Kurseeva, Valeria; Moskaleva, Marina; Valovik, Dmitry Asymptotical analysis of a nonlinear Sturm-Liouville problem: linearisable and non-linearisable solutions. (English) Zbl 1452.35201 Asymptotic Anal. 119, No. 1-2, 39-59 (2020). Summary: The paper focuses on a nonlinear eigenvalue problem of Sturm-Liouville type with real spectral parameter under first type boundary conditions and additional local condition. The nonlinear term is an arbitrary monotonically increasing function. It is shown that for small nonlinearity the negative eigenvalues can be considered as perturbations of solutions to the corresponding linear eigenvalue problem, whereas big positive eigenvalues cannot be considered in this way. Solvability results are found, asymptotics of negative as well as positive eigenvalues are derived, distribution of zeros of the eigenfunctions is presented. As a by-product, a comparison theorem between eigenvalues of two problems with different data is derived. Applications of the found results in electromagnetic theory are given. Cited in 5 Documents MSC: 35Q60 PDEs in connection with optics and electromagnetic theory 34L30 Nonlinear ordinary differential operators 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators Keywords:nonlinear Sturm-Liouville problem; asymptotical analysis; distribution of eigenvalues; comparison theorem; periodicity of eigenfunctions PDFBibTeX XMLCite \textit{V. Kurseeva} et al., Asymptotic Anal. 119, No. 1--2, 39--59 (2020; Zbl 1452.35201) Full Text: DOI