This paper deals with the second-order semi-linear degenerate elliptic equation yuyy+buy+Δxu+|u|α−1u=0,(x,y)∈Rn×(0,∞), where n≥1,α>1. We establish a Liouville theorem of stable solution of the degenerate equation mentioned above by using the energy method. The classification results for stable solutions belonging to C2 can be thought of as an analogue of the recent results of Farina for the Lane–Emden equation.