The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation

Abstract

In this paper we consider the existence and asymptotic behavior of k-convex solution to the boundary blow-up k-Hessian problemSk(D2u)=H(x)f(u) for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0, where k∈{1,2,⋯,N}, Sk(D2u) is the k-Hessian operator, Ω is a smooth, bounded, strictly convex domain in RN(N≥2), H∈C∞(Ω) is positive in Ω, but is not necessarily bounded on ∂Ω, and f is a smooth positive function that satisfies the so-called Keller-Osserman condition. Further results are obtained for the special case that Ω is a ball. Our approach to show the existence and asymptotic behavior, exploits the method of sub- and super-solutions and Karamata regular variation theory.