On a power-type coupled system of Monge-Amp\`{e}re equations

We study an elliptic system coupled by Monge-Amp\`{e}re equations: \begin{center} \left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^\alpha, & \hbox{in \Omega,} det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in \Omega,} u_{1}<0, u_{2}<0,& \hbox{in \Omega,} u_{1}=u_{2}=0, & \hbox{on \partial \Omega,} \end{array} \right. \end{center} here $\Omega$~is a smooth, bounded and strictly convex domain in~$\mathbb{R}^{N}$,~$N\geq2,~\alpha >0,~\beta >0$. When $\Omega$ 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 $\alpha,\beta$. When $\alpha>0$, $\beta>0$ and $\alpha\beta=N^2$ we also study a corresponding eigenvalue problem in more general domains.