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A note on a resonance problem

Published online by Cambridge University Press:  14 November 2011

Daniela Lupo  and
Sergio Solimini
Daniela Lupo
Affiliation:
Istituto di Matematica dell'Università degli Studi, 34100 Trieste, Italy
Sergio Solimini
Affiliation:
International School for Advanced Studies, 34014 Trieste, Italy
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Synopsis

In this paper, we prove the existence of at least one solution to the problem

where ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 1-2 , 1986 , pp. 1 - 7
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Solimini, S.. On the solvability of some elliptic partial differential equation with linear part at resonance J. Math. Anal. Appf., to appear.Google Scholar
2Ward, J. R.. A boundary value problem with periodic nonlinearity (preprint).Google Scholar