Abstract

In this article, we study the equation −Gαu+∇Gw⋅∇Gu=∥x∥s|u|p−1u,x=(x,y)∈RN=RN1×RN2, where Gα (resp., ∇G) is Grushin operator (resp. Grushin gradient), p>1 and s≥0. The scalar function w satisfies a decay condition, and ∥x∥ is the norm corresponding to the Grushin distance. Based on the approach by A. Farina [J. Math. Pures Appl. (9) 87, No. 5, 537–561 (2007; Zbl 1143.35041)], we establish a Liouville type theorem for the class of stable sign-changing weak solutions. In particular, we show that the nonexistence result for stable positive classical solutions in [C. Cowan and M. Fazly, Proc. Am. Math. Soc. 140, No. 6, 2003–2012 (2012; Zbl 1242.35024)] is still valid for the above equation.