Boundary blow-up solutions to real $(N-1)$-Monge-Amp\`{e}re equations with singular weights

TL;DR

This paper investigates boundary blow-up solutions for a class of real (N-1)-Monge-Ampère equations with singular weights in a ball, establishing the existence of infinitely many radial solutions under certain conditions.

Contribution

It introduces new existence results for boundary blow-up solutions of Monge-Ampère equations with singular weights, using sub- and super-solution methods.

Findings

Existence of infinitely many radial solutions proven.

Solutions exhibit boundary blow-up behavior.

Applicable to equations with singular weight functions.

Abstract

In this paper, we study a boundary blow-up problem for real $(N-1)$-Monge-Amp\`{e}re equations of the form \begin{equation} \nonumber \left \{ \begin{aligned} & \operatorname{\det}^{\frac{1}{N-1}}\left(\Delta zI-D^{2}z\right)=K(|x|)f(z) && \text{ in } \Omega, & z(x) \to \infty \text{ as } \dist(x,\partial\Omega) \to 0, \end{aligned} \right. \end{equation} where $\Omega$ denotes a ball in $\mathbb{R}^{N} ~ (N \geq 2)$. The weight function $K$ is allowed to be singular, and the nonlinearity $f$ is assumed to satisfy a Keller-Osserman type condition. We establish the existence of infinitely many radial $(N-1)$-convex solutions to the system by employing the method of sub- and super-solutions, in conjunction with a comparison principle.