Liouville type theorems for Kirchhoff sub-elliptic equations involving $\Delta_\lambda$-operators

Abstract

In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(\|\nabla_\lambda u\|^2)\Delta_\lambda u=w(x)f(u) \quad \mbox{in }\mathbb R^{N}, $$ where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $\Delta_\lambda$ is strongly degenerate and given by $$ \Delta_\lambda=\sum_{j=1}^N \frac{\partial}{\partial x_j}\bigg(\lambda_j^2(x)\frac{\partial }{\partial x_j}\bigg). $$ We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $\lambda_j$, $j=1,\ldots,N$.