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A Liouville theorem of degenerate elliptic equation and its application

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  • 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-linear degenerate elliptic equations.
    Mathematics Subject Classification: Primary: 35J61, 35J70; Secondary: 35B53, 35B44.


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