非线性项零点对平均曲率方程Dirichlet问题多解性的影响  

Effect of the zero points of nonlinear term on the multiplicity of Dirichlet problem of mean curvature equations

作　　者：何志乾[1] 苗亮英[2] 迟昊东 HE Zhi-Qian;MIAO Liang-Ying;CHI Hao-Dong(School of Mathematics and Physics,Qinghai University,Xining 810016,China;School of Mathematics and Statistics,Qinghai Minzu University,Xining 810007,China;School of Chemical Engineering,Qinghai University,Xining 810016,China)

机构地区：[1]青海大学数理学院,西宁810016 [2]青海民族大学数学与统计学院,西宁810007 [3]青海大学化工学院,西宁810016

出　　处：《四川大学学报（自然科学版）》2024年第6期37-41,共5页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(12461035,12301631);青海省自然科学基金(2023-ZJ-949Q)。

摘　　要：在相对论中质点运动状态等重要问题与Minkowski空间中的平均曲率方程密切相关,而可压缩流体的毛细现象及人眼角膜几何形状的刻画等问题则与Euclidean空间中的平均曲率方程密切相关.本文在Euclidean空间和Minkowski空间中考虑了平均曲率方程Dirichlet问题的正径向解的存在性和多解性.基于方程的特殊结构,本文首先将解的存在性问题转化为相应积分算子的不动点的存在性问题.然后,当方程的非线性项存在零点时,本文运用锥上的不动点定理证明了问题正解的存在性和多解性.本文的结果揭示了正径向解个数与非线性项零点个数的关系.Many problems in relativity theory,such as the state of particle motion,are closely related to the mean curvature equations in Minkowski space.Meanwhile,the capillary phenomenon for compressible fluid and the characterization of corneal geometry are closely related to the mean curvature equations in Euclidean space.In this paper,we study the existence and multiplicity of positive radial solutions of the Dirichlet prob⁃lem of mean curvature equations in Euclidean and Minkowski spaces.Firstly,based on the special structure of the mean curvature equations,the existence of positive radial solutions is transformed into the problem of existence of fixed points of the corresponding integral operator.Secondly,the existence and multiplicity of positive radial solutions are proved by using the fixed point theorem in cones under the condition that the non⁃linear term of equation has zero point.The obtained results reveal the effect of the number of zero points of the nonlinear term on the number of positive radial solutions of the Dirichlet problem.

关 键 词：平均曲率方程 正径向解 多解性 

分 类 号：O175.8[理学—数学]