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

Abstract

We examine the degenerate elliptic system $$-Δ_{s} u = v^p, \quad -Δ_{s} v= u^θ, \quad u,v>0 \quad\mbox{in }\; \mathbb{R}^N=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where }\;\;\;\; s \geq 0\;\; \mbox{and} \;\;p,θ>0.$$ We prove that the system has no smooth stable solution provided $p,\theta >0$ and $N_s<2+\alpha +\beta ,$ where $$α= \frac{2(p+1)}{pθ- 1} \quad\mbox{and} \quad β= \frac{2(θ+1)}{pθ- 1}.$$ 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 $0<p<1.$