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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> RN 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> f∈ C¹(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.