Liouville type theorems for Kirchhoff sub-elliptic equations involving $Δ_λ$-operators - Nguyen, Thi Thu Huong; Quyet, Dao Trong; Vu, Thi Hien Anh (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(\| abla_λ u\|^2)Δ_λ u=w(x)f(u)\quad in \mathbb R^{N}, $$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 $Δ_λ$ is strongly degenerate and given by $$Δ_λ=\sum_{j=1}^N \frac{\partial}{\partial x_j}\bigg(λ_j^2(x)\frac{\partial }{\partial x_j}\bigg).$$We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $λ_j$, $j=1,…,N$.