Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds

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 $${\mathrm{\Delta }}_{p}v+a\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^q{v}^r=0$$ 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 $p=2$.