A Liouville theorem of degenerate elliptic equation and its application

Abstract

In this paper, we apply the moving plane method to the following degenerate elliptic equation arising from isometric embedding,\begin{equation*}yu_{yy}+au_y+\Delta_x u+u^\alpha=0\text{ in } \mathbb R^{n+1}_+,n\geq 1.\end{equation*} We get a Liouville theorem for subcritical case and classify the solutions for critical case.As an application, we derive the a priori bounds for positive solutions of some semi-lineardegenerate elliptic equations.