Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T22:14:44.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbounded solution components for nonlinear Hill's equations

Published online by Cambridge University Press:  14 November 2011

Thomas Mrziglod
Thomas Mrziglod
Affiliation:
Mathematisches Institut, Universität zu Köln, D-50923 Köln, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1131 - 1167
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alama, SLi, Y. Y.. Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differential Equations 96 (1992), 89–115.CrossRefGoogle Scholar
2Courant, R.Hilbert, D.. Methods of Mathematical Physics, vol. I (New York: Interscience, 1953).Google Scholar
3Dieudonne, J.. Foundations of Modern Analysis, vol. I (New York, London: Academic Press, 1969).Google Scholar
4Eastham, M. S. P.. The Spectral Theory of Periodic Differential Equations (Edinburgh: Scottish Academic Press, 1973).Google Scholar
5Heinz, H. P.. Bifurcation from the essential spectrum for nonlinear perturbations of Hill's equation. In Differential Equations–Stability and Control, ed. Elaydi, S., 219–26 (New York: Marcel Dekker, 1990).Google Scholar
6Heinz, H. P.. Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝn. Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 237–70.CrossRefGoogle Scholar
7Heinz, H. P.. Lacunary bifurcation of multiple solutions of nonlinear eigenvalue problems. In Bifurcation and Chaos, ISMN 97, eds Seydel, R., Schneider, F. W., Kupper, T. and Troger, H., 161–9 (Basel: Birkhäuser, 1991).Google Scholar
8Heinz, H. P.. On the number of solutions of nonlinear Schrödinger equations and on unique continuation J. Differential Equations 116 (1995), 149–71.CrossRefGoogle Scholar
9Heinz, H. P.. Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problems. Nonlinear Anal. 21 (1993), 457–84.CrossRefGoogle Scholar
10Heinz, H. P., Kupper, T.Stuart, C. A.. Existence and bifurcation for nonlinear perturbations of the periodic Schrödinger equation. J. Differential Equations 100 (1992), 341–54.CrossRefGoogle Scholar
11Heinz, H. P.Stuart, C. A.. Solvability of nonlinear equations in spectral gaps of the linearization. Nonlinear Anal. 19 (1992), 124–64.CrossRefGoogle Scholar
12Hempel, R.Voigt, J.. The spectrum of a Schrödinger operator in L Pℝv) is p-independent. Comm. Math. Phys. 104 (1986), 243–50.CrossRefGoogle Scholar
13Küpper, T.. Verzweigungen aus dem wesentlichen Spektrum. GAMM-Mitteilungen, Heft 1 (1991), 11–22.Google Scholar
14Küpper, T.Mrziglod, Th.. On the bifurcation structure of nonlinear perturbations of Hill's equations at boundary points of the continuous spectrum. SIAM J. Math. Anal. 26 (1995).CrossRefGoogle Scholar
15Küpper, T.Stuart, C. A.. Bifurcation into gaps in the essential spectrum. J. Reine Angew. Math. 409 (1990), 1–34.Google Scholar
16Küpper, T.Stuart, C. A.. Gap-bifurcation for nonlinear perturbations of Hill's equation. J. Reine Angew. Math. 410 (1990), 23–52.Google Scholar
17Küpper, T.Stuart, C. A.. Bifurcation at boundary points of the continuous spectrum. In Progress in Nonlinear Differential Equations, Nonlinear Diffusion Equations and their Equilibrium States 3 (Boston: Birkhäuser, 1992).Google Scholar
18Küpper, T.Stuart, C. A.. Necessary and sufficient conditions for gap-bifurcation. Nonlinear Anal. 18(1992), 893–903.CrossRefGoogle Scholar
19Mrziglod, Th.. Global bifurcation for Hill's equation with nonlinear boundary conditions. Applicable Anal. 55 (1994).Google Scholar
20Mrziglod, Th.. Globale Verzweigungen bei nichtlinearen Hill'schem Gleichungen. Z. angew. Math. Meek 75 (1995), 471–2.Google Scholar
21Mrziglod, Th.. On the solution structure of nonlinear Hill's equation I, global results. Math. Nachrichten. (to appear).Google Scholar
22Mrziglod, Th. On the solution structure of nonlinear Hill's equation II, local results. Math. Nachrichten (to appear).Google Scholar
23Newman, M. H. A.. Elements of the Topology of Plane Sets of Points (Cambridge: Cambridge University Press, 1951).Google Scholar
24Rabinowitz, P.. Nonlinear Sturm-Liouville problems for second order differential equations. Comm. Pure Appl. Math. 23 (1970), 939–61.CrossRefGoogle Scholar
25Rabinowitz, P.. Some global results for nonlinear eigenvalue problems. J. Fund. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
26Ruppen, H. J.. The existence of infinitely many bifurcating branches. Proc. Roy. Soc. Edinburgh Sect. A 101(1985), 307–20.CrossRefGoogle Scholar
27Stuart, C. A.. Global properties of solutions of non-linear second order differential equations on the half-line. Ann. Scuola Norm. Sup. Pisa 2 (1975), 265–86.Google Scholar
28Stuart, C. A.. Bifurcation for Neuman problems without eigenvalues. J. Differential Equations 36 (1980), 391–407.CrossRefGoogle Scholar
29Stuart, C. A.. A global branch of solutions to a semilinear equation on an unbounded interval. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 273–82.CrossRefGoogle Scholar
30Stuart, C. A.. Guidance properties of nonlinear planar waveguides Arch. Rational Mech. Anal. 125 (1993), 145–200.CrossRefGoogle Scholar
31Tertikas, A.. Global bifurcation analysis and uniqueness for a semilinear problem. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 265–84.CrossRefGoogle Scholar
32Toland, J. F.. Global bifurcation for Neuman problems without eigenvalues. J. Differential Equations 44(1982), 82–110.CrossRefGoogle Scholar
33Walter, W.. Gewohnliche Differentialgleichungen (Berlin: Springer, 1976).CrossRefGoogle Scholar
34Whyburn, G. Th.. Topological Analysis (Princeton: Princeton University Press, 1958).Google Scholar
35Zeidler, E.. Nonlinear Functional Analysis and its Applications, vol. I (New York: Springer, 1985).CrossRefGoogle Scholar