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A multiplicity result for a class of elliptic boundary value problems

Published online by Cambridge University Press:  14 November 2011

Herbert Amann
Affiliation:
Mathematisches Institut, Universität Zürich, Freiestrasse 36, CH-8032 Zürich, Switzerland
Peter Hess
Affiliation:
Mathematisches Institut, Universität Zürich, Freiestrasse 36, CH-8032 Zürich, Switzerland

Synopsis

We consider a mildly nonlinear elliptic boundary value problem depending on a parameter. Given appropriate hypotheses concerning the asymptotic behaviour of the nonlinearity, we derive lower bounds on the number of solutions. The results complement an earlier theorem due to Kazdan and Warner [6].

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Amann, H.. Multiple positive fixed points of asymptotically linear maps. J. Functional Analysis 17 (1974), 174213.Google Scholar
2Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620709.CrossRefGoogle Scholar
3Ambrosetti, A. and Prodi, G.. On the inversion of some difierentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl., 93 (1972), 231247.CrossRefGoogle Scholar
4Berger, M. S. and Podolak, E.. On the solutions of a nonlinear Dirichlet problem. Indiana Univ. Math. J. 24 (1975), 837846.CrossRefGoogle Scholar
5Hess, P. and Ruf, B.. On a superlinear elliptic boundary value problem. Math. Z. 164 (1978), 914.CrossRefGoogle Scholar
6Kazdan, J. L. and Warner, F. W.. Remarks on some quasilinear elliptic equations. Comm. Pure Appl Math. 28 (1975), 567597.Google Scholar
Note added in proof, 31 July 1979. After this paper had been submitted for publication the authors became aware of an article by Dancer, E. N., On the ranges of certain weakly nonlinear elliptic partial differentialequations, J. Math. Pure Appl. 57 (1978), 351366. In that paper a similar result is proved by closely related considerations.Google Scholar