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{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = γ|v|^p~&&\text{in} ~ Ω, \\\ \det D^2 v~& = μ|u|^{n^2/p}~&&\text{in} ~ Ω, \\\ u=v &= 0~&&\text{on}~ \partialΩ\end{alignedat} \right. \end{equation*} on bounded, smooth and uniformly convex domains $Ω\subset R^n$ provided that $p$ is close to $n\geq 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $Ω\subset R^n$.

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