We investigate the existence and multiplicity of periodic solutions of the $ p $-Laplacian equation $ \left(\phi_p(x')\right)'+f(t, x) = 0 $. 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 $ f $. We use a approach of rotation number in the $ p $-polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of $ p $-Laplacian equation in terms of the gap between the rotation numbers of referred piecewise $ p $-linear systems at zero and infinity.
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