In this paper, we establish the existence and multiplicity for periodic solutions of the nonlinear system associated with a tapping mode cantilever Atomic Force Microscope (AFM) with Lennard-Jones potential and an external harmonic excitation. The technique used to solve the nonlinear system is based in the classical nonlinear technique of lower and upper solutions in reverse order. Finally, we show some numerical simulations using the Poincaré map to present regions where the existence is guaranteed.