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} (-\Delta_p)^s u -\mu\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(\alpha)-2}u}{|x|^{\alpha}}+f \;\mbox{ in }\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} where $1 0$, $p^*_s(\alpha):= p(d-\alpha)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\mathcal{D}^{s,p}(\mathbb{R}^d))^*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $\alpha>0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $\alpha=0$ 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}.