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

Abstract

We study an elliptic system coupled by Monge-Ampère equations: $$ \begin{cases} \det D^{2}u_{1}={(-u_{2})}^\alpha & \hbox{in $\Omega,$} \\ \det D^{2}u_{2}={(-u_{1})}^\beta & \hbox{in $\Omega,$} \\ u_{1}\le 0,\ u_{2}\le 0 & \hbox{in $\Omega,$}\\ u_{1}=u_{2}=0 & \hbox{on $ \partial \Omega,$} \end{cases} $$ here $\Omega$ is a smooth, bounded and strictly convex domain in $\mathbb{R}^{N}$, $N\geq2$, $\alpha \ge 0$, $\beta \ge 0$. 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 $\alpha\ge 0$, $\beta\ge 0$ and $\alpha\beta=N^2$ we also study a corresponding eigenvalue problem in more general domains.