A variational approach to resonance for asymmetric oscillators

D. Bonheure; C. Fabry

doi:10.3934/cpaa.2007.6.163

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2007, Volume 6, Issue 1: 163-181. Doi: 10.3934/cpaa.2007.6.163

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A variational approach to resonance for asymmetric oscillators

D. Bonheure^1, and
C. Fabry^1,

1.
Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve

Received: December 31, 2005

Revised: May 31, 2006

Published: February 2007

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Abstract

We consider in this note the equation

$x'' + \alpha x^+ - \beta x^- + g(x) =p(t),$

where $x^+ =$ max{$x,0$} is the positive part of $x$, $x^- $ =max{$-x,0$} its negative part and $\alpha,\beta$ are positive parameters. We assume that $g :\mathbb R \to \mathbb R$ is continuous and bounded on $\mathbb R$, $p:\mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of $2\pi$-periodic solutions when $(\alpha,\beta)$ belongs to one of the curves of the Fučík spectrum corresponding to $2\pi$-periodic boundary conditions.

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Mathematics Subject Classification: Primary: 34B15, 34C25; Secondary: 70K30.

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