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On the periodic Fučik spectrum and a superlinear Sturm–Liouville equation

Published online by Cambridge University Press:  14 November 2011

Djairo G. Figueiredo
Affiliation:
IMECC, Universidade Estadual di Campinas, 13081 Campinas, S.P., Brasil
Bernhard Ruf
Affiliation:
Dip. di Matematica, Università degli Studi, Via Saldini 50, 20133 Milano, Italy

Synopsis

In the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.

In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. 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 in the Fučik spectrum.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1993

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