Uniform estimates and Brezis-Merle type inequalities for the $k$-Hessian equation

TL;DR

This paper develops new inequalities and estimates for the $k$-Hessian equation, advancing understanding of solution behavior and boundary conditions in nonlinear PDEs.

Contribution

It introduces a Brezis-Merle type inequality and an Alexandrov-Bakelman-Pucci estimate for the $k$-Hessian equation, along with a concentration-compactness principle.

Findings

Proved a Brezis-Merle type inequality for $k$-convex functions.

Established an Alexandrov-Bakelman-Pucci estimate for intermediate Hessian equations.

Developed a concentration-compactness principle for blow-up analysis.

Abstract

In this paper, we prove a Brezis-Merle type inequality for $k$-convex functions vanishing on the boundary. As an application, we establish an Alexandrov-Bakelman-Pucci type estimate for the intermediate Hessian equation. Furthermore, we establish a concentration-compactness principle for the blow-up behavior of solutions to the mean field type $k$-Hessian equation.