Universal gradient estimates for solutions of ${\mathrm{\Delta }}_{p,f}u+a{u}^{\sigma }\ln u=0$ on complete Riemannian manifolds

In this paper, we consider the weighted $p$-Laplacian equation $${\mathrm{\Delta }}_{p,f}u+a{u}^{\sigma }\ln u=0$$ defined on a complete smooth metric measure space under the conditon that the $m$-Bakry-Émery Ricci curvature has a lower bound, where $a$, $\sigma$ are two nonzero real constants. By applying the Nash-Moser iteration, we obtain sharp gradient estimates and thereby establish Liouville theorems for the above equation.