Uniqueness for a system of Monge-Amp\`ere equations

Abstract

In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Amp\`ere equations \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = \gamma |v|^p~&&\text{in} ~ \Omega, \\\ \det D^2 v~& = \mu |u|^{n^2/p}~&&\text{in} ~ \Omega, \\\ u=v &= 0~&&\text{on}~ \partial\Omega \end{alignedat} \right. \end{equation*} on bounded, smooth and uniformly convex domains $\Omega\subset R^n$ provided that $p$ is close to $n\geq 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $\Omega\subset R^n$.