| Title:
|
Gradient estimates for a nonlinear equation $\Delta_fu+cu^{-\alpha}=0$ on complete noncompact manifolds (English) |
| Author:
|
Zhang, Jing |
| Author:
|
Ma, Bingqing |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
19 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
73-84 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
gradient estimates |
| Keyword:
|
positive solution |
| Keyword:
|
Bakry-Emery Ricci tensor |
| MSC:
|
35J60 |
| MSC:
|
58J05 |
| idZBL:
|
Zbl 1242.58011 |
| idMR:
|
MR2855392 |
| . |
| Date available:
|
2011-10-31T08:15:31Z |
| Last updated:
|
2013-10-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141680 |
| . |
| Reference:
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| . |
