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 formul