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

Let $N \geq 2$, $s\in (0,1)$, and $p \in (1, \infty)$ with $N>sp$. For the fractional $p$-Laplacian, the power function $|x|^{-γ}$ with $0<γ<\frac{N}{p-1}$ satisfies $$(-Δ_p)^s \big(|x|^{-γ}\big)=Φ(γ)|x|^{-γ(p-1)-sp} \, \text{ in }\mathbb{R}^N\setminus\{0\}.$$ 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 $\frac{N-sp}{p}$. This maximum $\barμ:=Φ(\frac{N-sp}{p})$ equals one half of the sharp fractional Hardy constant. Consequently, for every $0<μ<\barμ$, the equation $Φ(γ)=μ$ has exactly two roots $0<γ_{-}(μ)<\frac{N-sp}{p}<γ_{+}(μ)<\frac{N-sp}{p-1}$. These roots determine the homogeneous solutions $|x|^{-γ_{-}(μ)}$ and $|x|^{-γ_{+}(μ)}$ of the Hardy equation \begin{equation*} (-Δ_p)^s u - μ\frac{u^{p-1}}{|x|^{sp}}=0 \quad \text{in }\mathbb R^N\setminus\{0\}, \; u> 0 \text{ in } \mathbb{R}^N \setminus \{0\}. \end{equation*} For $0<μ<\barμ$, we then study the following critical Hardy-Sobolev equation \begin{align}\tag{$\mathcal{P}$}\label{P} (-Δ_p)^s u-μ\frac{u^{p-1}}{|x|^{sp}} = u^{\frac{Np}{N-sp}-1} \text{ in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N. \end{align} Let $p \geq 2$ and $U$ be a positive, radially symmetric, nonincreasing finite-energy weak solution to \eqref{P}. We prove that there exists $C>0$ for which \begin{align*} U(x) \leq C|x|^{-γ_-(μ)}\;\text{as }|x| \rightarrow 0,\quad \text{ and } \quad U(x)\le C|x|^{-γ_+(μ)} \; \text{as }|x| \rightarrow \infty. \end{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.