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Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds

Liang Zhao

Author Affiliations

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

Boundary Value Problems 2013, 2013:190  doi:10.1186/1687-2770-2013-190

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/190


Received:31 May 2013
Accepted:9 August 2013
Published:27 August 2013

© 2013 Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the gradient estimates for positive solutions to the following parabolic Lichnerowicz equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M1">View MathML</a>

on complete noncompact Riemannian manifolds, where h, p, q, A, B are real constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3">View MathML</a>.

MSC: 58J05, 58J35.

Keywords:
Lichnerowicz equation; positive solutions; Harnack inequality

1 Introduction

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M4">View MathML</a>

(1.1)

where h, p, q, A, B are real constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3">View MathML</a>.

Gradient estimates play an important role in the study of PDE, especially the Laplace equation and heat equation. Li [1] derived the gradient estimates and Harnack inequalities for positive solutions of nonlinear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M8">View MathML</a> on Riemannian manifolds. The author in [1] also obtained a theorem of Liouville-type for positive solutions of the nonlinear elliptic equation. Later, Yang [2] gave the gradient estimates for the solution to the elliptic equation with singular nonlinearity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M9">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M10">View MathML</a>, c are two real constants. More precisely, the author [2] obtained the following result.

Theorem 1.1 (Yang [2])

LetMbe a noncompact complete Riemannian manifold of dimensionnwithout boundary. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11">View MathML</a>be a geodesic ball of radius 2Raround<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12">View MathML</a>. We denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M13">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M14">View MathML</a>, to be a lower bound of the Ricci curvature on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M16">View MathML</a>for all tangent fieldξon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18">View MathML</a>is a positive smooth solution of the equation (1.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M19">View MathML</a>, cbeing two real constants. Then we have:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M20">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18">View MathML</a>satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M22">View MathML</a>

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M23">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M25">View MathML</a>are some universal constants independent of geometry ofM.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M26">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M18">View MathML</a>satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M28">View MathML</a>

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M23">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M25">View MathML</a>are some universal constants independent of geometry ofM.

For some interesting gradient estimates in this direction, we can refer to [3-7].

Recently, Song and Zhao [8] studied a generalized elliptic Lichnerowicz equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M32">View MathML</a>

(1.3)

on compact manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>. The authors in [8] got the local gradient estimate for the positive solutions of (1.3). Moreover, they considered the following parabolic Lichnerowicz equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M34">View MathML</a>

(1.4)

on manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a> and obtained the Harnack differential inequality.

Theorem 1.2 (Song and Zhao [8])

LetMbe a compact Riemannian manifold without boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M37">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38">View MathML</a>is any positive solution of (1.4) onMwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M40">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M41">View MathML</a>. Denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M42">View MathML</a>, suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M46">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M47">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M48">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M49">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M50">View MathML</a>

While the author considered the gradient estimates on compact Riemannian manifolds in Theorem 1.2, it is natural to study this problem on complete noncompact manifolds. Motivated by the work above, we present our main results as follows.

Theorem 1.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>be a complete noncompactn-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M52">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M54">View MathML</a>in the metric ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M55">View MathML</a>around<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12">View MathML</a>. Assume thatuis a positive solution of (1.1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M57">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M58">View MathML</a>. Then

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M61">View MathML</a>

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M64">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M65">View MathML</a>, c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M67">View MathML</a>, δare positive constants with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M70">View MathML</a>, we can get the following global gradient estimates for the nonlinear parabolic equation (1.1).

Corollary 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>be a complete noncompactn-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M72">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M73">View MathML</a>. Assume thatuis a positive solution of (1.1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M57">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M75">View MathML</a>. Then

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M78">View MathML</a>

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M81">View MathML</a>

δare positive constants with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M85">View MathML</a> in Corollary 1.4, we get a Li-Yau-type gradient estimate.

Corollary 1.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>be a complete noncompactn-dimensional Riemannian manifold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38">View MathML</a>is a positive solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M89">View MathML</a>

on complete noncompact manifolds, whereh, q, Bare real constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M91">View MathML</a>

(1.5)

As an application, we have the following Harnack inequality.

Theorem 1.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>be a complete noncompactn-dimensional Riemannian manifold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M36">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M38">View MathML</a>is a positive solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M95">View MathML</a>

on complete noncompact manifolds, whereh, q, Bare real constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M97">View MathML</a>. Then for any points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M99">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M100">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M101">View MathML</a>, we have the following Harnack inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M102">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M104">View MathML</a>.

