Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

被引：6

作者：

Abolarinwa, Abimbola ^{[1
]}

Salawu, Sulyman O. ^{[1
]}

Onate, Clement A. ^{[1
]}

机构：

[1] Landmark Univ, Dept Phys Sci, Omu Aran, Kwara State, Nigeria

来源：

HELIYON | 2019年 / 5卷 / 11期

关键词：

Mathematics; Riemannian manifolds; Elliptic equations; Liouville theorem; Gradient estimates; Yamabe problem; LIOUVILLE TYPE THEOREMS; HARNACK INEQUALITIES; RIEMANNIAN-MANIFOLDS; POSITIVE SOLUTIONS;

D O I：

10.1016/j.heliyon.2019.e02784

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equation Delta(f)u + p(x)u + q(x)u(alpha) = 0 on complete smooth metric measure spaces (M-N, g,e(-f) dv) with x-Bakry-Emery Ricci tensor bounded from below, where alpha is an arbitrary real constant, p(x) and q(x) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (M-N, g,e(-f) dv) with nonpositive weighted scalar curvature.

引用

页数：6

共 22 条

[1]

Abolarinwa A., 2015, Elctron. J. Differ. Equ, V2015, P1

[2] Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation [J].

Abolarinwa, Abimbola .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2019, 473 (01) :297-312

[3] GRADIENT ESTIMATES FOR A NONLINEAR ELLIPTIC EQUATION ON COMPLETE NONCOMPACT RIEMANNIAN MANIFOLD [J].

Abolarinwa, Abimbola .

JOURNAL OF MATHEMATICAL INEQUALITIES, 2018, 12 (02) :391-402

[4] NONLINEAR ELLIPTIC-EQUATIONS ON COMPACT RIEMANNIAN-MANIFOLDS AND ASYMPTOTICS OF EMDEN EQUATIONS [J].

BIDAUTVERON, MF ;

VERON, L .

INVENTIONES MATHEMATICAE, 1991, 106 (03) :489-539

[5] Positive solutions of Yamabe type equations on complete manifolds and applications [J].

Brandolini, L ;

Rigoli, M ;

Setti, AG .

JOURNAL OF FUNCTIONAL ANALYSIS, 1998, 160 (01) :176-222

[6] A Liouville-Type Theorem for Smooth Metric Measure Spaces [J].

Brighton, Kevin .

JOURNAL OF GEOMETRIC ANALYSIS, 2013, 23 (02) :562-570

[7]

Case JS, 2015, J DIFFER GEOM, V101, P467

[8] 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

[9] GLOBAL AND LOCAL BEHAVIOR OF POSITIVE SOLUTIONS OF NON-LINEAR ELLIPTIC-EQUATIONS [J].

GIDAS, B ;

SPRUCK, J .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1981, 34 (04) :525-598

[10] Gradient estimates and Harnack inequalities for Yamabe-type parabolic equations on Riemannian manifolds [J].

Ha Tuan Dung .

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 2018, 60 :39-48

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