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This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations:$$\begin{aligned}&amp; -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\&amp; \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$   − M ( ∫ R N ξ ( z ) | ∇ G u | 2 d z ) div G ( ξ ( z ) ∇ G u ) = η ( z ) | u | p − 1 u , z = ( x , y ) ∈ R N = R N 1 × R N 2 , where    $M(t)=a+bt^{k}$  M ( t ) = a + b t k ,    $t\geq0$  t ≥ 0 , with    $a,b,k\geq0$  a , b , k ≥ 0 ,    $a+b&gt;0$  a + b &gt; 0 ,    $k=0$  k = 0 if and only if    $b=0$  b = 0 . Let    $N=N_{1}+N_{2}\geq2$  N = N 1 + N 2 ≥ 2 ,    $p&gt;1+2k$  p &gt; 1 + 2 k and    $\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}$  ξ ( z ) , η ( z ) ∈ L loc 1 ( R N ) ∖ { 0 } be nonnegative functions such that    $\xi(z)\leq C\|z\|_{G}^{\theta}$  ξ ( z ) ≤ C ∥ z ∥ G θ and    $\eta(z)\geq C'\|z\|_{G}^{d}$  η ( z ) ≥ C ′ ∥ z ∥ G d for large    $\|z\|_{G}$  ∥ z ∥ G with    $d&gt;\theta-2$  d &gt; θ − 2 . Here    $\alpha\geq0$  α ≥ 0 and    $\|z\|_{G}=(|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}$  ∥ z ∥ G = ( | x | 2 ( 1 + α ) + | y | 2 ) 1 2 ( 1 + α ) .    $\operatorname{div}_{G}$  div G (resp.,    $\nabla_{G}$  ∇ G ) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on  k ,  θ ,  d , and    $N_{\alpha}=N_{1}+(1+\alpha)N_{2}$  N α = N 1 + ( 1 + α ) 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.

Stable weak solutions to weighted Kirchhoff equations of Lane–Emden type