一类含平均曲率算子的拟线性微分方程Robin问题正解的存在性和多解性  

Existence and Multiplicity of Positive Solutions for a Class of Robin Problem of Quasilinear Differential Equation with Mean Curvature Operator

作　　者：何志乾 张浩然 HE Zhiqian;ZHANG Haoran(Teaching and Research Department of Basic Courses,Qinghai University,Xi'ning 810016,China;School of Water Conservancy and Electric Power,Qinghai University,Xi'ning 810016,China)

机构地区：[1]青海大学基础部,青海西宁810016 [2]青海大学水利电力学院,青海西宁810016

出　　处：《山西大学学报（自然科学版）》2022年第6期1470-1475,共6页Journal of Shanxi University(Natural Science Edition)

基　　金：青海省自然科学基金(2021-ZJ-957Q);国家自然科学基金(12061064);青海大学教育教学研究项目(JY202138)。

摘　　要：可压缩流体的毛细现象及人眼睛角膜的几何形状的刻画等重要应用问题与一类欧氏空间中含平均曲率算子的拟线性微分方程Robin问题直接相关,本文研究了该问题正解的存在性和多解性。首先,利用平均曲率方程的特殊结构将求微分方程正解的问题转化为证明相应积分算子不动点的问题。其次,当非线性项在原点和无穷远处分别满足超线性或次线性增长时,运用锥上的不动点定理证明了该Robin问题正解的存在性和多解性。文章所得结论推广和改进了已有工作的相关结果,为更好地研究这类问题的定性性质提供理论依据。The application problems,such as the characterization of the capillary phenomenon for compressible fluid and the corneal geometry are directly related to a class of quasilinear Robin problem with mean curvature operator in Euclidean space.In this paper,the existence and multiplicity of positive solutions of the problem are studied.Firstly,the problem of finding positive solutions of mean curvature equations is transformed into the fixed point problem of proving corresponding integral operators by using the special structure of these equations.Secondly,when the nonlinear term satisfies superlinear or sublinear growth at the origin and at infinity,the existence and multiplicity of positive solutions of the Robin problem are proved by using the fixed point theorem on the cone.The conclusions of this paper extend and improve the results of previous work,and provide a theoretical basis for better research on the qualitative properties of this kind of problems.

关 键 词：平均曲率算子 不动点指数 正解 多解性 

分 类 号：O175.8[理学—数学]