Gradient estimate for solutions of the equation \(\Delta _pv +av^{q}=0\) on a complete Riemannian manifold

In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation \(\Delta _pv +av^{q}=0\) ${\mathrm{\Delta }}_{p}v+a{v}^q=0$ defined on a complete Riemannian manifolds (M, g) where \(p>1\) $p>1$ , a and q are constants and \(\Delta _p(v)=\textrm{div}(|\nabla v|^{p-2}\nabla v)\) ${\mathrm{\Delta }}_p(v)={\text{div}(|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v)$ is the p-Laplace operator. Under some assumptions on a, p and q, we derive gradient estimates and Liouville type theorems for such positive solutions.