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

<p style='text-indent:20px;'>We examine the following degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{s} u \! = v^p, \quad v\! u^\theta, \;\; u, v>0 \;\;\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. $\end{document} </tex-math></disp-formula></p><p prove that system has no stable solution provided <inline-formula><tex-math id="M1">\begin{document}$ p, >0 $\end{document}</tex-math></inline-formula> and id="M2">\begin{document}$ N_s: N_1+(1+s)N_2< 2 + \alpha \beta, where</p><p id="FE2"> \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \beta \frac{2(\theta +1)}{p\theta 1}. style='text-indent:20px;'>This result is an extension of some results in [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we establish a new integral estimate for id="M3">\begin{document}$ id="M4">\begin{document}$ v (see Proposition 1.1), which crucial to deal with case id="M5">\begin{document}$ 0 < p 1. $\end{document}</tex-math></inline-formula></p>