Existence and non-existence of solutions for a class of Monge-Ampere equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:246

Article

Zhang, Zhitao¹  Wang, Kelei¹ 

[1] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China

关键词: Monge-Ampere equations;    Moving plane;    Implicit Function Theorem;    Leray-Schauder degree theory;    Bifurcation;   

DOI  :  10.1016/j.jde.2009.01.004

来源: Elsevier

【 摘 要 】

We study the boundary Value problems for Monge-Ampere equations: det D(2)u = e(-u) in Omega subset of R-n, n >= 1, u vertical bar(partial derivative Omega) = 0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a Solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter det D(2)u = e(-tu) in Omega, u vertical bar(partial derivative Omega) = 0, t >= 0 . By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point: by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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