Cheng—Yau logarithmic gradient estimates for positive solutions to $\Delta_f u+p\frac{|\nabla u|^2}{u}+\mu(x)u^{r}=0$ on smooth metric measure spaces

In this paper, through the combination of the Nash—Moser iteration and the Saloff-Coste-type Sobolev inequality, we establish Cheng—Yau logarithmic gradient estimates for solutions to the following nonlinear elliptic equation $$\Delta_f u+p\frac{|\nabla u|^2}{u}+\mu(x)u^{r}=0,$$ which is closely related to the gradient generalized $\tau$-quasi-Einstein metrics. Especially, we obtain the Liouville property and Harnack inequality for this equation when $\mu(x)$ is constant.