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

Abstract

For $p>1, $ and $\phi_p (s) = |s| ^{p-2} s,$ we considertheequation$(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = f(t,x),$where $ x^{+}=\max\{x,0\}$; $x^{-} =\max\{-x,0\},$ in asituation 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 Lipschitzianin $x,$ $T$-periodic in$t,$ bounded on$\mathbf R^2,$ and having limits $f_{\pm}(t)$ for $x \to \pm \infty,$ thelimits being uniform in $t.$ Denoting by $v $ a solution of thehomogeneous 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)where $ I \stackrel{def}{=} [0,T].$In particular, we prove the existence of $T$-periodic solutions atresonancewhen $Z$ has $2z$ zeros in the interval $[0,T/n),$ all zeros beingsimple, and $z$being different from $1.$