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A variational approach to resonance for asymmetric oscillators

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  • 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.

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


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