In this paper, we consider the nonlinear elliptic equation $$Δ_fv^τ+λv=0$$ on a complete smooth metric measure space with $m$-Bakry-Émery Ricci curvature bounded from below, where $τ>0$ and $λ$ are constant. We obtain some new local gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem and a Harnack inequality, and the global gradient estimates for such solutions. Our results generalize and improve the estimates in Wang (J. Differential Equations 260:567-585, 2016) and Zhao (Arch. Math. (Basel) 114:457-469, 2020).

In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian: \begin{document}$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $\end{document} on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.

In this paper, we employ the Nash–Moser iteration technique and the Saloff–Coste’s Sobolev inequality to study the local and global properties of positive solutions to the Trudinger equation ${\Delta }_{p,f}{u}^{{\displaystyle \frac{1}{p-1}}}+b{u}^q+c{u}^r=0$ on a complete smooth metric measure space with m-Bakry-Émery Ricci curvature bounded from below, where $b,c\in \mathbb{R}$, $p>1$, and $q\le r$ are real constants. We first give universal gradient estimates for the above equation under certain assumptions on b, c, p, q, and r. As their natural corollary, Harnack inequalities and Liouville-type theorems for positive solutions are obtained. Later, we consider the explicit global gradient estimates for such entire solutions through the global gradient estimates obtained.