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In this paper, we apply the moving plane method to the following high orderdegenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in }\mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator$A=y\partial_y^2+a\partial_y+\Delta_x,a\geq 1$. We get a Liouville theorem forsubcritical case and classify the solutions for the critical case.

A Liouville theorem for high order degenerate elliptic equations