{"title":"A class of semilinear hypoelliptic Robin problems and Morse theory","authors":"Kazuaki Taira","doi":"10.1007/s11565-024-00529-1","DOIUrl":null,"url":null,"abstract":"<div><p>(1) Background: This paper is devoted to the study of a class of semilinear elliptic boundary value problems with hypoelliptic (degenerate) Robin condition that includes as particular cases the Dirichlet, Neumann and regular Robin problems. (2) Methods: We give a rigorous proof of main theorem, which is based heavily on the theory of linear elliptic boundary value problems in the framework of <span>\\(L^{p}\\)</span> Sobolev spaces. (3) Results: We extend earlier theorems due to Ambrosetti–Lupo and Struwe to the hypoelliptic Robin case via Morse theory. (4) Conclusions: The main purpose of this paper is to understand the essence of a modern version of the classical Lyapunov–Schmidt procedure through a concrete approach to semilinear hypoelliptic Robin problems.</p></div>","PeriodicalId":35009,"journal":{"name":"Annali dell''Universita di Ferrara","volume":"70 4","pages":"1545 - 1605"},"PeriodicalIF":0.0000,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali dell''Universita di Ferrara","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s11565-024-00529-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
(1) Background: This paper is devoted to the study of a class of semilinear elliptic boundary value problems with hypoelliptic (degenerate) Robin condition that includes as particular cases the Dirichlet, Neumann and regular Robin problems. (2) Methods: We give a rigorous proof of main theorem, which is based heavily on the theory of linear elliptic boundary value problems in the framework of \(L^{p}\) Sobolev spaces. (3) Results: We extend earlier theorems due to Ambrosetti–Lupo and Struwe to the hypoelliptic Robin case via Morse theory. (4) Conclusions: The main purpose of this paper is to understand the essence of a modern version of the classical Lyapunov–Schmidt procedure through a concrete approach to semilinear hypoelliptic Robin problems.
期刊介绍：
Annali dell''Università di Ferrara is a general mathematical journal publishing high quality papers in all aspects of pure and applied mathematics. After a quick preliminary examination, potentially acceptable contributions will be judged by appropriate international referees. Original research papers are preferred, but well-written surveys on important subjects are also welcome.
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