Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T02:59:51.900Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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