Oscillatory property of solutions to nonlinear eigenvalue problems

Shibata, Tetsutaro: Oscillatory property of solutions to nonlinear eigenvalue problems. Electronic journal of qualitative theory of differential equations 67. pp. 1-9. (2019)

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Abstract

This paper is concerned with the nonlinear eigenvalue problem −u 00(t) = λ (u(t) + g(u(t))), u(t) > 0, t ∈ I := (−1, 1), u(±1) = 0, where g(u) = u p sin(u q ) (0 ≤ p < 1, 0 < q ≤ 1) and λ > 0 is a bifurcation parameter. It is known that, for a given α > 0, there exists a unique solution pair (λ(α), uα) ∈ R+ × C 2 (I) satisfying α = kuαk∞ (= uα(0)). We establish the precise asymptotic formula for L r -norm kuαkr (1 ≤ r < ∞) of the solution uα as α → ∞ to show the evidence that uα(t) is oscillatory as α → ∞. We also obtain the asymptotic formula for λ in L r -framework, which has different property from that for diffusive logistic equation of population dynamics.

Item Type: Article
Journal or Publication Title: Electronic journal of qualitative theory of differential equations
Date: 2019
Number: 67
Page Range: pp. 1-9
ISSN: 1417-3875
DOI: https://doi.org/10.14232/ejqtde.2019.1.67
Uncontrolled Keywords: Oszcilláció - differenciálegyenlet, Bifurkáció
Additional Information: Bibliogr.: p. 8-9. ; összefoglalás angol nyelven
Date Deposited: 2019. Sep. 30. 12:24
Last Modified: 2019. Sep. 30. 12:24
URI: http://acta.bibl.u-szeged.hu/id/eprint/62291

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