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

By Xuemei Zhang, Meiqiang Feng. Journal of Differential Equations, 2019

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Abstract In this paper we consider the existence and asymptotic behavior of k-convex solution to the boundary blow-up k-Hessian problem S k ( D 2 u ) = H ( x ) f ( u )  for  x ∈ Ω , u ( x ) → + ∞  as  dist ( x , ∂ Ω ) → 0 , where k ∈ { 1 , 2 , ⋯ , N } , S k ( D 2 u ) is the k-Hessian operator, Ω is a smooth, bounded, strictly convex domain in R N ( 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

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