Boundary behavior of [formula omitted]-convex solutions for singular [formula omitted]-Hessian equations

Abstract

We discuss the existence and boundary behavior of k-convex solution to the singular k-Hessian problem Sk(D2u(x))=b(x)f(−u(x)),x∈Ω,u(x)=0,x∈∂Ω,where Sk(D2u)(k∈{1,2,…,n}) is the k-Hessian operator, Ω⊂Rn(n≥2) is a smooth bounded strictly convex domain. Here the weight function b(x) is not necessarily bounded on ∂Ω. Another interest is that f(u)→∞ as u→0. Our approach mainly relies on Karamata’s regular variation theory and the construction of suitable sub- and super-solutions.