The homogeneity function of the fractional p-Laplacian and optimal asymptotic exponents for critical Hardy-Sobolev equations

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 μ:=Φ(N-sp/p) equals one half of the sharp fractional Hardy constant. Consequently, for every 0<μ<μ, 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 {equation*} (-Δ_p)^s u - μ{uᵖ⁻¹}{|x|ˢᵖ}=0 in R^N\{0\}, \; u> 0 in R^N \{0\}. {equation*} For 0<μ<μ, we then study the following critical Hardy-Sobolev equation {align}{P}{P} (-Δ_p)^s u-μ{uᵖ⁻¹}{|x|ˢᵖ} = uNp/N-sp-1 in R^N, \; u > 0 in R^N. {align} Let p ≥ 2 and U be a positive, radially symmetric, nonincreasing finite-energy weak solution to {P}. We prove that there exists $C>0$ for which {align*} U(x) ≤ C|x|-γ_-(μ)\;as |x| → 0, and U(x)≤ C|x|-γ_+(μ) \; as |x| → ∞. {align*} 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.