Optimal regularity of positive solutions of the Hénon-Hardy equation and related equations  

作　　者：Zongming Guo Fangshu Wan 

机构地区：[1]College of Mathematics and Information Science,Henan Normal University,Xinxiang 453007,China [2]School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China [3]School of Mathematical Sciences,Anhui University,Hefei 230601,China

出　　处：《Science China Mathematics》2024年第10期2283-2302,共20页中国科学（数学）（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11571093);supported by the Fundamental Research Funds for the Central Universities (Grant No. WK0010000064);Anhui Provincial Natural Science Foundation (Grant No. BJ0010000026)。

摘　　要：We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.

关 键 词：H´enon-Hardy equation positive solutions optimal regularity singular point asymptotic expansions 

分 类 号：O175[理学—数学]