一类非线性椭圆型方程的一个Liouville定理

摘要/Abstract

摘要：

设(Mⁿ, g)是一个n维的完备黎曼流形, 其Ricci曲率满足Ric_M(x)≥-A(1+r²(x)ln²(2+r(x))), 其中A是非负常数, r(x)表示点x∈M到某固定点x₀∈M的测地距离.则M上方程Δu+Su+Ku^α=0在下述条件: 在M上S≤0; 在M上K<0且有常数a>0使在一个紧集之外K≤-a²;常数α>1”下的C²-非负解只有零解.

关键词: 黎曼流形,Ricci曲率, 椭圆型方程, Liouville定理.

Abstract:

Let (Mⁿ, g) be a complete Riemannian manifold of dimention n with Ricci curvature Ric_M(x)≥-A(1+r²(x)ln²(2+r(x))), where A is a nonnegative constant and r(x) is the geodesic distance from x to some fixed point x₀∈M.\$ In this paper, we show that there is no nonnegative C² solution but zero to the equation Δu+Su+Ku^α=0 on M under such conditions as: S≤0 on M; K<0 on M and K≤-a²<0 outside some compact set for a constant a>0 and any constant α>1.

Key words: Riemannian manifold, Ricci curvature, Elliptic equation, Liouville theorem.

中图分类号:

58G30

胡泽军. 一类非线性椭圆型方程的一个Liouville定理[J]. 数学物理学报, 2000, 20(4): 474-479.

Hu Zejun. A Liouville Theorem for a Class of Nonlinear Elliptic Equations[J]. Acta mathematica scientia,Series A, 2000, 20(4): 474-479.

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