完备黎曼流形上方程Δv+v^(r)−v^(s)=0解的梯度估计  

Gradient Estimate for Solutions ofΔv+v^(r)−v^(s)=0 on a Complete Riemannian Manifold

作　　者：王友德 张艾琦[1] Wang Youde;Zhang Aiqi(School 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 Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China)

机构地区：[1]广州大学数学与信息科学学院,广州510006 [2]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 [3]中国科学院大学数学科学学院,北京100049

出　　处：《数学理论与应用》2023年第3期1-22,共22页Mathematical Theory and Applications

摘　　要：在本文中,我们讨论定义于完备黎曼流形(M,g)上的椭圆方程Δv+v^(r)−v^(s)=0正解的梯度估计,其中r和s是常数.当(M,g)满足Ric≥−(n−1)κ时(其中n≥2是M的维数,κ是非负常数),在适当的几何和分析条件下,我们采用NashMoser迭代技巧导出该方程正解的Cheng-Yau型梯度估计,并证明当(M,g)的Ricci曲率非负时,若r<s,并且1<r<n+3/n−1或1<s<n+3/n−1,则该方程除了v≡1以外无其它正解.In this paper we consider the gradient estimates on the positive solutions to the elliptic equationΔv+v^(r)−v^(s)=0,defined on a complete Riemannian manifold(M,g),where r and s are two real constants.When(M,g)satisfies Ric≥−(n−1)κ(where n≥2 is the dimension of M and κ is a nonnegative constant),we employ the NashMoser iteration technique to derive a ChengYau type gradient estimate for the positive solutions to the above equation under some suitable geometric and analysis conditions.Moreover,it is shown that when the Ricci curvature of M is nonnegative,this elliptic equation does not admit any positive solutions except for v≡1 if r<s and 1<r<n+3/n−1 or 1<s<n+3/n−1.

关 键 词：椭圆方程 黎曼流形 梯度估计 

分 类 号：O186.12[理学—数学]