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

Abstract: In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$Δ<em>{V}u^{{m}+μ(x)u+p(x)u^α=0} , \quad m&gt;1$$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 $Δu<sup>{m}+p(x)u<sup>α=0$. And $V=f$, $μ=c, p=0$, the equation becomes $Δ</em>{f}u^{m}+cu=0$.