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Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

Published online by Cambridge University Press:  14 November 2011

Hans-Peter Heinz
Hans-Peter Heinz
Affiliation:
Fachbereich Mathematik der JohannesGutenberg-Universitat Mainz, Postfach 3980, D-6500 Mainz, Germany
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Synopsis

We consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 3-4 , 1991 , pp. 237 - 270
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Benci, V. and Rabinowitz, P. H.. Critical point theorems for indefinite functionals. Invent. Math. 52 (1979), 241–273.CrossRefGoogle Scholar
2Chen, C. N.. Uniqueness and bifurcation for solutions of nonlinear Strum-Liouville eigenvalue problems Arch. Rational Mech. Anal. 111 (1) (1990), 51–85.Google Scholar
3Corduneanu, C.. Almost periodic functions (New York: Interscience, 1968).Google Scholar
4Eastham, M. S. P.. The spectral theory of periodic differential equations (Edinburgh and London: Scottish Academic Press, 1973).Google Scholar
5Favard, J.. Leçons sur les fonctions presque-périodiques (Paris: Gauthier-Villars, 1933).Google Scholar
6Heinz, H.-P.. Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems. J. Differential Equations 64 (1986), 79–108.CrossRefGoogle Scholar
7Heinz, H.-P.. Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems. J. Differential Equations 66 (1987), 263–299.CrossRefGoogle Scholar
8Heinz, H.-P.. Bifurcation from the essential spectrum and variational techniques. In Differential Equations and Applications, Vol. I, ed. Aftabizadeh, A. R., p. 432–440 (Athens, Ohio: Ohio University Press, 1989).Google Scholar
9Heinz, H.-P.. Bifurcation from the essential spectrum for nonlinear perturbations of Hill's equation. In Differential Equations: Stability and Control, ed. Elaydi, S., pp. 217–26 (Dekker: N.Y., 1990).Google Scholar
10Küpper, T.. The lowest point of the continuous spectrum as a bifurcation point. J. Differential Equations 34 (1979), 212–217.CrossRefGoogle Scholar
11Küpper, T. and Stuart, C. A., Bifurcation into gaps in the essential spectrum J. reine ungew. Math. 409 (1990), 1–34.Google Scholar
12Küpper, T. and Stuart, C. A.. Gap-bifurcation for nonlinear perturbations of Hill's equation J. reine angew. Math. 410 (1990), 23–52.Google Scholar
13Magnus, R. J.. On perturbations of a translationally invariant differential equation. Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 1–25.CrossRefGoogle Scholar
14Reed, M. and Simon, B.. Methods of modern mathematical physics, Vol. II (Orlando, Florida: 1975), Vol. IV (New York:, 1978).Google Scholar
15Ruppen, H.J.. The existence of infinitely many bifurcating branches. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 307–320.CrossRefGoogle Scholar
16Stuart, C. A.. Bifurcation from the essential spectrum. In Proceedings of Equadiff 82, Lecture Notes in Mathematics 1017 (Berlin: Springer, 1983).Google Scholar
17Stuart, C. A.. Bifurcations for Dirichlet problems without eigenvalues. Proc. London Math. Soc. 45 (1982), 169–192.CrossRefGoogle Scholar
18Stuart, C. A.. Bifurcation from the continuous spectrum in the L 2-theory of elliptic equations on ∝n. In Recent Methods in Nonlinear Analysis and Applications, Proc. of SAFA IV (Napoli: Lignori, 1981).Google Scholar
19Stuart, C. A.. Bifurcation in L P(∝N) for a semilinear elliptic equation. Proc. London Math. Soc. (3) 57(1988), 511–541.CrossRefGoogle Scholar
20Stuart, C. A.. Bifurcation of homoclinic orbits and bifurcation from the essential spectrum SIAM J. Math. Anal. 20 (1989), 1145–1171.CrossRefGoogle Scholar