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