多重调和Dirichlet问题奇异正解的对称性  

Symmetry of Singular Positive Solutions for Polyharmonic Problems with Dirichlet Boundary Condition

作　　者：刘忠原 Zhong Yuan LIU(School of Mathematics and Statistics,Henan University,Kaifeng 475004,P.R.China)

机构地区：[1]河南大学数学与统计学院,开封475004

出　　处：《数学学报（中文版）》2022年第1期115-122,共8页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11971147);中国博士后科学基金资助项目(2019M662475)。

摘　　要：我们在穿孔单位球上研究下面多重调和Dirichlet问题{(-△)^(k)u=f(u),在B\{0}内,u>0,在B\{0}内,u=■u/■v=…=■^(k-1)u/■v^(k-1)=0,在■B上,其中,B是R^(N)中的单位球,v是■B的单位外法向量,N>2k,k≥2.在f满足适当假设条件下,如果0是不可去奇点,我们利用移动平面法得到奇异正解的径向对称性.由此,我们可以得到临界双调和Dirichlet问题正解的不存在性.We study the following polyharmonic Dirichlet problems in a punctured unit ball {(-△)^(k)u=f(u),in B\{0},u>0,in B\{0},u=■u/■v=…=■^(k-1)u/■v^(k-1)=0,on ■B where B is the unit ball in R^(N),v is the unit outward normal vector of ■B,N>2 k,k≥2.Under certain assumptions on f,we use the moving plane method to show radial symmetry of any singular positive solution provided that 0 is a nonremovable singularity point.As an application,we can obtain nonexistence of positive solutions for a critical Dirichlet biharmonic problem.

关 键 词：奇异正解的对称性 多重调和问题 移动平面法 

分 类 号：O175.25[理学—数学]