{"title":"规定分数 Q曲率问题中存在性和紧凑性的统一结果","authors":"Yan Li, Zhongwei Tang, Heming Wang, Ning Zhou","doi":"10.1007/s00030-024-00927-6","DOIUrl":null,"url":null,"abstract":"<p>In this paper we study the problem of prescribing fractional <i>Q</i>-curvature of order <span>\\(2\\sigma \\)</span> for a conformal metric on the standard sphere <span>\\(\\mathbb {S}^n\\)</span> with <span>\\(\\sigma \\in (0,n/2)\\)</span> and <span>\\(n\\ge 3\\)</span>. Compactness and existence results are obtained in terms of the flatness order <span>\\(\\beta \\)</span> of the prescribed curvature function <i>K</i>. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when <span>\\(\\beta \\in [n-2\\sigma ,n)\\)</span> for all <span>\\(\\sigma \\in (0,n/2)\\)</span>. This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for <span>\\(\\beta \\in (n-2\\sigma ,n)\\)</span>.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unified results for existence and compactness in the prescribed fractional Q-curvature problem\",\"authors\":\"Yan Li, Zhongwei Tang, Heming Wang, Ning Zhou\",\"doi\":\"10.1007/s00030-024-00927-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we study the problem of prescribing fractional <i>Q</i>-curvature of order <span>\\\\(2\\\\sigma \\\\)</span> for a conformal metric on the standard sphere <span>\\\\(\\\\mathbb {S}^n\\\\)</span> with <span>\\\\(\\\\sigma \\\\in (0,n/2)\\\\)</span> and <span>\\\\(n\\\\ge 3\\\\)</span>. Compactness and existence results are obtained in terms of the flatness order <span>\\\\(\\\\beta \\\\)</span> of the prescribed curvature function <i>K</i>. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when <span>\\\\(\\\\beta \\\\in [n-2\\\\sigma ,n)\\\\)</span> for all <span>\\\\(\\\\sigma \\\\in (0,n/2)\\\\)</span>. This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for <span>\\\\(\\\\beta \\\\in (n-2\\\\sigma ,n)\\\\)</span>.</p>\",\"PeriodicalId\":501665,\"journal\":{\"name\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00030-024-00927-6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-024-00927-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Unified results for existence and compactness in the prescribed fractional Q-curvature problem
In this paper we study the problem of prescribing fractional Q-curvature of order \(2\sigma \) for a conformal metric on the standard sphere \(\mathbb {S}^n\) with \(\sigma \in (0,n/2)\) and \(n\ge 3\). Compactness and existence results are obtained in terms of the flatness order \(\beta \) of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when \(\beta \in [n-2\sigma ,n)\) for all \(\sigma \in (0,n/2)\). This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for \(\beta \in (n-2\sigma ,n)\).