应用奇点理论研究一类非线性边值问题的分支

On the Bifurcation of a Class of Nonlinear BVP Via Singularity Theory

用奇点理论的方法研究了一类带分支参数λ的非线性边值问题,这类问题形如(u,λ)=u″+au′+bu+F(u,λ)=0,边值条件为u(0)=u(π)=0,其中a≠0,非线性项F:(R×R,0)→(R,0)是一个余维有限的分支问题.在一定条件下给出了这类问题平衡解的局部分支性质,包括分支解的存在性和分支解的个数.

In this paper, a class of nonlinear BVP （boundary value problem） with bifurcation parameter ,λ is studied by using singularity theory. We study the nonlinear differential system φ（u,λ）=u＋au′＋bu＋F（u,λ）=0 with conditions： u（0）=u（π）=0, where a≠0 and the nonlinear term F：（R×R,0）→（R,0） is a bifurcation problem with finite codimension. The local bifurcation properties of equilibrium solution of the system follows from our results, including the information of the existence and the numbers of the bifurcation solutions.