Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions

We study the critical fractional $p$-Hardy-Sobolev equation \begin{equation}\tag{$\mathcal{P}$}\label{a-main} (-Δ_p)^s u -μ\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(α)-2}u}{|x|^α}+f \;\mbox{ in }\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} where $1<p<\infty