Asymptotical analysis of a nonlinear Sturm–Liouville problem: Linearisable and non-linearisable solutions

Article type: Research Article

Authors: Kurseeva, Valeria | Moskaleva, Marina | Valovik, Dmitry^{; *}

Affiliations: Department of Mathematics and Supercomputing, Penza State University, Krasnaya Street 40, 440026, Penza, Russian Federation. E-mails: 79273698109@ya.ru, m.a.moskaleva1@gmail.com, dvalovik@mail.ru

Correspondence: [*] Corresponding author. E-mail: dvalovik@mail.ru.

Abstract: 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.

Keywords: Nonlinear Sturm–Liouville problem, asymptotical analysis, distribution of eigenvalues, comparison theorem, periodicity of eigenfunctions

DOI: 10.3233/ASY-191565

Journal: Asymptotic Analysis, vol. 119, no. 1-2, pp. 39-59, 2020