  In this paper, we provide a new routine to employ the Nash-Moser iteration
technique to analyze the local and global properties of positive solutions to
the equation $$\Delta_pv + a|\nabla v|^qv^r =0$$ on a complete Riemannian
manifold with Ricci curvature bounded from below, where $p>1$, $q$, $r$ and $a$
are some real constants. Assuming certain conditions on $a,\, p,\, q$ and $r$,
we can derive universal and succinct Cheng-Yau type logarithmic gradient
estimates for such solutions. In particular, we give the obvious expressions of
constants in the logarithmic gradient estimate for entire solutions to the
above equation (see \thmref{t10}). The gradient estimates enable us to obtain
some Liouville-type theorems, Harnack inequalities and some local estimates
near singularities for positive solutions. Some of our results are new even in
the case the domain is an Euclidean space and $p=2$.
