Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-12T21:40:11.110Z Has data issue: false hasContentIssue false
  • English
  • Français

On a conjecture related to the number of solutions of a nonlinear Dirichlet problem

Published online by Cambridge University Press:  14 November 2011

A. C. Lazer  and
P. J. McKenna
A. C. Lazer
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221, U.S.A.(Visiting University of Miami 1981–82)
P. J. McKenna
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 3-4 , 1983 , pp. 275 - 283
Copyright
Copyright © Royal Society of Edinburgh 1983

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

1Amann, H. and Hess, P.. A multiplicity result for a class of elliptic boundary value problems. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 145–151.CrossRefGoogle Scholar
2Ambrosetti, A. and Prodi, G.. On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Math. Pura Appl 93 (1972), 231–247.Google Scholar
3Dancer, E. N.. On the ranges of certain weakly nonlinear elliptic partial differential equations. J. Math. Pures Appl. 57 (1978), 351–366.Google Scholar
4Dolph, C. L.. Nonlinear integral equations of the Hammerstein type. Trans. Amer. Math. Soc. 60 (1949), 289–307.CrossRefGoogle Scholar
5Hammerstein, A.. Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math. 54 (1929), 117–176.Google Scholar
6Hart, D. C. and McKenna, P. J.. On the number of solutions of a generalized Ambrosetti–Prodi problem, to appear.Google Scholar
7Kazdan, J. L. and Warner, F. W.. Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28 (1875), 837–846.Google Scholar
8Keller, H. B.. Some positive problems suggested by nonlinear heat generation. In Bifurcation and Nonlinear Eigenvalue Problems (Keller, J. B. and Antman, S. eds.) (New York: Benjamin, 1969).Google Scholar
9Lazer, A. C. and McKenna, P. J.. On the number of solutions of a nonlinear Dirichlet problem. J. Math. Anal. Appl. 84 (1981), 282–294.Google Scholar
10Manes, A. and Micheletti, A. M.. Un estensione della teoria variazionale classica degli auto-valori per operatori ellittici del secundo ordine. Boll. Un. Mat. Ital. 7 (1973), 285–301.Google Scholar
11Sattinger, D. H.. Topics in stability and bifurcation theory. Lecture Notes in Mathematics 309 (Berlin: Springer, 1973).Google Scholar
12Berestycki, H.. Le nombre de solutions de certains problems semi-lineaires elliptiques. J. Functional Anal. 40 (1981), 1–29.Google Scholar
13Berestycki, H. and Lions, P.-L.. Sharp existence results for a class of semilinear elliptic problems. Bol. Soc. Brasil. Mat. 12 (1981), 9–14 (also MRC Technical Report 2111, Madison, Wisconsin, U.S.A., August 1980).Google Scholar
14Figueiredo, D. G. de. Lectures on boundary value problems of the Ambrosetti–Prodi type. Proceedings of 12th Brazilian Seminar of Analysis, Sao Jose dos campos S.P. (Oct. 1980), 231–291.Google Scholar