Gradient estimate for solutions of the equation ${\mathrm{\Delta }}_{p}v+a{v}^q=0$ on a complete Riemannian manifold

In this paper, we use Nash-Moser iteration method to study the local and global behaviours of positive solutions to the nonlinear elliptic equation ${\mathrm{\Delta }}_{p}v+a{v}^q=0$ defined on a complete Riemannian manifolds $(M,g)$ where $p>1$, $a$ and $q$ are constants. Under some assumptions on $a$, $p$ and $q$, we derive gradient estimates and Liouville type theorems for such positive solutions.