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In this paper, we use Nash-Moser iteration method to study the local andglobal behaviours of positive solutions to the nonlinear elliptic equation$\Delta_pv +av^{q}=0$ defined on a complete Riemannian manifolds $(M,g)$ where$p>1$, $a$ and $q$ are constants. Under some assumptions on $a$, $p$ and $q$,we derive gradient estimates and Liouville type theorems for such positivesolutions.

Gradient estimate for solutions of the equation $\Delta_p v+av^{q}=0$ on a complete Riemannian manifold