Existence and non-existence of solutions for a class of Monge-Ampère equations. (English) Zbl 1165.35023

Zhang, Zhitao; Wang, Kelei

J. Differ. Equations 246, No. 7, 2849-2875 (2009).

Summary: We study the boundary value problems for Monge-Ampère equations: \(\det D^2u=e^{-u}\) in \(\Omega \subset \mathbb R^n\), \(n \geq 1\), \(u|_{\partial \Omega }=0\). First we prove by the argument of moving plane that any solution on the ball is radially symmetric. Then we show that there exists a critical radius such that if the radius of a ball is smaller than this critical value then a solution exists, 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 \(\det D^2u=e^{-tu}\) in \(\Omega\), \(u|_{\partial \Omega }=0\), \(t\geq 0\). 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.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B45	A priori estimates in context of PDEs
35B32	Bifurcations in context of PDEs

Keywords:

Monge-Ampère equations; moving plane; implicit function theorem; Leray-Schauder degree theory; bifurcation

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