Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

We examine the degenerate elliptic system $$-{\mathrm{\Delta }}_{s}u=v^p,-{\mathrm{\Delta }}_{s}v=u^{\theta },u,v>0\text{quadin}{\mathbb{R}}^N={\mathbb{R}}^{N_1}\times {\mathbb{R}}^{N_2},\text{quadwhere}s\ge 0andp,\theta >0.$$ We prove that the system has no smooth stable solution provided $p,\theta >0$ and $N_s< 2 + \alpha + \beta,$where$$\alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}.$$ This result is an extension of some result in \cite{ MY}. In particular, we establish a new the integral estimate for $u$ and $v$ ;(see Proposition 1.1), which is crucial to deal with the case $0<p<1.$