On a power-type coupled system of Monge-Amp\`{e}re equations

TL;DR

This paper investigates a coupled system of Monge-Ampère equations with power nonlinearities, establishing existence, uniqueness, and nonexistence of solutions in convex domains, and explores eigenvalue problems for specific parameter conditions.

Contribution

It introduces new results on the existence and uniqueness of solutions for a coupled Monge-Ampère system, including nonexistence results and eigenvalue analysis under certain parameter constraints.

Findings

Existence of solutions in the unit ball for certain parameters.

Nonexistence of radial convex solutions under specific conditions.

Eigenvalue problem analysis for the case when eta = N^2.

Abstract

We study an elliptic system coupled by Monge-Amp\`{e}re equations: \begin{center} $\left\{ \begin{array}{ll} 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,$} \end{array} \right.$ \end{center} 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.