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On a variational characterization of the Fučík spectrum of the Laplacian and a superlinear Sturm–Liouville equation

Published online by Cambridge University Press:  12 July 2007

Eugenio Massa
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via Saldini 50, 20133 Milano, Italy (eugenio@mat.unimi.it)
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Abstract

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In the first part of this paper, a variational characterization of parts of the Fučík spectrum for the Laplacian in a bounded domain Ω is given. The proof uses a linking theorem on sets obtained through a suitable deformation of subspaces of H1 (Ω). In the second part, a nonlinear Sturm–Liouville equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but a finite number of eigenvalues. It is proved that, under certain conditions, this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values related to the Fucík spectrum.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2004