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

Abstract

AbstractThis paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: $$\begin{aligned}& -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\& \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$ − 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 , $a+b>0$ a + b > 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 , $p>1+2k$ p > 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 $d>\theta-2$ d > 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.