Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T21:17:54.576Z Has data issue: false hasContentIssue false
  • English
  • Français

The existence of infinitely many solutions all bifurcating from λ = 0

Published online by Cambridge University Press:  14 November 2011

Wolfgang Rother
Wolfgang Rother
Affiliation:
Mathematisches Institut der Universität Bayreuth, Postfach 10 12 51, W-8580 Bayreuth, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the non-linear differential equation

and state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 3-4 , 1991 , pp. 295 - 303
Copyright
Copyright © Royal Society of Edinburgh 1991

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

1Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 359–381.CrossRefGoogle Scholar
2Küpper, T. and Riemer, D.. Necessary and sufficient conditions for bifurcation from the continuous spectrum. Nonlinear anal. 3 (1979), 555–561.CrossRefGoogle Scholar
3Rother, W.. Bifurcation of nonlinear elliptic equations on ℝN. Bull. London Math. Soc. 21 (1989), 567–572.CrossRefGoogle Scholar
4Rother, W.. Bifurcation for a semilinear elliptic equation on ℝN with radially symmetric coefficients. Manuscripta math. 65 (1989), 413–426.CrossRefGoogle Scholar
5Ruppen, H.-J.. The existence of infinitely many bifurcation branches. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 307–320.CrossRefGoogle Scholar
6Stuart, C. A.. Bifurcation pour des problèmes de Dirichlet et de Neumann sans valeurs propres. C.R. Acad. Sci. Paris. 288 (1979), 761–764.Google Scholar
7Stuart, C. A.. Bifurcation for variational problems when the linearisation has no eigenvalues. J. Fund. Anal. 38 (1980), 169–187.CrossRefGoogle Scholar
8Stuart, C. A.. Bifurcation from the continuous spectrum in the L2-theory of elliptic equations on ℝN. Recent Methods in Nonlinear Analysis and Applications. Proceedings of SAFA IV, Liguori, Napoli (1981).Google Scholar
9Stuart, C. A.. Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. (3) 45 (1982), 169–192.CrossRefGoogle Scholar
10Stuart, C. A.. Bifurcation from the essential spectrum. Lecture Notes in Mathematics 1017 (1983), 575–596.CrossRefGoogle Scholar
11Stuart, C. A.. A variational approach to bifurcation in L p on an unbounded symmetrical domain. Math. Ann. 263 (1983), 51–59.CrossRefGoogle Scholar
12Stuart, C. A.. Bifurcation in L P(ℝN) for a semilinear elliptic equation. Proc. London Math. Soc. (3)57 (1988), 511–541.CrossRefGoogle Scholar
13Toland, J. F.. Global bifurcation for Neumann problems without eigenvalues. J. Differential Equations 44 (1982), 82–110.CrossRefGoogle Scholar
14Toland, J. F.. Positive solutions of nonlinear elliptic equations – existence and nonexistence of solutions with radially symmetry in L p(∝N). Trans. Amer. Math. Soc. 282 (1984), 335–354.Google Scholar
15Zhou, H.-S. and Zhu, X.-P.. Bifurcation from the essential spectrum of superlinear elliptic equations. Appl. Analysis. 28 (1988), 51–61.Google Scholar