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

Abstract

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