Liouville-type theorems for positive solutions to ${\mathrm{\Delta }}_{p_1,\cdots ,p_r}v+f(v)=0$ in ${\mathbb{R}}^m$

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.