In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{ V}u^{ m}+\mu(x)u+p(x)u^{ \alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{ E}mery curvature is bounded from below by $-(k-1)K$ with $K \geq 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $\mu=0$ the equation becomes $\Delta u^{ m}+p(x)u^{ \alpha}=0$. And $V=f$, $\mu=c, p=0 $, the equation becomes $\Delta_{ f}u^{ m}+cu=0 $.
Categories: math.AP math.DG