摘要：The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations:0.1 $$\begin{aligned} \begin{aligned}[b] & -M \biggl( \int _{{\mathbb{R}} ^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)e^{u}, \\ &\quad z=(x,y) \in R^{N}=R^{N_{1}}\times R^{N_{2}} \end{aligned} \end{aligned}$$ and0.2 $$\begin{aligned} \begin{aligned}[b] & M \biggl( \int _{\mathbb{R}^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)u^{-q}, \\ &\quad z=(x,y) \in {\mathbb{R}} ^{N}={\mathbb{R}} ^{N_{1}}\times {\mathbb{R}} ^{N_{2}}, \end{aligned} \end{aligned}$$ where $M(t)=a+bt^{k}$, $t\geq 0$, with $a>0$, $b, k\geq 0$, $k=0$ if and only if $b=0$. $q>0$ and $\omega (z), f(z)\in L^{1}_{\mathrm{loc}}({\mathbb{R}} ^{N})$ are nonnegative functions satisfying $\omega (z)\leq C_{1}\ z \ _{G}^{\theta }$ and $f(z)\geq C_{2}\ z\ _{G}^{d}$ as $\ z\ _{G} \geq R_{0}$ with $d>\theta -2$, $R_{0}$, $C_{i}$ ($i=1,2$) are some positive constants, here $\alpha \geq 0$ and $\ z\ _{G}=( x ^{2(1+ \alpha )}+ y ^{2})^{\frac{1}{2(1+\alpha )}}$ is the norm corresponding to the Grushin distance. $N_{\alpha }=N_{1}+(1+\alpha )N_{2}$ is the homogeneous dimension of ${\mathbb{R}} ^{N}$. $\operatorname{div}_{G}$ (resp., $\nabla _{G}$) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and $N_{\alpha }$, the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated..
