, and precise asymptotic behavior at infinity for finite-energy weak solutions. Under a suitable decay condition, we also obtain radial symmetry for the full range q>p−1. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for C1,1 solutions in the pointwise sense.
Related
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- Large Solutions for Fractional Laplacian on Infinite Cylindrical Domains
- Hölder regularity of solutions to parabolic equations associated with fractional p-Laplacian type operators
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- An accurate parallel finite difference method for the variable-order fractional Laplacian in heterogeneous media
BibTeX
@misc{xu2026direct,
title = {{The direct moving plane method for weak solutions of the fractional $p$-Laplacian}},
author = {Meiqing Xu and Hui Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02225},
eprint = {2609.02225},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02225},
}
RIS
TY - UNPB
TI - The direct moving plane method for weak solutions of the fractional p-Laplacian
AU - Meiqing Xu
AU - Hui Yang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02225
UR - https://arxiv.org/abs/2609.02225
AB - In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional p-Laplacian in the singular range 1<p≤2. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fractional p-Laplacian equations in a bounded domain. We also consider the nonlocal quasilinear Lane--Emden equation (−Δ)^s_pu=u^q in ℝ^n. In the Sobolev critical case, we establish radial symmetry, monotonicity, and precise asymptotic behavior at infinity for finite-energy weak solutions. Under a suitable decay condition, we also obtain radial symmetry for the full range q>p−1. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for C^1,1 solutions in the pointwise sense.
ER -