On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

作     者：Qiu Xiang BIAN Jing CHEN Jing YANG Qiu Xiang BIAN;Jing CHEN;Jing YANG

作者机构：School of ScienceJiangsu University of Science and TechnologyZhenjiang 212003P.R.China School of Mathematics and Statistics and Hubei Key Laboratory Mathematical SciencesCentral China Normal UniversityWuhan 430079P.R.China 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2023年第39卷第10期

页      面：1951-1979页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Supported by NSFC(Grant Nos.12226324 11961043 11801226) 

主　　题：Infinitely many solutions prescribed boundary mean curvature finite reduction local Pohozaev identities 

摘      要：This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−*** the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r00 and K˜(r0,y˜0)0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily *** our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).