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

We examine the degenerate elliptic system $$-\Delta_{s} u = v^p, \quad -\Delta_{s} v= u^\theta, \quad u,v>0 \quad\mbox{in }\; \mathbb{R}^N=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where }\;\;\;\; s \geq 0\;\; \mbox{and} \;\;p,\theta >0.$$ 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 $0 < p < 1.$