A Liouville theorem for weighted p-Laplace operator on smooth metric measure spaces

被引：5

作者：

Wang, Lin Feng ^{[1
]}

Zhang, Ze Yu ^{[1
]}

Zhao, Liang ^{[2
]}

Zhou, Yu Jie ^{[1
]}

机构：

[1] Nantong Univ, Sch Sci, Nantong 226007, Jiangsu, Peoples R China

[2] Nanjing Univ Aeronaut & Astronaut, Dept Math, Nanjing 210016, Jiangsu, Peoples R China

来源：

MATHEMATICAL METHODS IN THE APPLIED SCIENCES | 2017年 / 40卷 / 04期

关键词：

Bakry-Emery curvature; smooth metric measure space; weighted p-Laplace operator; Moser iteration; MEAN-CURVATURE FLOW; RIEMANNIAN-MANIFOLDS; HARMONIC FUNCTIONS; GRADIENT ESTIMATE; EQUATIONS;

D O I：

10.1002/mma.4031

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove that on a smooth metric measure space with m-Bakry-mery curvature bounde from below by -(m - 1) K for some constant K >= 0 (i. e., Ric(f,m) >= -(m - 1)K), the following degenerate elliptic equation Delta(f,p)u = -lambda(f,p) vertical bar u vertical bar(p=2)u has no nonconstant positive solution when p > 1 and constant lambda(f,p) satisfies lambda(f,p) > p(-p)K(p/2)(m - 1)(p). Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng-Yau's method, which was used by Wang-Zhu in 2012 to derive a same Liouville theorem when 1 < p <= 2, Ric(f,m) >= -(m-1) K and the sectional curvature is bounded from below. Copyright (C) 2016 JohnWiley & Sons, Ltd.

引用

页码：992 / 1002

页数：11

共 15 条

[1]

Bakry Dominique, 2005, Theta Ser. Adv. Math., V4, P115

[2] DIFFERENTIAL EQUATIONS ON RIEMANNIAN MANIFOLDS AND THEIR GEOMETRIC APPLICATIONS [J].

CHENG, SY ;

YAU, ST .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1975, 28 (03) :333-354

[3] The inverse mean curvature flow and the Riemannian Penrose Inequality [J].

Huisken, G ;

Ilmanen, T .

JOURNAL OF DIFFERENTIAL GEOMETRY, 2001, 59 (03) :353-437

[4]

Kotschwar B, 2009, ANN SCI ECOLE NORM S, V42, P1

[5]

Li P., 2006, 14 ESCOLA GEOMETRIA

[6] Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds [J].

Li, XD .

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, 2005, 84 (10) :1295-1361

[7]

Moser R, 2007, J EUR MATH SOC, V9, P77

[8]

Perelman G., ARXIVMATHDG0211159

[9]

SALOFFCOSTE L, 1992, J DIFFER GEOM, V36, P417

[10] REGULARITY FOR A MORE GENERAL-CLASS OF QUASILINEAR ELLIPTIC-EQUATIONS [J].

TOLKSDORF, P .

JOURNAL OF DIFFERENTIAL EQUATIONS, 1984, 51 (01) :126-150

← 1 2 →