Gradient estimates for a nonlinear equation Δ f u + c u - α = 0 on complete noncompact manifolds

Jing Zhang; Bingqing Ma

Communications in Mathematics (2011)

  • Volume: 19, Issue: 1, page 73-84
  • ISSN: 1804-1388

Abstract

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Let ( M , g ) be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation Δ f u + c u - α = 0 in M , where α , c are two real constants and α > 0 , f is a smooth real valued function on M and Δ f = Δ - ∇ f ∇ . When N is finite and the N -Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that ∞ -Bakry-Emery Ricci tensor is bounded from below and | ∇ f | is bounded from above, we also obtain a gradient estimate for positive solutions of the above equation. It extends the results of Yang [Yang, Y.Y. Gradient estimates for the equation Δ u + c u - α = 0 on Riemannian manifolds Acta. Math. Sin. 26(B) 2010 1177–1182].

How to cite

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Zhang, Jing, and Ma, Bingqing. "Gradient estimates for a nonlinear equation $\Delta _fu+cu^{-\alpha }=0$ on complete noncompact manifolds." Communications in Mathematics 19.1 (2011): 73-84. <http://eudml.org/doc/196367>.

@article{Zhang2011,
abstract = {Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation $\Delta _fu+cu^\{-\alpha \}=0$ in $M$, where $\alpha $, $c$ are two real constants and $\alpha >0$, $f$ is a smooth real valued function on $M$ and $\Delta _f=\Delta -\nabla f\nabla $. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty $-Bakry-Emery Ricci tensor is bounded from below and $|\nabla f|$ is bounded from above, we also obtain a gradient estimate for positive solutions of the above equation. It extends the results of Yang [Yang, Y.Y. Gradient estimates for the equation $\Delta u+cu^\{-\alpha \}=0$ on Riemannian manifolds Acta. Math. Sin. 26(B) 2010 1177–1182].},
author = {Zhang, Jing, Ma, Bingqing},
journal = {Communications in Mathematics},
keywords = {gradient estimates; positive solution; Bakry-Emery Ricci tensor; gradient estimates; positive solution; Bakry-Emery Ricci tensor},
language = {eng},
number = {1},
pages = {73-84},
publisher = {University of Ostrava},
title = {Gradient estimates for a nonlinear equation $\Delta _fu+cu^\{-\alpha \}=0$ on complete noncompact manifolds},
url = {http://eudml.org/doc/196367},
volume = {19},
year = {2011},
}

TY - JOUR
AU - Zhang, Jing
AU - Ma, Bingqing
TI - Gradient estimates for a nonlinear equation $\Delta _fu+cu^{-\alpha }=0$ on complete noncompact manifolds
JO - Communications in Mathematics
PY - 2011
PB - University of Ostrava
VL - 19
IS - 1
SP - 73
EP - 84
AB - Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation $\Delta _fu+cu^{-\alpha }=0$ in $M$, where $\alpha $, $c$ are two real constants and $\alpha >0$, $f$ is a smooth real valued function on $M$ and $\Delta _f=\Delta -\nabla f\nabla $. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty $-Bakry-Emery Ricci tensor is bounded from below and $|\nabla f|$ is bounded from above, we also obtain a gradient estimate for positive solutions of the above equation. It extends the results of Yang [Yang, Y.Y. Gradient estimates for the equation $\Delta u+cu^{-\alpha }=0$ on Riemannian manifolds Acta. Math. Sin. 26(B) 2010 1177–1182].
LA - eng
KW - gradient estimates; positive solution; Bakry-Emery Ricci tensor; gradient estimates; positive solution; Bakry-Emery Ricci tensor
UR - http://eudml.org/doc/196367
ER -

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