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

We study the boundary value problems for Monge–Ampère equations: detD2u=e−u in Ω⊂Rn, n⩾1, u|∂Ω=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 detD2u=e−tu in Ω, u|∂Ω=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.