On the Classification of Solutions to a Weighted Elliptic System Involving the Grushin Operator

Abstract

We investigate here the following weighted degenerate elliptic system

$$\begin{aligned} &-\Delta _{s} u =\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{ \frac{\alpha }{2(s+1)}} v^{p}, \quad -\Delta _{s} v= \Big(1+\| \mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha }{2(s+1)}}u^{\theta }, \\ &\quad u,v>0 \quad \mbox{in }\; \mathbb{R}^{N}:=\mathbb{R}^{N_{1}}\times \mathbb{R}^{N_{2}}, \end{aligned}$$

where $\Delta _{s}=\Delta _{x}+|x|^{2s}\Delta _{y}$, is the Grushin operator, $s$, $\alpha \geq 0$ and $1< p\leq \theta$. Here

$$ \|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^{2}\Big)^{\frac{1}{2(s+1)}} \; \mbox{and}\quad \mathbf{x}:=(x, y)\in \mathbb{R}^{N}:=\mathbb{R}^{N_{1}} \times \mathbb{R}^{N_{2}}. $$

In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results (Duong and Phan in J. Math. Anal. Appl. 454(2):785–801, 2017; Hajlaoui et al. in Discrete Contin. Dyn. Syst. 37:265–279, 2017).