Boundary blow-up solutions to real $(N-1)$-Monge-Ampère equations with singular weights
Kiran Kumar Saha, Sweta Tiwari
TL;DR
This work addresses the boundary blow-up problem for the real $$(N-1)$$-Monge-Ampère equation with a singular weight $K(|x|)$ and a Keller-Osserman type nonlinearity $f$, seeking radial $$ (N-1) $$-convex large solutions in a ball. The authors reduce the problem to a radial ordinary differential equation, establish existence and uniqueness for an initial value problem, and prove a comparison principle to compare radial solutions. They then construct sub- and super-solutions under a Keller-Osserman framework, using auxiliary functions and a precise control of $K$ near the boundary, to obtain infinitely many radial large solutions. The main contribution is the multiplicity of radial $$ (N-1) $$-convex boundary blow-up solutions for a Hessian-type equation with singular weights, extending boundary blow-up theory to this operator class and providing a robust sub-/super-solution methodology. This advances the understanding of Hessian-type boundary blow-up phenomena and offers a framework applicable to related geometric and analytic problems.
Abstract
In this paper, we study a boundary blow-up problem for real $(N-1)$-Monge-Ampère equations of the form \begin{equation} \nonumber \left \{ \begin{aligned} & \operatorname{\det}^{\frac{1}{N-1}}\left(ΔzI-D^{2}z\right)=K(|x|)f(z) && \text{ in } Ω, & z(x) \to \infty \text{ as } \dist(x,\partialΩ) \to 0, \end{aligned} \right. \end{equation} where $Ω$ 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.