Liouville-type theorems for positive solutions to $\Delta_{p_{1}, \cdots, p_{r}} v+f(v)=0 \text { in } \mathbb{R}^{m}$

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Abstract

In this paper, we employed the Bernstein method to prove some Liouville-type theorems of the equation $\Delta_{p_{1}, \cdots, p_{r}} v+f(v)=0 \text {. Here, } \Delta_{p_{1}, \cdots, p_{r}} v:=\operatorname{div}\left(\sum_{i=1}^{r}|\nabla v|^{p_{i}-2} \nabla v\right)$. 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.

Keywords

Liouville-type theorems / (p₁, ...,p_r)-Laplacian / nonlinear elliptic equations

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