Two-channel global compactness and existence for critical GJMS equations on hyperbolic space

[2411.14719] Two-channel global compactness and existence for critical GJMS equations on hyperbolic space Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2411.14719 (math) [Submitted on 22 Nov 2024 ( v1 ), last revised 27 Aug 2026 (this version, v4)] Title: Two-channel global compactness and existence for critical GJMS equations on hyperbolic space Authors: Jungang Li , Zhiwei Wang View PDF HTML (experimental) Abstract: Let $P_m$ be the GJMS operator of order $2m$ on real hyperbolic space, $n>2m$. For the critical equation \[ P_m u+a(x)u=|u|^{4m/(n-2m)}u \quad\text{on }\HH^n, \] where $0\leq a\in L^{n/(2m)}(\HH^n)$, we prove a global compactness theorem for arbitrary, possibly sign-changing Palais--Smale sequences. Every such sequence decomposes into a solution of the equation, finitely many Euclidean profiles concentrating at vanishing scales, finitely many solutions of the limiting GJMS equation transported to infinity by divergent hyperbolic isometries, and a remainder converging strongly in the energy space. The quadratic form, the $L^{2n/(n-2m)}$ norm and the energy split along this decomposition. The main new point is the treatment of the half-space profiles arising at the conformal boundary of the Poincaré ball. After conformal lifting, they are precisely finite-energy solutions on $\HH^n$ escaping at a fixed hyperbolic scale; hence they do not form a third type of profile. For sign-changing sequences, Moreau's polar-cone decomposition and positivity of the limiting Green operators give a sharp two-quantum energy bound and a compactness range below the two-bubble level. As an application, we obtain one solution for sufficiently concentrated nonnegative potentials and, under an additional smallness condition, a second solution. Comments: 28 pages Subjects: Analysis of PDEs (math.AP) MSC classes: 35J91, 35B33, 46E35, 47B65, 58E05, 58J05 Cite as: arXiv:2411.14719 [math.AP] (or arXiv:2411.14719v4 [math.AP] for this version) https://doi.org/10.48550/arXiv.2411.14719 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jungang Li [ view email ] [v1] Fri, 22 Nov 2024 04:16:56 UTC (36 KB) [v2] Fri, 15 May 2026 03:59:56 UTC (34 KB) [v3] Sat, 8 Aug 2026 15:17:58 UTC (31 KB) [v4] Thu, 27 Aug 2026 05:44:06 UTC (33 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2024-11 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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