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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (4): 65-73.doi: 10.6040/j.issn.1671-9352.0.2022.052

• • 上一篇    

非齐次边界条件下弹性梁方程正解的多解性

孙晓玥   

  1. 西安电子科技大学数学与统计学院, 陕西 西安 710126
  • 发布日期:2023-03-27
  • 作者简介:孙晓玥(1999—),女,硕士研究生,研究方向为常微分方程边值问题. E-mail:sxy437760307@163.com
  • 基金资助:
    国家自然科学基金资助项目(12061064)

Multiplicity of positive solutions for elastic beam equations under inhomogeneous boundary conditions

SUN Xiao-yue   

  1. School of Mathematics and Statistics, Xidian University, Xian 710126, Shaanxi, China
  • Published:2023-03-27

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摘要： 研究带非齐次边界条件的两端简单支撑的弹性梁方程{Y (4)(x)=f(x,y), x∈(0,1),y(0)=0, y(1)=b, y″(0)=0, y″(1)=0多个正解的存在性,其中f∈C(［0,1］×［0,∞),［0,∞)), b>0,且对给定的x∈［0,1］, f(x,s)关于s单调递增。在适当的条件下,证明存在b*>0,使得当0*时至少存在两个正解;当b=b*时至少存在一个正解;当b>b*时无正解。该结果的证明基于上下解方法和拓扑度理论。

关键词: 非齐次, 简单支撑, 上下解, 拓扑度

Abstract: This paper studies the existence of multiple positive solutions for elastic beam equations simply supported at both ends with inhomogeneous boundary conditions{Y (4)(x)=f(x,y), x∈(0,1),y(0)=0, y(1)=b, y″(0)=0, y″(1)=0,where f∈C(［0,1］×［0,∞),［0,∞)), b>0, and f(x,s) is a monotone increasing function with respect to s for a fixed x∈［0,1］. Under appropriate conditions, there exists b*>0 such that the problem has at least two positive solutions for 0*, at least one positive solution for b=b*, and no positive solution for b>b*. The proof of the main results is based on the upper and lower solution method and topological degree theory.

Key words: inhomogeneous, simple support, upper and lower solution, topological degree

中图分类号: 

  • O175.8
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