In this talk, we use the Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation

$\Delta_pv +av^{q}=0$

defined on a complete Riemannian manifolds $(M,g)$ where $p>1$, $a,\ q$ are constants and $\Delta_p(v)=div(|\nabla v|^{p-2}\nabla v)$ is the $p$-Laplace operator. Under some assumptions on $a$, $p$ and $q$, we derive gradient estimates and Liouville type theorems for such positive solutions.