On the large solutions to a class of k-Hessian problems

Abstract In this paper, we consider the k -Hessian problem S k (D 2 u) = b (x) f (u) in Ω, u = +∞ on ∂ Ω, where Ω is a C ∞ -smooth bounded strictly (k − 1) -convex domain in R N R^N with N ≥ 2, b ∈ C ∞ (Ω) is positive in Ω and may be singular or vanish on ∂ Ω, f ∈ C [0, ∞) ∩ C 1 (0, ∞) (or f ∈ C 1 (R) f C^1 (R) ) is a positive and increasing function. We establish the first expansions (equalities) of k -convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂ Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂ Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k -convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂ Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.