Liouville theorems and new gradient estimates for positive solutions to $\Delta_pv+a(v+b)^q=0$ on a complete manifold

In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation $\Delta_pv+a(v+b)^q=0$ defined on a complete Riemannian manifold $\left(M,g\right)$ with Ricci lower bound, where $p>1$ is a constant and $\Delta_pv=\mathrm{div}\left(\left|\nabla v\right|^{p-2}\nabla v\right)$ is the usual $p$-Laplace operator. Under certain assumptions on $a$, $p$ and $q$, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on $a$, $b$, $p$ and $q$ we show whether or not the exact Cheng-Yau $\log$-gradient estimates for the positive solutions to $\Delta_pv+av^q=0$ on $\left(M,g\right)$ with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau $\log$-gradient estimates are established.