Critical fractional p-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions
Abstract
We study the critical fractional p-Hardy-Sobolev equation tag$𝒫$labela−main (−Δp)s u −μ(|u|p−2u)/|x|sp=(|u|p*s(α)−2u)/|x|α+f in ℝd, u∈ 𝒟s,p(ℝd), where 1<p<∞, 0<s<1, 0≤α<sp<d, μ>0, p*s(α):= p(d−α)/(d−sp) is the critical Hardy-Sobolev exponent, and f is a nontrivial nonnegative functional in (𝒟s,p(ℝd))*. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When α>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}.
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BibTeX
@misc{biswas2026critical,
title = {{Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions}},
author = {Nirjan Biswas and Souptik Chakraborty and Debangana Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08510},
eprint = {2609.08510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08510},
}
RIS
TY - UNPB
TI - Critical fractional p-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions
AU - Nirjan Biswas
AU - Souptik Chakraborty
AU - Debangana Mukherjee
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08510
UR - https://arxiv.org/abs/2609.08510
AB - We study the critical fractional p-Hardy-Sobolev equation tag$𝒫$labela−main (−Δ_p)^s u −μ(|u|^p−2u)/|x|^sp=(|u|^p^*_s(α)−2u)/|x|^α+f in ℝ^d, u∈ 𝒟^s,p(ℝ^d), where 1<p<∞, 0<s<1, 0≤α<sp<d, μ>0, p^*_s(α):= p(d−α)/(d−sp) is the critical Hardy-Sobolev exponent, and f is a nontrivial nonnegative functional in (𝒟^s,p(ℝ^d))^*. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When α>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}.
ER -