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

In this paper we study the positive solutions of \(\begin{aligned} \Delta _pv+b|\nabla v|^q+cv^r =0 \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{p}v+b{|\mathrm{\nabla }v|}^q+c{v}^r=0\end{array}$$ on a complete Riemannian manifold (M, g) with Ricci curvature bounded from below, where \(p>1\) $p>1$ , \(q,\, r, \, b\) $q,r,b$ and c are some real constants. If \(bc\ge 0\) $bc\ge 0$ , 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. We also derive explicit expressions of upper bounds for the entire solutions to the above equation. As applications, we reveal some universal local and global properties of positive solutions to the equation. In particular, we obtain new regions of (q, r) for Liouville properties.