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

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.