A Liouville theorem for high order degenerate elliptic equations

Abstract

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^α\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator $A=y{\partial }_y^2+a{\partial }_y+{\Delta }_x,a\ge 1$. We get a Liouville theorem for subcritical case and classify the solutions for the critical case.