The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations
Abstract
Let N ≥ 2, s∈ (0,1), and p ∈ (1, ∞) with N>sp. For the fractional p-Laplacian, the power function |x|−γ with 0<γ<N/(p−1) satisfies (−Δp)s (|x|−γ)=Φ(γ)|x|−γ(p−1)−sp in ℝN∖{0}. Starting from the power function formula of Del Pezzo and Quaas (2026), we derive a paired integral representation of Φ, and show that Φ is strictly increasing and then strictly decreasing, with a unique maximum at (N−sp)/p. This maximum μ̅:=Φ((N−sp)/p) equals one half of the sharp fractional Hardy constant. Consequently, for every 0<μ<μ̅, the equation Φ(γ)=μ has exactly two roots 0<γ−(μ)<(N−sp)/p<γ+(μ)<(N−sp)/(p−1). These roots determine the homogeneous solutions |x|−γ−(μ) and |x|−γ+(μ) of the Hardy equation (−Δp)s u − μ(up−1)/|x|sp=0 in ℝN∖{0}, u> 0 in ℝN ∖ {0}. For 0<μ<μ̅, we then study the following critical Hardy-Sobolev equation tag$𝒫$labelP (−Δp)s u−μ(up−1)/|x|sp = uNp/(N−sp)−1 in ℝN, u > 0 in ℝN. Let p ≥ 2 and U be a positive, radially symmetric, nonincreasing finite-energy weak solution to \eqref{P}. We prove that there exists C>0 for which U(x) ≤ C|x|−γ-(μ) as |x| → 0, and U(x)≤ C|x|−γ+(μ) as |x| → ∞. Further, we show that γ-(μ) and γ+(μ) are the optimal exponents describing the singularity at the origin and the decay at infinity respectively.
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BibTeX
@misc{biswas2026homogeneity,
title = {{The homogeneity function of the fractional $p$-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations}},
author = {Nirjan Biswas and Debangana Mukherjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37212},
eprint = {2610.37212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37212},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations
AU - Nirjan Biswas
AU - Debangana Mukherjee
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37212
UR - https://arxiv.org/abs/2610.37212
AB - Let N ≥ 2, s∈ (0,1), and p ∈ (1, ∞) with N>sp. For the fractional p-Laplacian, the power function |x|^−γ with 0<γ<N/(p−1) satisfies (−Δ_p)^s (|x|^−γ)=Φ(γ)|x|^−γ(p−1)−sp in ℝ^N∖{0}. Starting from the power function formula of Del Pezzo and Quaas (2026), we derive a paired integral representation of Φ, and show that Φ is strictly increasing and then strictly decreasing, with a unique maximum at (N−sp)/p. This maximum μ̅:=Φ((N−sp)/p) equals one half of the sharp fractional Hardy constant. Consequently, for every 0<μ<μ̅, the equation Φ(γ)=μ has exactly two roots 0<γ_−(μ)<(N−sp)/p<γ_+(μ)<(N−sp)/(p−1). These roots determine the homogeneous solutions |x|^−γ_−(μ) and |x|^−γ_+(μ) of the Hardy equation (−Δ_p)^s u − μ(u^p−1)/|x|^sp=0 in ℝ^N∖{0}, u> 0 in ℝ^N ∖ {0}. For 0<μ<μ̅, we then study the following critical Hardy-Sobolev equation tag$𝒫$labelP (−Δ_p)^s u−μ(u^p−1)/|x|^sp = u^Np/(N−sp)−1 in ℝ^N, u > 0 in ℝ^N. Let p ≥ 2 and U be a positive, radially symmetric, nonincreasing finite-energy weak solution to \eqref{P}. We prove that there exists C>0 for which U(x) ≤ C|x|^−γ_-(μ) as |x| → 0, and U(x)≤ C|x|^−γ_+(μ) as |x| → ∞. Further, we show that γ_-(μ) and γ_+(μ) are the optimal exponents describing the singularity at the origin and the decay at infinity respectively.
ER -