方程∆u+au^(p+1)=0的梯度估计及其刘维尔定理  

Gradient Estimates and Liouville Theorems forΔu+au^(p+1)=0

作　　者：彭博 王友德 魏国栋[5] Peng Bo;Wang Youde;Wei Guodong(School of Mathematical Sciences,UCAS,Beijing 100049,China;Institute of Mathematics,The Academy of Mathematics and Systems of Sciences,Chinese Academy of Sciences,Beijing 100190,China;College of Mathematics and Information Sciences,Guangzhou University,Guangzhou 510006,China;Hua LooKeng Key Laboratory of Mathematics,Institute of Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China;School of Mathematics(Zhuhai),Sun Yatsen University,Zhuhai 519082,China)

机构地区：[1]中国科学院大学数学科学学院,北京100049 [2]中国科学院数学与系统科学研究院数学研究所,北京100190 [3]广州大学数学与信息科学学院,广州510006 [4]中国科学院数学与系统科学研究院数学研究所华罗庚数学重点实验室,北京100190 [5]中山大学数学学院(珠海),珠海519082

出　　处：《数学理论与应用》2023年第1期32-43,共12页Mathematical Theory and Applications

基　　金：National Natural Science Foundation of China(Nos.11731001,11971400,12101619);Guangdong Basic and Applied Basic Research Foundation(No.2020A1515011019);Science and Technology Projects of Guangzhou(No.202102020743)。

摘　　要：本文用经典李丘的方法并结合非常精细的分析给出定义在非紧完备黎曼流形(M,g)上的一类非线性椭圆方程Δu+au^(p+1)=0的一种统一且简单的梯度估计方法,这里常数a,p满足a>0,p<4/n或a<0,p>0.当a>0时,我们拓展了使得上述方程梯度估计成立的p的取值范围,改进了文献[11,13]中的一些结果,补充了dim(M)=2情形的结果.当a<0及p>0时,因为不需要假设正解是有界的,我们改进了Ma,Huang和Luo在[13]中得到的结果.当(M,g)的里奇曲率非负时,我们得到了上述方程的刘维尔定理.In this paper,we employ LiYau’s method and delicate analysis techniques to provide a unified and simple approach to the gradient estimate of the positive solution to the nonlinear elliptic equationΔu+au^(p+1)=0 defined on a complete noncompact Riemannian manifold(M,g)where a>0 and p<4/n or a<0 and p>0 are two constants.For the case a>0,we extend the range of p and improve some results in[11,13]and supplement the results for the case dim(M)=2.For the case a<0 and p>0,we improve or perfect the previous results due to Ma,Huang and Luo[13]since one does not need to suppose the positive solutions are bounded.When the Ricci curvature of(M,g)is nonnegative,we also obtain a Liouvilletype theorem for the above equation.

关 键 词：梯度估计 非线性椭圆方程 刘维尔定理 

分 类 号：O186.12[理学—数学] O175.25[理学—基础数学]