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

We study an elliptic system coupled by Monge--Amp\`{e}re equations:$$\begin{cases} \det D^{2}u_{1}={(-u_{2})}^\alpha & \hbox{in $\Omega,$} \\ \det D^{2}u_{2}={(-u_{1})}^\beta & \hbox{in $\Omega,$} \\ u_{1}<0,\ u_{2}<0& \hbox{in $\Omega,$}\\ u_{1}=u_{2}=0 & \hbox{on $ \partial \Omega,$} \end{cases}$$%here $\Omega$~is a smooth, bounded and strictly convex domainin~$\mathbb{R}^{N}$, $N\geq2$, $\alpha >0$, $\beta >0$. When $\Omega$ isthe unit ball in $\mathbb{R}^{N}$, we use index theory of fixedpoints for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions undersome 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