用奇点理论研究两类非线性边值问题的分支解

On the Bifurcation Solutions of Some Nonlinear BVPs:Using Singularity Theory

在本文中我们用奇点理论的方法研究了两类带分支参数λ的非线性边值问题．这两类方程形如Φ（ｕ，λ）＝ｕ″＋Ｆ（ｕ，λ）＝０和Φ（ｕ，λ）＝ｕ″＋ｕ＋Ｆ（ｕ，λ）＝０（带几种常见的边值条件），而其中的非线性项Ｆ（ｕ，λ）是含有分支参数的余维有限的奇点．本文的结果包括这两类问题的分支解的存在性及分支解的个数等等．

In this paper, two classes of nonlinear BVPs (boundary value problems) with bifurcation parameter λ are studied by using singularity theory. We study the second-order nonlinear differential equations Φ (u, λ) = u“+ F (u,λ)= 0and Φ(u, λ) = u”+u+F(u , λ) = 0 with some usual conditions of boubdary value ,where the nonlinear terms F(u,λ) are singularities which are finite codimensional .When the singularities F(u ,λ) are of some“good types”,the information of the existence of the bifurcation solutions of the nonlinear BVPs follows from our results.