Uniqueness for a system of Monge–Ampère equations

In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge–Ampère equations\begin{cases}\operatorname {det} D^2 u = \gamma \lvert v \rvert p & \textrm{in} \; \Omega , \\\operatorname {det} D^2 v = \mu {\lvert u \rvert }^{n^ 2 / p} & \textrm{in} \; \Omega , \\\quad u = v = 0 & \textrm{on} \; \partial \Omega\end{cases}on bounded, smooth and uniformly convex domains $\Omega \subset \mathbb{R}^n$ provided that $p$ is close to $n \geq 2$. When $p = n$, we show that the uniqueness holds for general bounded convex domains $\Omega \subset \mathbb{R}^n$.

Keywords

system of Monge–Ampère equations, uniqueness, eigenvalue problem

2010 Mathematics Subject Classification

35A02, 35J70, 35J96, 47A75

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The research of the author was supported in part by the National Science Foundation under grant DMS-1764248.

Received 11 December 2019

Accepted 9 June 2020

Published 1 December 2021