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

Abstract

We examine the following weighted degenerate elliptic equation involving the Grushin operator:

$$\begin{aligned} \Delta _s u+\vartheta _{s}(x') |u|^{\theta -1}u =0\;\;\; \text{ in }\,\, \mathbb {R}^N,\;\;N>2, \;\; \theta >1, \end{aligned}$$

where \(x'=(x_{1},...,x_{m})\in \mathbb {R}^m,\) \(1\le m\le N,\) \(\vartheta _{s} \in C(\mathbb {R}^m, \mathbb {R})\) is a continuous positive function satisfying

$$\begin{aligned} \displaystyle {\lim _{|x'|_{s}\rightarrow \infty }}\frac{\vartheta _{s}(x')}{|x'|_{s}^{\alpha }}>0,\;\;\; \text{ for } \text{ some }\,\,\alpha >-2, \end{aligned}$$

and \(\Delta _s\) is an operator of the form

$$\begin{aligned} \Delta _s:=\sum _{i=1}^k \partial _{x_{i}}(s_{i}^2\partial _{x_{i}}). \end{aligned}$$

Under some general hypotheses of the functions \(s_i,\;i=1,\dots , k,\) 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).