Gradient estimates for a weighted nonlinear equation on complete noncompact manifolds

Ma, Huang and Luo [12] considered $\Delta u+cu^{\alpha}=0(\alpha<0)$ with $\operatorname{Ric}_{ij}\geq-Kg_{ij}$, and obtained some gradient estimates. In the present paper, we investigate the weighted nonlinear equation $\Delta_{f}u+cu^{-\alpha}=0$ with $\operatorname{Ric}_{f}^{N}\geq-K$, where $f$ is a smooth real-valued function on a complete noncompact Riemannian manifold $(M^{n},g)$, $\alpha>0$ and $c$ are two real constants, and we achieve some gradient estimates for positive solutions of this weighted nonlinear equation. The results posed in this paper can be regarded as a natural generalization of the results in [12].