Periodic solutions of <i>p</i>-Laplacian equations via rotation numbers

Abstract

We investigate the existence and multiplicity of periodic solutions of the \begin{document}$ p $\end{document} -Laplacian equation \begin{document}$ \left(\phi_p(x')\right)'+f(t, x) = 0 $\end{document} . Both asymptotically linear and partially superlinear nonlinearities are studied, in absence of global existence and uniqueness conditions on the solutions of the associated Cauchy problems and the sign assumption on \begin{document}$ f $\end{document} . We use a approach of rotation number in the \begin{document}$ p $\end{document} -polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincare-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of \begin{document}$ p $\end{document} -Laplacian equation in terms of the gap between the rotation numbers of referred piecewise \begin{document}$ p $\end{document} -linear systems at zero and infinity.