We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms when nonlinearities cross some eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem, and the contraction mapping principle on p-Lebesgue space. We obtain the third result by Leray–Schauder degree theory.

中文翻译：

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一些具有跳跃非线性的椭圆边值问题的多重解

我们研究了具有跳跃非线性的一维椭圆狄利克雷边值问题的多重解。当非线性与某些特征值交叉时，我们根据源项获得三个定理。我们通过椭圆特征值问题的特征值和对应的归一化特征函数，以及p -Lebesgue空间上的收缩映射原理，得到第一个定理和第二个定理。我们通过 Leray-Schauder 度数理论得到第三个结果。