# American Institute of Mathematical Sciences

March  2007, 6(1): 163-181. doi: 10.3934/cpaa.2007.6.163

## A variational approach to resonance for asymmetric oscillators

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

Received  January 2006 Revised  June 2006 Published  December 2006

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.

Citation: D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure & Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163
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