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Gradient estimates for the equation Δu + cu −α = 0 on Riemannian manifolds
- Source :
- Acta Mathematica Sinica, English Series. 26:1177-1182
- Publication Year :
- 2010
- Publisher :
- Springer Science and Business Media LLC, 2010.
-
Abstract
- Let (M, g) be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity: $$ \Delta u\left( x \right) + cu^{ - \alpha } \left( x \right) = 0 in M $$ , where α > 0, c are two real constants. When c < 0 and M is a bounded smooth domain in ℝn, the above equation is known as the thin film equation, which describes a steady state of the thin film (see Guo-Wei [Manuscripta Math., 120, 193–209 (2006)]). The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233–256 (1991)], where the nonlinearity is the positive power of u.
Details
- ISSN :
- 14397617 and 14398516
- Volume :
- 26
- Database :
- OpenAIRE
- Journal :
- Acta Mathematica Sinica, English Series
- Accession number :
- edsair.doi...........b4b9efb4860baaf510260abc0b67bf5d