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

Abstract

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.

Keywords

• Liouville type theorem
• Stable weak solutions
• Weighted Kirchhoff equations
• Grushin operator
• Lane–Emden nonlinearity