On the universal local and global properties of positive solutions to ${\mathrm{\Delta }}_{p}v+b\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^q+c{v}^r=0$ on complete Riemannian manifolds

In this paper we study the positive solutions to a nonlinear elliptic equation $${\mathrm{\Delta }}_{p}v+b\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^q+c{v}^r=0$$ defined on a complete Riemannian manifold $(M,g)$ with Ricci curvature bounded from below, where $p>1$, $q,r,b$ and $c$ are some real constants. If $p>1$ is given and $bc\ge 0$, we provide a new routine to give some regions of $(q,r)$ such that the Cheng-Yau’s logarithmic gradient estimates hold true exactly on such given regions. 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 above equation. As applications, we reveal some universal local and global properties of positive solutions to the equation. On the other hand, we extend some results due to \cite{MR1879326} to the case the domain of the equation is a complete manifold and obtain wider ranges of $(q,r)$ for Liouville properties.