Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-11T14:36:47.423Z Has data issue: false hasContentIssue false

Two-parameter nonlinear Sturm-Liouville problems

Published online by Cambridge University Press:  20 January 2009

Tetsutaro Shibata
Affiliation:
The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study two-parameter nonlinear Sturm-Liouville problems. We shall establish the continuity of the variational eigencurve λ(μ) and asymptotic formulas of λ(μ)as μ → ∞, μ → π2

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Amann, H., Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976), 281295.CrossRefGoogle Scholar
2.Berestycki, H., Le nombre de solutions de certain problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), 129.CrossRefGoogle Scholar
3.Binding, P. and Browne, P. J., Asymptotics of eigencurves for second order ordinary differential equations. I, J. Differential Equations 88 (1990), 3045.CrossRefGoogle Scholar
4.Binding, P. and Browne, P. J., Asymptotics of eigencurves for second order ordinary differential equations. II, J. Differential Equations 89 (1991), 224243.CrossRefGoogle Scholar
5.Binding, P. and Volkmer, H., Eigencuves for two-parameter Sturm-Liouville equations, SIAM Rev. 38 (1996), 2748.CrossRefGoogle Scholar
6.Brezis, H., Analyse Fonctionnelle (Masson, Paris, 1983).Google Scholar
7.Browne, P. J. and Sleeman, B. D., Nonlinear multiparameter Sturm-Liouville problems, J. Differential Equations 34 (1979), 139146.CrossRefGoogle Scholar
8.Browne, P. J. and Sleeman, B. D., Non-linear multiparameter eigenvalue problems for ordinary differential equations, J. Math. Anal. Appl. 77 (1980), 425432.CrossRefGoogle Scholar
9.Browne, P. J. and Sleeman, B. D., Bifurcation from eigenvalues in non-linear multiparameter Sturm-Liouville problems, Glasgow Math. J. 21 (1980), 8590.CrossRefGoogle Scholar
10.Chow, S. N. and Hale, J. K., Methods of Bifurcation Theory (Springer, New York, 1982).CrossRefGoogle Scholar
11.Faierman, M., Two-parameter eigenvalue problems in ordinary differential equations (Longman House, Essex, UK, 1991).Google Scholar
12.Faierman, M., Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of second order, Trans. Amer. Math. Soc. 168 (1972), 152.CrossRefGoogle Scholar
13.Faierman, M., Asymptotic formulae for the eigenvalues for a two-parameter system of ordinary differential equations of second order, Canad. Math. Bull. 17 (1975), 657665.CrossRefGoogle Scholar
14.Gómez, J. L., Multiparameter local bifurcation based on the linear part, J. Math. Anal. Appl. 138 (1989), 358370.CrossRefGoogle Scholar
15.Heinz, H. P., Nodal properties and variational characterization of solutions to nonlinear Sturm-Liouville problems, J. Differential Equations 62 (1986), 299333.CrossRefGoogle Scholar
16.Langer, R. E., Asymptotic theories for linear ordinary differential equations depending upon a parameter, J. SIAM 7 (1959), 298305.Google Scholar
17.Rynne, B. P., Bifurcation from eigenvalues in nonlinear multiparameter problems, Nonlinear Anal. 15(1990), 185198.CrossRefGoogle Scholar
18.Rynne, B. P., Uniform convergence of multiparameter eigenfunction expansions, J. Math. Anal. Appl. 147 (1990), 340350.CrossRefGoogle Scholar
19.Shibata, T., Asymptotic behavior of the variational eigenvalues for semilinear Sturm-Liouville problems, Nonlinear Anal. 18 (1992), 929935.CrossRefGoogle Scholar
20.Shibata, T., Spectral properties of two parameter nonlinear Sturm-Liouville problem, Proc. Roy. Soc. Edinburgh Ser. A 123A (1993), 10411058.CrossRefGoogle Scholar
21.Shibata, T., Spectral asymptotics of two parameter nonlinear Sturm-Liouville problem, Results in Math. 24 (1993), 308317.CrossRefGoogle Scholar
22.Volkmer, H., Asymptotic spectrum of multiparameter eigenvalue problems, Proc. Edinburgh Math. Soc. (2) 39 (1996), 119132.CrossRefGoogle Scholar
23.Zeidler, E., Ljusternik-Schnirelman theory on general level sets, Math. Nachr. 129 (1986), 235259.CrossRefGoogle Scholar