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Asymptotic Behavior of Singular Solution to the k-hessian equation with a Matukuma-Type Source

Wuhan University Journal of Natural Sciences 2024年第3期29卷 242-256页

作者： LIU Jinyu WANG Biao CHANG Caihong College of Science Xi'an University of Science and TechnologyXi'an 710054ShaanxiChina

This paper is concerned with radially positive solutions of the k-hessian equation involving a Matukuma-type source S_(k)(D^(2)(-φ))=|x|^(λ-2)/(1+|x|^(2))^(λ2)φ^(q),x∈Ω,where S_(k)(D^(2)(-φ))is the k-hessian operator,q>k>1,λ>0,n>2k,k∈N,andΩis a suitable bounded do-main in R~*** turns out that there are two different types of radially positive solutions for k>1,i.e.,M-solution(singular at r=0)and Esolution(regular at r=0),which is distinct from the case when k=*** 1k](n-2k),we apply an iterative approach to improve accuracy of asymptotic expansions of M-solution step by step to the desired *** contrast to the case k=1,we require a more precise range of parameters due to repeated application of Taylor expansions,which also makes asymptotic expansions need more delicate investigation.

关键词： k-hessian equation singular solutions asymptotic expansion

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Asymptotic Behaviour of hessian equation with Boundary Blow Up

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Acta Mathematicae Applicatae Sinica 2017年第3期33卷 575-586页

作者： Lei-na ZHAO Zi-jian LIU College of Mathematics and Statistics/College of Transportation Chongqing Jiaotong UniversityChongqing400074China

We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.

关键词： boundary blowup problem nondegeneracy k-hessian equation

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Boundary Regularity for k-hessian equations

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Acta Mathematica Sinica,English Series 2023年第12期39卷 2393-2413页

作者： You LI Meng Ni LI Yan Nan LIU School of Mathematics and Computational Sciences Xiangtan UniversityHunan 411105P.R.China School of Mathematics Southeast UniversityNanjing 211189P.R.China School of Mathematics and Statistics Beijing Technology and Business UniversityBeijing 100048P.R.China

In this paper we focus on the boundary regularity for a class of k-hessian equations which can be degenerate and(or)singular on the boundary of the *** by the case of Monge-Ampere equations,we first construct sub-solutions,then apply the characteristic of the global Holder continuity for convex functions,and finally use the maximum principle to obtain the boundary Holder continuity for the solutions of the k-hessian ***,finding such sub-solutions is very difficult due to the complexity of the k-hessian *** particular,we employ the symmetric mean to overcome the difficulties.

关键词： Dirichlet problem k-hessian equation symmetric mean

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