Gradient estimates for positive weak solution to $\Delta_pu+au^{\sigma}=0$ on Riemannian manifolds

In this paper, we study gradient estimates for positive weak solutions to the following $p$-Laplacian equation $$\Delta_pu+au^{\sigma}=0$$ on a Riemannian manifold, where $p>1$ and $a,\sigma$ are two nonzero real constants. By virtue of the Morser iteration technique, we derive some gradient estimates, which show that when the Ricci curvature is nonnegative, the above equation does not admit positive weak solutions under some scopes of $p$.