摘要：This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: $$\begin{aligned}& -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\& \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$ where $M(t)=a bt^{k}$ , $t\geq0$ , with $a,b,k\geq0$ , $a b>0$ , $k=0$ if and only if $b=0$ . Let $N=N_{1} N_{2}\geq2$ , $p>1 2k$ and $\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}$ be nonnegative functions such that $\xi(z)\leq C\ z\ _{G}^{\theta}$ and $\eta(z)\geq C'\ z\ _{G}^{d}$ for large $\ z\ _{G}$ with $d>\theta-2$ . Here $\alpha\geq0$ and $\ z\ _{G}=( x ^{2(1 \alpha)} y ^{2})^{\frac{1}{2(1 \alpha)}}$ . $\operatorname{div}_{G}$ (resp., $\nabla_{G}$ ) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k, θ, d, and $N_{\alpha}=N_{1} (1 \alpha)N_{2}$ , the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function M could be zero, which implies that the above problem is degenerate.
