This paper analyzes the structure of the set of nodal solutions, i.e., solutions changing sign, of a class of one-dimensional superlinear indefinite boundary value problems with indefinite weight functions in front of the spectral parameter. Quite surprisingly, the associated high-order eigenvalues may not be concave as is the case for the lowest one. As a consequence, in many circumstances, the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried out is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminates the analysis.

本文分析了一类具有一维权函数在频谱参数前的一维超线性不定边界值问题的节点解集（即解改变符号）的结构。令人惊讶的是，相关的高阶特征值可能并不像最低的特征值那样是凹的。结果，在许多情况下，节点解可以从平凡解的三个甚至四个分叉点分叉。本文结合了分析和数值工具。进行的分析是数学分析如何辅助问题的数值研究的范例，而数值研究同时证实并阐明了分析。