{"paper_id":"18c861ef-5df4-40c6-a602-b85fab7618eb","body_text":"Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon Mohameden Ahmedou, Mohamed Ben Ayed, Khalil El Mehdi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4294521/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 14 Nov, 2024 Read the published version in Nonlinear Differential Equations and Applications NoDEA → Version 1 posted 9 You are reading this latest preprint version Abstract In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in [28, 27]. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a non-zero weak limit, which necessarily constitutes a solution of the Nirenberg problem itself. Our focus lies in providing a comprehensive description of such blowing up solutions, including precise determinations of blow-up points and blow-up rates. Additionally, we compute the topological contribution of these solutions to the difference in topology between the level sets of the associated Euler-Lagrange functional. Such an analysis is intricate due to the potential degeneracy of the involved solutions. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate. AMS subject classification: 58J05, 35A01, 58E05. Nirenberg problem Blow-up analysis Partial Differential Equations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 14 Nov, 2024 Read the published version in Nonlinear Differential Equations and Applications NoDEA → Version 1 posted Editorial decision: Revision requested 04 Sep, 2024 Reviews received at journal 04 Sep, 2024 Reviews received at journal 09 Jul, 2024 Reviewers agreed at journal 29 Apr, 2024 Reviewers agreed at journal 25 Apr, 2024 Reviewers invited by journal 22 Apr, 2024 Editor assigned by journal 21 Apr, 2024 Submission checks completed at journal 20 Apr, 2024 First submitted to journal 19 Apr, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-4294521\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":294458432,\"identity\":\"a80751ef-b64d-422d-a73c-a24afa4a0260\",\"order_by\":0,\"name\":\"Mohameden Ahmedou\",\"email\":\"data:image/png;base64,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\",\"orcid\":\"\",\"institution\":\"University of Giessen\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Mohameden\",\"middleName\":\"\",\"lastName\":\"Ahmedou\",\"suffix\":\"\"},{\"id\":294458434,\"identity\":\"f4a1d865-85c9-495a-a9cf-8ede71b52aae\",\"order_by\":1,\"name\":\"Mohamed Ben Ayed\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"University of Sfax\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Mohamed\",\"middleName\":\"Ben\",\"lastName\":\"Ayed\",\"suffix\":\"\"},{\"id\":294458436,\"identity\":\"4ffc4e42-dbe4-443d-b318-c0f56afcd32e\",\"order_by\":2,\"name\":\"Khalil El Mehdi\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"University of Nouakchott Al Aasriya\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Khalil\",\"middleName\":\"El\",\"lastName\":\"Mehdi\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2024-04-19 17:29:25\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-4294521/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-4294521/v1\",\"draftVersion\":[],\"editorialEvents\":[{\"content\":\"https://doi.org/10.1007/s00030-024-01004-8\",\"type\":\"published\",\"date\":\"2024-11-14T15:57:04+00:00\"}],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":69285488,\"identity\":\"c2ec5bce-91bf-42ce-87b3-d7daf5545f09\",\"added_by\":\"auto\",\"created_at\":\"2024-11-18 19:26:00\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":378961,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"AhmedouBenAyedElMehdiBlowup.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-4294521/v1_covered_e2368697-b74f-4759-bb2a-6b49e4892b3e.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"\\u003cp\\u003eNirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon\\u003c/p\\u003e\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":true,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"nonlinear-differential-equations-and-applications-nodea\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"ndea\",\"sideBox\":\"Learn more about [Nonlinear Differential Equations and Applications NoDEA](http://link.springer.com/journal/30)\",\"snPcode\":\"30\",\"submissionUrl\":\"https://submission.nature.com/new-submission/30/3\",\"title\":\"Nonlinear Differential Equations and Applications NoDEA\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"em\",\"reportingPortfolio\":\"Springer Hybrid\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false},\"keywords\":\"Nirenberg problem, Blow-up analysis, Partial Differential Equations\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-4294521/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-4294521/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eIn this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in [28, 27]. 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