In this paper, we employed the Bernstein method to prove some Liouville-type theorems of the equation ${\mathrm{\Delta }}_{p_1,\cdots ,p_r}v+f(v)=0$. Here, ${\mathrm{\Delta }}_{p_1,\cdots ,p_r}v:=\mathrm{d}\mathrm{i}\mathrm{v}(\sum _{i=1}^r|\mathrm{\nabla }v{|}^{p_i-2}\mathrm{\nabla }v)$. This could be regarded as a natural generalization of the $p$-Laplacian and the $(p,q)$-Laplacian. As applications, we derived Liouville-type theorems of positive solutions to some generalized static Fisher-KPP equation, Allen-Cahn equation, static Newell-Whitehead equation, and Lichnerowicz equation.

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation

$${\mathrm{\Delta }}_f{v}^{\tau }+\lambda (x)v^l=0$$

on metric measure spaces with $m$-Bakry-Emery Ricci curvature bounded from below. Here $\tau >0$ and $l$ were constants, and $\lambda (x)$ was allowed to change sign. As applications, we also obtained a Liouville-type result and Harnack's inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of $\tau$ and $l$.