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

Abstract

We study an elliptic system coupled by Monge-Ampère equations: \begin{center} $\left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^α, & \hbox{in $Ω,$} det~D^{2}u_{2}={(-u_{1})}^β, & \hbox{in $Ω,$} u_{1}<0, u_{2}<0,& \hbox{in $Ω,$} u_{1}=u_{2}=0, & \hbox{on $ \partial Ω,$} \end{array} \right.$ \end{center} here $\Omega$~is a smooth, bounded and strictly convex domain in~${\mathbb{R}}^N$,~$N\ge 2,\alpha >0,\beta >0$. When $\Omega$ 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.

References in corpus (1)

• Radial Convex Solutions of Boundary Value Problems for Systems of Monge-Ampere equations