Liouville theorem for Lane-Emden equation of Baouendi-Grushin operators

Abstract

In this paper, we establish a Liouville theorem for solutions to the Lane-Emden equation involving Baouendi-Grushin operator:$-({\Delta }_{x}u+{(\alpha +1)}^2|x{|}^{2\alpha }{\Delta }_{y}u)=|u{|}^{p-1}u,$ where $(x,y)\in R^N=R^m\times R^n$ with $m\ge 1$, $n\ge 1$, and $\alpha \ge 0$. We focus on solutions that are stable outside a compact set. Specifically, we prove that for $p>1$, when p is smaller than the Joseph–Lundgren exponent and differs from the Sobolev exponent, $u=0$ is the unique $C^2$ solution stable outside a compact set. This work extends the results obtained by Farina (J. Math. Pures Appl., 87 (5) (2007), 537–561).