The Brezis–Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains  

作　　者：Yan Sheng SHEN 

机构地区：[1]School of Mathematical Sciences,Jiangsu University,Zhenjiang 212013,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2023年第11期2181-2206,共26页数学学报（英文版）

基　　金：Natural Science Foundation of China(Grant No.12071185)。

摘　　要：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 Poincaréinequality in unbounded cylindrical domains,we first study the asymptotic property of the first eigenvalueλp,s(ωδ)with respect to the domainωδ.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.C 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 p-Laplacian setting.

关 键 词：Brezis–Nirenberg problem fractional Poincaréinequality fractional p-Laplacian unbounded cylinder type domains concentration–compactness principle 

分 类 号：O177[理学—数学]