Liouville results for stable solutions of weighted elliptic equations involving the Grushin operator

We examine the following weighted degenerate elliptic equation involving the Grushin operator: \(\begin{aligned} \Delta _s u+\vartheta _{s}(x') |u|^{\theta -1}u =0\;\;\; \text{ in }\,\, \mathbb {R}^N,\;\;N>2, \;\; \theta >1, \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{s}u+{\vartheta }_s(x^{\prime }){|u|}^{\theta -1}u=0\text{in}{\mathbb{R}}^N,N>2,\theta >1,\end{array}$$ where \(x'=(x_{1},...,x_{m})\in \mathbb {R}^m,\) $x^{\prime }=(x_1,...,x_m)\in {\mathbb{R}}^m,$ \(1\le m\le N,\) $1\le m\le N,$ \(\vartheta _{s} \in C(\mathbb {R}^m, \mathbb {R})\) ${\vartheta }_s\in C({\mathbb{R}}^m,\mathbb{R})$ is a continuous positive function satisfying \(\begin{aligned} \displaystyle {\lim _{|x'|_{s}\rightarrow \infty }}\frac{\vartheta _{s}(x')}{|x'|_{s}^{\alpha }}>0,\;\;\; \text{ for } \text{ some }\,\,\alpha >-2, \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle \underset{|x^{\prime }{|}_s\to \infty }{lim}\frac{{\vartheta }_s(x^{\prime })}{|x^{\prime }{|}_s^{\alpha }}>0,\text{for}\text{some}\alpha >-2,}\end{array}$$ and \(\Delta _s\) ${\mathrm{\Delta }}_s$ is an operator of the form \(\begin{aligned} \Delta _s:=\sum _{i=1}^k \partial _{x_{i}}(s_{i}^2\partial _{x_{i}}). \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_s:=\sum _{i=1}^k{\partial }_{x_i}(s_i^2{\partial }_{x_i}).\end{array}$$ Under some general hypotheses of the functions \(s_i,\;i=1,\dots , k,\) $s_i,i=1,\cdots ,k,$ we establish some new Liouville type theorems for stable solutions of this equation for a large classe of weights. Our results recover and considerably improve the previous works (Mtiri in Acta Appl Math 174:7, 2021; Farina and Hasegawa in Proc Royal Soc Edinburgh 150:1567, 2020).