We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation vxx - gv + n (x) F (v) = 0, previously considered by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function n (x). Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight n. We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincaré-Bikhoff fixed point theorem.
