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 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.