一类带非线性边界条件的微分方程正解的存在性及多解性

Existence and Multiplicity of Positive Solutions for a Class of Differential Equations with Nonlinear Boundary Conditions

本文研究了二阶非线性微分方程边值问题正解的存在性及多解性, 其中b是正参数,k > 0,a∈C([0,1],(0,∞)),f,g∈C([0,∞), (0,∞)).在f,g满足适当条件下证得存在一个正数b∗, 使得当0 ∗时,(P)至少存在两个正解;当b=b∗时,(P)存在一个正解,当b > b∗时,(P)不存在正解.主要结果的证明基于拓扑度理论和上下解方法。

In this paper, we are concerned with the existence and multiplicity of positive solutionsfor second order nonlinear differential equations boundary value problems where b is a positive parameter,k > 0,a∈C([0,1],(0,∞)),f,g∈C([0,∞), (0,∞)).When f and g satisfy the proper conditions, we prove that there exists a positive number b∗, such that (P) has zero, exactly one and at least two positive solutions according to b > b∗,b=b∗ and 0 ∗, respectively. The proof of the main results is based on topological theory and the method of upper and lower solutions.