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In this paper we study the positive solutions to a nonlinear ellipticequation $$\Delta_pv+b|\nabla v|^q+cv^r =0$$ defined on a complete Riemannianmanifold $(M,g)$ with Ricci curvature bounded from below, where $p>1$, $q,\, r,\, b$ and $c$ are some real constants. If $p>1$ is given and $bc\geq 0$, weprovide a new routine to give some regions of $(q, r)$ such that theCheng-Yau's logarithmic gradient estimates hold true exactly on such givenregions. In particular, we derive the upper bounds of the constants $c(n, p, q,r)$ in the Cheng-Yau's gradient estimates for the entire solutions to the aboveequation. As applications, we reveal some universal local and global propertiesof positive solutions to the equation. On the other hand, we extend someresults due to \cite{MR1879326} to the case the domain of the equation is acomplete manifold and obtain wider ranges of $(q,r)$ for Liouville properties.

On the universal local and global properties of positive solutions to $\Delta_pv+b|\nabla v|^q+cv^r=0$ on complete Riemannian manifolds