  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 $p$.
