Local and global sharp gradient estimates for weighted $p$-harmonic functions

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\lambda_{1,p}$ on $M$, namely $$e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla v)=-\lambda_{1,p}v^{p-1}.$$ in the distribution sense. We first give a local gradient estimate for $v$ provided the $m$-dimmensional Bakry-\'Emery curvature $Ric_f^{m}$ bounded from below. Consequently, we show that when $Ric_f^m\geq0$ then $v$ is constant if $v$ is of sublinear growth. At the same time, we prove a Harnack inequality for weighted $p$-harmonic functions. Moreover, we show global sharp gradient estimates for weighted $p$-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue $\lambda_{1,p}$ obtains its maximal value. Our achievements generalize several results proved ealier by Li-Wang, Munteanu-Wang,...(\cite{LW1, LW2, MW1, MW2})