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 $${\mathrm{\Delta }}_V{u}^m+\mu (x)u+p(x)u^{\alpha }=0,m>1$$on any smooth metric measure space whose $k$-Bakry-Émery curvature is bounded from below by $-(k-1)K$ with $K\ge 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $\mu =0$ the equation becomes $\mathrm{\Delta }{u}^m+p(x)u^{\alpha }=0$. And $V=f$, $\mu =c,p=0$, the equation becomes ${\mathrm{\Delta }}_f{u}^m+cu=0$.