Liouville-type theorem for Kirchhoff equations involving Grushin operators

Abstract

AbstractThe aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations: $$\begin{aligned} \begin{aligned}[b] & -M \biggl( \int _{{\mathbb{R}} ^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)e^{u}, \\ &\quad z=(x,y) \in R^{N}=R^{N_{1}}\times R^{N_{2}} \end{aligned} \end{aligned}$$ − M ( ∫ R N ω ( z ) | ∇ G u | 2 d z ) div G ( ω ( z ) ∇ G u ) = f ( z ) e u , z = ( x , y ) ∈ R N = R N 1 × R N 2 and $$\begin{aligned} \begin{aligned}[b] & M \biggl( \int _{\mathbb{R}^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)u^{-q}, \\ &\quad z=(x,y) \in {\mathbb{R}} ^{N}={\mathbb{R}} ^{N_{1}}\times {\mathbb{R}} ^{N_{2}}, \end{aligned} \end{aligned}$$ M ( ∫ R N ω ( z ) | ∇ G u | 2 d z ) div G ( ω ( z ) ∇ G u ) = f ( z ) u − q , z = ( x , y ) ∈ R N = R N 1 × R N 2 , where $M(t)=a+bt^{k}$ M ( t ) = a + b t k , $t\geq 0$ t ≥ 0 , with $a>0$ a > 0 , $b, k\geq 0$ b , k ≥ 0 , $k=0$ k = 0 if and only if $b=0$ b = 0 . $q>0$ q > 0 and $\omega (z), f(z)\in L^{1}_{\mathrm{loc}}({\mathbb{R}} ^{N})$ ω ( z ) , f ( z ) ∈ L loc 1 ( R N ) are nonnegative functions satisfying $\omega (z)\leq C_{1}\|z \|_{G}^{\theta }$ ω ( z ) ≤ C 1 ∥ z ∥ G θ and $f(z)\geq C_{2}\|z\|_{G}^{d}$ f ( z ) ≥ C 2 ∥ z ∥ G d as $\|z\|_{G} \geq R_{0}$ ∥ z ∥ G ≥ R 0 with $d>\theta -2$ d > θ − 2 , $R_{0}$ R 0 , $C_{i}$ C i ($i=1,2$ i = 1 , 2 ) are some positive constants, here $\alpha \geq 0$ 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 ) is the norm corresponding to the Grushin distance. $N_{\alpha }=N_{1}+(1+\alpha )N_{2}$ N α = N 1 + ( 1 + α ) N 2 is the homogeneous dimension of ${\mathbb{R}} ^{N}$ R N . $\operatorname{div}_{G}$ div G (resp., $\nabla _{G}$ ∇ G ) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and $N_{\alpha }$ N α , the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.