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 $1<p<\infty$, $0<s<1$, $0\le \alpha <sp<d$, $\mu >0$, $p_s^{\ast }(\alpha ):=p(d-\alpha )/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $({\mathcal{D}}^{s,p}({\mathbb{R}}^d){)}^{\ast }$. 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}.

arXiv comment47 pages. Comments are welcome!