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 $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$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 $.