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In this paper, we study gradient estimates for positive weak solutions to thefollowing $p$-Laplacian equation $$\Delta_pu+au^{\sigma}=0$$ on a Riemannianmanifold, where $p>1$ and $a,\sigma$ are two nonzero real constants. By virtueof the Morser iteration technique, we derive some gradient estimates, whichshow that when the Ricci curvature is nonnegative, the above equation does notadmit positive weak solutions under some scopes of $p$.

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