Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$Δ_{V}u^{m}+μ(x)u+p(x)u^α=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-Émery 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 $V=0$, $μ=0$ the equation becomes $Δu^{m}+p(x)u^α=0$. And $V=f$, $μ=c, p=0 $, the equation becomes $Δ_{f}u^{m}+cu=0 $.