奇异四阶三点特征值问题正解的存在性

Positive solutions of singular fourth-order three-point eigenvalue problem

摘要:

本文主要研究了非线性奇异四阶三点特征值问题 u^{(4)}(t)=\lambda a(t)f(t,u(t)),t\in [0,1], u(0)=u'(\eta)=u''(1)=u'''(0)=0 正解的存在性. 其中\lambda是正的参数,\eta\in[\frac{1}{2},1)为常数.通过使用锥上的不动点定理获得了此问题的一个和多个正解的存在性.本文主要强调在非线性项f和a的假设条件下,我们给出了存在正解的\lambda的取值范围.尤其是,非线性项里的函数a(t)是奇异的.

Abstract:

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular fourth-order three-point eigenvalue problem u^{(4)}(t)=\lambda a(t)f(t,u(t)),t\in [0,1], u(0)=u'(\eta)=u''(1)=u'''(0)=0, where \lambda is a positive parameter and \eta\in[\frac{1}{2},1) is a constant. By using the fixed point theorem of cone expansion-compression type duo to Krasnosel'skii, we establish various results on the existence of single and multiple positive solutions to the eigenvalue problem. Under various assumptions on functions f and a, we give explicitly the intervals for parameter \lambda in which the existence of positive solutions is guaranteed. Especially, we allow the function a(t) of nonlinear term to have suittable singularities.