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 $\mathrm{\Omega },$} det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in $\mathrm{\Omega },$} u_{1}<0, u_{2}<0,& \hbox{in $\mathrm{\Omega },$} u_{1}=u_{2}=0, & \hbox{on $\mathrm{\partial }\mathrm{\Omega },$} \end{array} \right.\end{center} here\Omega$~is a smooth, bounded and strictly convex domain in~${\mathbb{R}}^N$,~$N\ge 2,\alpha >0,\beta >0$. When $\mathrm{\Omega }$ 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.