{"title":"The Brezis–Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains","authors":"Yan Sheng Shen","doi":"10.1007/s10114-023-2108-8","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional <i>p</i>-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue <span>\\({\\lambda _{p,s}}(\\widehat {{\\omega _\\delta }})\\)</span> with respect to the domain <span>\\((\\widehat {{\\omega _\\delta }})\\)</span>. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi–Perera–Squassina–Yang [The Brezis–Nirenberg problem for the fractional <i>p</i>-Laplacian. <i>C</i>alc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192–199] to the fractional <i>p</i>-Laplacian setting.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2108-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue \({\lambda _{p,s}}(\widehat {{\omega _\delta }})\) with respect to the domain \((\widehat {{\omega _\delta }})\). Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi–Perera–Squassina–Yang [The Brezis–Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis–Nirenberg type results of Ramos–Wang–Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192–199] to the fractional p-Laplacian setting.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.