  In this paper, we consider the non-linear general $p$-Laplacian equation
$\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure
spaces. Assume that a Sobolev inequality holds true on $M$ and an integral
Ricci curvature is small, we first prove a local gradient estimate for the
equation. Then, as its applications, we prove several Liouville type results on
manifolds with lower bounds of Ricci curvature. We also derive new local
gradient estimates provided that the integral Ricci curvature is small enough.
