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Conicting nonlinearities and spectral lacunae bounded on one side by eigenvalues

Published online by Cambridge University Press:  12 July 2007

Hans-Jörg Ruppen
Affiliation:
Cours de Mathématiques Spéciales, Ecole Polytechnique Fédérale de Lausanne, Station 4, 1015 Lausanne, Switzerland ([email protected])

Abstract

This paper deals with eigenvalue problems of the form where 0 < σ < τ and V(x) is such that the spectrum of −u″ consists of eigenvalues λ1, λ2,…situated below the continuous spectrum [δ,+∞[.

We analyse the existence of (multiple) solutions for λ < λ1 as well as for λ > λ1 when λ is in a spectral lacuna.

The existence of solutions depends on the weight of μ > 0. Moreover, when λ increases (while μ is kept fixed), some solutions are lost when crossing eigenvalues.

The above results are derived with the help of an abstract approach based on variational techniques for multiple solutions. This approach can even be applied to a wider class of problems, the one presented herein being only a model problem.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005

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