Boundary Value Problems (Jan 2020)
Liouville-type theorem for Kirchhoff equations involving Grushin operators
Abstract
Abstract The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations: 0.1 − 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 $$\begin{aligned} \begin{aligned}[b] & -M \biggl( \int _{{\mathbb{R}} ^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)e^{u}, \\ &\quad z=(x,y) \in R^{N}=R^{N_{1}}\times R^{N_{2}} \end{aligned} \end{aligned}$$ and 0.2 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 , $$\begin{aligned} \begin{aligned}[b] & M \biggl( \int _{\mathbb{R}^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)u^{-q}, \\ &\quad z=(x,y) \in {\mathbb{R}} ^{N}={\mathbb{R}} ^{N_{1}}\times {\mathbb{R}} ^{N_{2}}, \end{aligned} \end{aligned}$$ where M ( t ) = a + b t k $M(t)=a+bt^{k}$ , t ≥ 0 $t\geq 0$ , with a > 0 $a>0$ , b , k ≥ 0 $b, k\geq 0$ , k = 0 $k=0$ if and only if b = 0 $b=0$ . q > 0 $q>0$ and ω ( z ) , f ( z ) ∈ L loc 1 ( R N ) $\omega (z), f(z)\in L^{1}_{\mathrm{loc}}({\mathbb{R}} ^{N})$ are nonnegative functions satisfying ω ( z ) ≤ C 1 ∥ z ∥ G θ $\omega (z)\leq C_{1}\|z \|_{G}^{\theta }$ and f ( z ) ≥ C 2 ∥ z ∥ G d $f(z)\geq C_{2}\|z\|_{G}^{d}$ as ∥ z ∥ G ≥ R 0 $\|z\|_{G} \geq R_{0}$ with d > θ − 2 $d>\theta -2$ , R 0 $R_{0}$ , C i $C_{i}$ ( i = 1 , 2 $i=1,2$ ) are some positive constants, here α ≥ 0 $\alpha \geq 0$ and ∥ z ∥ G = ( | x | 2 ( 1 + α ) + | y | 2 ) 1 2 ( 1 + α ) $\|z\|_{G}=(|x|^{2(1+ \alpha )}+|y|^{2})^{\frac{1}{2(1+\alpha )}}$ is the norm corresponding to the Grushin distance. N α = N 1 + ( 1 + α ) N 2 $N_{\alpha }=N_{1}+(1+\alpha )N_{2}$ is the homogeneous dimension of R N ${\mathbb{R}} ^{N}$ . div G $\operatorname{div}_{G}$ (resp., ∇ G $\nabla _{G}$ ) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and N α $N_{\alpha }$ , the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.
Keywords
