Boundary behavior of k-convex solutions for singular k-Hessian equations

Abstract We discuss the existence and boundary behavior of k -convex solution to the singular k -Hessian problem S k ( D 2 u ( x ) ) = b ( x ) f ( − u ( x ) ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω , where S k ( D 2 u ) ( k ∈ { 1 , 2 , … , n } ) is the k -Hessian operator, Ω ⊂ R n ( 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.