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Gradient estimates for the equation Δu + cu α = 0 on Riemannian manifolds

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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.

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Correspondence to Yun Yan Yang.

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Partly supported by National Natural Science Foundation of China (Grant Nos. 1060106, 10811120558) and the program for NCET

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Yang, Y.Y. Gradient estimates for the equation Δu + cu α = 0 on Riemannian manifolds. Acta. Math. Sin.-English Ser. 26, 1177–1182 (2010). https://doi.org/10.1007/s10114-010-7531-y

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