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 -

References

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  1. Calabi, E., 10.1215/S0012-7094-58-02505-5, Duke Math. J. 25 1957 45–46 (1957) MR0092069DOI10.1215/S0012-7094-58-02505-5
  2. Chen, L., Chen, W.Y., 10.1007/s10455-008-9141-9, Ann. Glob. Anal. Geom. 35 2009 397–404 (2009) Zbl1177.35040MR2506242DOI10.1007/s10455-008-9141-9
  3. Chen, L., Chen, W.Y., Gradient estimates for positive smooth f -harmonic functions, Acta Math. Sci. 30(B) 2010 1614–1618 (2010) Zbl1240.58019MR2778630
  4. Cheng, S.Y., Yau, S.T., 10.1002/cpa.3160280303, Commun. Pure. Appl. Math. 28 1975 333–354 (1975) Zbl0312.53031MR0385749DOI10.1002/cpa.3160280303
  5. Guo, Z.M., Wei, J.C., 10.1007/s00229-006-0001-2, Manuscripta Math. 120 2006 193–209 (2006) MR2234248DOI10.1007/s00229-006-0001-2
  6. Hsu, S.Y., Gradient estimates for a nonlinear parabolic equation under Ricci, arXiv: 0806.4004 
  7. Huang, G.Y., Ma, B.Q., 10.1007/s00013-009-0091-7, Arch. Math. (Basel) 94 2010 265–275 (2010) Zbl1194.58020MR2602453DOI10.1007/s00013-009-0091-7
  8. Li, J.Y., 10.1016/0022-1236(91)90110-Q, J. Funct. Anal. 100 1991 233–256 (1991) Zbl0746.58078MR1125225DOI10.1016/0022-1236(91)90110-Q
  9. Li, X.D., Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 2005 1295–1361 (2005) Zbl1082.58036MR2170766
  10. Ma, L., 10.1016/j.jfa.2006.06.006, J. Funct. Anal. 241 2006 374–382 (2006) Zbl1112.58023MR2264255DOI10.1016/j.jfa.2006.06.006
  11. Ma, L., Liu, B.Y., Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math. 14 2008 393–401 (2008) Zbl1156.35065MR2443979
  12. Ma, L., Liu, B.Y., 10.2140/pjm.2009.240.343, Pacific J. Math. 240 2009 343–361 (2009) Zbl1162.35059MR2485469DOI10.2140/pjm.2009.240.343
  13. Qian, Z.M., 10.1023/A:1008698923961, Potential Analysis 8 1998 137–142 (1998) Zbl0930.58012MR1618434DOI10.1023/A:1008698923961
  14. Wei, G.F., Wylie, W., Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geometry 83 2009 377–405 (2009) Zbl1189.53036MR2577473
  15. Yang, Y.Y., 10.1090/S0002-9939-08-09398-2, Proc. Amer. Math. Soc. 136 2008 4095–4102 (2008) Zbl1151.58013MR2425752DOI10.1090/S0002-9939-08-09398-2
  16. Yang, Y.Y., 10.1007/s10114-010-7531-y, Acta. Math. Sin. 26(B) 2010 1177–1182 (2010) Zbl1203.58006MR2644055DOI10.1007/s10114-010-7531-y

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