2 Proof of Theorem 1.3

Assume that u is a positive solution to (1.1). Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M105">View MathML</a>, then w satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M106">View MathML</a>

(2.1)

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M33">View MathML</a>be a complete noncompactn-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M52">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M73">View MathML</a>in the metric ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M55">View MathML</a>around<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M12">View MathML</a>. Letwbe a positive solution of (2.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M113">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M114">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69">View MathML</a>.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M116">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M69">View MathML</a>. By the Bochner formula, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M118">View MathML</a>

(2.2)

By a direct computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M119">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M120">View MathML</a>

and we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M121">View MathML</a>

Therefore, by equalities (2.2) and (2.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M122">View MathML</a>

This implies that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M123">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M124">View MathML</a>

Therefore, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M125">View MathML</a>

which completes the proof of Lemma 2.1. □

We take a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M126">View MathML</a> cut-off function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M127">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M129">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M131">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M132">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M133">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M127">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M136">View MathML</a>

for some absolute constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M137">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M138">View MathML</a> the distance between x and p in M. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M139">View MathML</a>

Using an argument of Cheng and Yau [9], we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M140">View MathML</a> with support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11">View MathML</a>. Direct calculation shows that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M11">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M143">View MathML</a>

(2.4)

By the Laplacian comparison theorem in [10],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M144">View MathML</a>

(2.5)

In inequality (2.5), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M145">View MathML</a>, then △φ can be controlled by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M146">View MathML</a>, so in any case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M147">View MathML</a>, where c is some positive constant.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M148">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149">View MathML</a> be a point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M150">View MathML</a>, at which φF attains its maximum value P, and we assume that P is positive (otherwise the proof is trivial). At the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M152">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M153">View MathML</a>

This inequality, together with inequalities (2.4) and (2.5), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M154">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M155">View MathML</a>

At <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M149">View MathML</a>, by Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M157">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M158">View MathML</a>

here we used

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M159">View MathML</a>

Following Davies [11] (see also Negrin [12]), we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M160">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M161">View MathML</a>

Next, we consider the following two cases:

(1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M44">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M164">View MathML</a>

multiplying both sides of the inequality above by , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M165">View MathML</a>

So, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M166">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M167">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M168">View MathML</a>

Now, (1) of Theorem 1.3 can be easily deduced from the inequality above;

(2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M63">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M171">View MathML</a>

multiplying both sides of the inequality above by , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M172">View MathML</a>

So, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M173">View MathML</a>

Similarly, we can obtain (2) of Theorem 1.3.

Proof of Theorem 1.6 For any points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M99">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M176">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M101">View MathML</a>, we take a curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M178">View MathML</a> parameterized with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M180">View MathML</a>. One gets from Corollary 1.5 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M181">View MathML</a>

which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M182">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M183">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M104">View MathML</a>. □

Competing interests

The author declares that they have no competing interests.

Author’s contributions

The author completed the paper. The author read and approved the final manuscript.

Acknowledgements

The author would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of the paper. Moreover, the author would like to thank his supervisor Professor Kefeng Liu for his constant encouragement and help. This work is supported by the Postdoctoral Science Foundation of China (2013M531342) and the Fundamental Research Funds for the Central Universities (NS2012065).

References

  1. Li, JY: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal.. 100, 233–256 (1991). Publisher Full Text OpenURL

  2. Yang, YY: Gradient estimates for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M186">View MathML</a> on Riemannian manifolds. Acta Math. Sin.. 26, 1177–1182 (2010). Publisher Full Text OpenURL

  3. Chen, L, Chen, WY: Gradient estimates for positive smooth f-harmonic functions. Acta Math. Sci., Ser. B. 30, 1614–1618 (2010)

  4. Huang, GY, Li, HZ: Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian. arXiv:1203.5482v1 [math.DG] (2012) OpenURL

  5. Li, XD: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl.. 84, 1295–1361 (2005)

  6. Zhang, J, Ma, BQ: Gradient estimates for a nonlinear equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/190/mathml/M188">View MathML</a> on complete noncompact manifolds. Commun. Math.. 19, 73–84 (2011)

  7. Zhu, XB: Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc. Am. Math. Soc.. 139, 1637–1644 (2011). Publisher Full Text OpenURL

  8. Song, XF, Zhao, L: Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. Z. Angew. Math. Phys.. 61, 655–662 (2010). PubMed Abstract | Publisher Full Text OpenURL

  9. Cheng, SY, Yau, ST: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math.. 28, 333–354 (1975). Publisher Full Text OpenURL

  10. Aubin, T: Nonlinear Analysis on Manifolds, Springer, New York (1982)

  11. Davies, EB: Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989)

  12. Negrin, ER: Gradient estimates and a Liouville type theorem for the Schrödinger operator. J. Funct. Anal.. 127, 198–203 (1995). Publisher Full Text OpenURL