Cheng-Yau type gradient estimates for $\Delta_{f} v^{\tau}+\lambda(x) v^{l}=0$ on smooth metric measure spaces

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Abstract

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation

$\Delta_{f} v^{\tau}+\lambda(x) v^{l}=0$

on metric measure spaces with m-Bakry-Emery Ricci curvature bounded from below. Here τ > 0 and l were constants, and λ(x) was allowed to change sign. As applications, we also obtained a Liouville- type result and Harnack’s inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of τ and l.

Keywords

nonlinear elliptic equation / gradient estimate / Liouville theorem / Harnack inequality

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