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

Abstract: 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 $1&lt;p&lt;\infty$, $0&lt;s\&lt;1$, $0\leqα&lt;sp&lt;d$, $$μ\&gt;0$$, $p^*_s(α):= p(d-α)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\mathcal{D}<sup>{s,p}(\mathbb{R}<sup>d))<sup>*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $α&gt;0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $α=0$ has a different structure: in addition to Hardy profiles, pure Sobolev profiles may occur when the centre of concentration escapes from the Hardy singularity relative to its scale. We give a direct centre-scale analysis of these two concentration regimes and obtain the corresponding energy decomposition and profile separation. As an application, under an explicit smallness assumption on $f$, we first obtain a positive solution for \eqref{a-main} with negative energy. We then construct a nonlinear path based on hidden convexity whose energy remains strictly below the first bubbling threshold. A minimax argument, combined with the global compactness theorem, then yields a second distinct positive solution for \eqref{a-main}.