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A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian
Abstract
We establish a quantitative Gidas--Ni--Nirenberg theorem for the pure fractional Dirichlet problem (−Δ)s u=k(x)g(u) in B1, u=0 in ℝn∖ B1, 0<s<1. For positive bounded solutions under a two-sided L∞ normalization, we prove 𝒟(u)≤ C 𝒟(k)γ, 0<γ<1, where 𝒟 measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas--Ni--Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.
Related
- Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications
- Ground states for Brezis-Nirenberg-type fractional elliptic systems with sublinear absorption and subcritical perturbation
- A fractional Brezis-Nirenberg problem with multipolar Hardy potentials
- Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions
- Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices
BibTeX
@misc{shen2026quantitative,
title = {{A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian}},
author = {Qizhen Shen and Jiongduo Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14618},
eprint = {2609.14618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14618},
}
RIS
TY - UNPB
TI - A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian
AU - Qizhen Shen
AU - Jiongduo Xie
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14618
UR - https://arxiv.org/abs/2609.14618
AB - We establish a quantitative Gidas--Ni--Nirenberg theorem for the pure fractional Dirichlet problem (−Δ)^s u=k(x)g(u) in B_1, u=0 in ℝ^n∖ B_1, 0<s<1. For positive bounded solutions under a two-sided L^∞ normalization, we prove 𝒟(u)≤ C 𝒟(k)^γ, 0<γ<1, where 𝒟 measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas--Ni--Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.
ER -