On a power-type coupled system of Monge-Ampère equations

We study an elliptic system coupled by Monge-Ampère equations: \begin{center} $\left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^α, & \hbox{in $Ω,$} det~D^{2}u_{2}={(-u_{1})}^β, & \hbox{in $Ω,$} u_{1}<0, u_{2}<0,& \hbox{in $Ω,$} u_{1}=u_{2}=0, & \hbox{on $ \partial Ω,$} \end{array} \right.$ \end{center} here $Ω$~is a smooth, bounded and strictly convex domain in~$\mathbb{R}^{N}$,~$N\geq2,~α>0,~β>0$. When $Ω$ is the unit ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on $α,β$. When $α>0$, $β>0$ and $αβ=N^2$ we also study a corresponding eigenvalue problem in more general domains.

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