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Equations with a $p$-Laplacian and an asymmetric nonlinear term

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  • For $p>1, $ and $\phi_p (s) = |s| ^{p-2} s,$ we consider the equation

    $(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = f(t,x),$

    where $ x^{+}=\max\{x,0\}$; $x^{-} =\max\{-x,0\},$ in a situation of resonance or near resonance for the period $T,$ i.e. when $\alpha,\beta$ satisfy exactly or approximately the equation

    $\frac{\pi_p }{\alpha^{1/p}} + \frac{\pi_p}{\beta^{1/p}} = \frac{T}{n},$

    for some integer $n.$ We assume that $f$ is continuous, locally Lipschitzian in $x,$ $T$-periodic in $t,$ bounded on $\mathbf R^2,$ and having limits $f_{\pm}(t)$ for $x \to \pm \infty,$ the limits being uniform in $t.$ Denoting by $v $ a solution of the homogeneous equation

    $(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = 0,$

    we study the existence of $T$-periodic solutions by means of the function

    $ Z (\theta) = \int_{\{t\in I | v_{\theta }(t)>0\}} f_{+}(t)v(t + \theta) dt + \int_{\{t\in I | v_{\theta }(t)<0\}} f_-(t) v (t + \theta) dt,$

    where $ I \stackrel{def}{=} [0,T].$ In particular, we prove the existence of $T$-periodic solutions at resonance when $Z$ has $2z$ zeros in the interval $[0,T/n),$ all zeros being simple, and $z$ being different from $1.$

    Mathematics Subject Classification: 33A34, 34C25, 34C29.

    Citation:

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