Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T21:36:35.417Z Has data issue: false hasContentIssue false
  • English
  • Français

A global branch of positive solutions above the continuous spectrum for problems with indefinite nonlinearities

Published online by Cambridge University Press:  14 November 2011

Tassilo Küpper  and
Achilles Tertikas
Tassilo Küpper
Affiliation:
Institute of Mathematics, University of Cologne, Weyertal 86–90, D-50931 Köln, Germany
Achilles Tertikas
Affiliation:
Department of Mathematics, University of Crete, P.O. Box 1470, Iraklion, Crete, Greece
Get access Rights & Permissions [Opens in a new window]

Extract

We prove the existence and bifurcation of a global branch of positive solutions for a nonlinear Neumann eigenvalue problem on the half axis [0, ∞). The nonlinearity is assumed to have a superlinear growth multiplied by a weight function changing sign. This leads to the existence of nontrivial solutions above the continuous spectrum of the linearised problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 465 - 482
Copyright
Copyright © Royal Society of Edinburgh 1996

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, S. and Li, Y. Y.. Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differential Equations 96 (1992), 89–115.CrossRefGoogle Scholar
2Benci, V. and Fortunate, D.Does bifurcation from the essential spectrum occur? Comm. Partial Differential Equations 6 (1981), 249–72.CrossRefGoogle Scholar
3Bongers, A., Heinz, H. P. and Küpper, T.. Existence and bifurcation theorems for nonlinear eigenvalue problems on unbounded domains. J. Differential Equations 54 (1983), 327–57.CrossRefGoogle Scholar
4Brown, K. J., Cosner, C. and Fleckinger, J.. Principal eigenvalues for problems with indefinite weight function on ℝ″. Proc. Amer. Math. Soc. 109 (1989), 147–55.Google Scholar
5Brown, K. J. and Lin, S. S.. On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function. J. Math. Anal. Appl. 75 (1980), 112–20.CrossRefGoogle Scholar
6Brown, K. J. and Tertikas, A.. On the bifurcation of radially symmetric steady-state solutions arising in population genetics. SIAM J. Math. Anal. 22 (1991), 400–13.CrossRefGoogle Scholar
7Brown, K. J. and Tertikas, A.. The existence of principal eigenvalues for problems with indefinite weight function on ℝ″. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 561–9.CrossRefGoogle Scholar
8Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Fund. Anal. 8 (1970), 321–40.CrossRefGoogle Scholar
9Gidas, B. and Spruck, J.. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), 883–901.CrossRefGoogle Scholar
10Küpper, T.. Verzweigung aus den wesentlichen Spektrum. GAMM-Mitteilungen Heft 1 (1991), 11–22.Google Scholar
11Küpper, T., Heinz, H. P. and Stuart, C. A.. Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrodinger equation. J. Differential Equations 100 (1992), 341–54.Google Scholar
12Küpper, T. and Mrziglod, T.. On the bifurcation structure of nonlinear perturbations of Hill's equation at boundary points of the continuous spectrum. SIAM J. Math. Anal. 26 (1995), 1284–305.CrossRefGoogle Scholar
13Küpper, T. and Stuart, C. A.. Bifurcation into gaps in the essential spectrum. J. Reine Angew. Math. 409(1990), 1–34.Google Scholar
14Küpper, T. and Stuart, C. A.. Gap-bifurcation for nonlinear perturbations of Hill's equation. J. Reine Angew. Math. 410 (1990), 23–52.Google Scholar
15Mrziglod, T.. Unbounded solution components for nonlinear Hill's equations. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar
16Stuart, C. A.. Bifurcation for variational problems when the linearisation has no eigenvalue. J. Fund. Anal. 38(1980), 169–87.CrossRefGoogle Scholar
17Tertikas, A.. Global bifurcation of positive solutions in ℝ″. In Progress in Nonlinear Differential Equations and their Applications 7, 513–536 (Boston: Birkhäuser, 1991).Google Scholar
18Tertikas, A.. Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in ℝ″. Differential Integral Equations 8, 4 (1995), 829–48.CrossRefGoogle Scholar