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A variational method for the existence of bounded solutions of a sublinear forced oscillator

Published online by Cambridge University Press:  14 April 2004

Rafael Ortega  and
Gianmaria Verzini
Rafael Ortega
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 180071 Granada, Spain. E-mail: rortega@ugr.es
Gianmaria Verzini
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy. E-mail: gianmaria.verzini@polimi.it
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Abstract

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

Keywords

bounded solutionminimaxNehari's method34C1134B1449J35
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 88 , Issue 3 , May 2004 , pp. 775 - 795
Copyright
2004 London Mathematical Society

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Footnotes

The research of the second author was partially supported by MURST, Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’.