A Liouville theorem of degenerate elliptic equation and its application

Genggeng Huang

doi:10.3934/dcds.2013.33.4549

Article Contents

2013, Volume 33, Issue 10: 4549-4566. Doi: 10.3934/dcds.2013.33.4549

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A Liouville theorem of degenerate elliptic equation and its application

Genggeng Huang^1,

1.
School of Mathematical Science, Fudan University, Shanghai, 200433

Received: October 31, 2012

Revised: January 31, 2013

Published: September 2013

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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-linear degenerate elliptic equations.

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Mathematics Subject Classification: Primary: 35J61, 35J70; Secondary: 35B53, 35B44.

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