Oscillatory property of solutions to nonlinear eigenvalue problems
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 soluti...
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| Format: | Serial |
| Published: |
2019
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| Series: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Oszcilláció - differenciálegyenlet, Bifurkáció |
| doi: | 10.14232/ejqtde.2019.1.67 |
| Online Access: | http://acta.bibl.u-szeged.hu/62291 |
| Summary: | 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. |
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| Physical Description: | 1-9 |
| ISSN: | 1417-3875 |