We consider one-dimensional p-Laplacian eigenvalue problems of the form- ?p u = (? - q) | u |p - 1 sgn u, on (0, b), together with periodic or separated boundary conditions, where p > 1, ?p is the p-Laplacian, q ? C1 [0, b], and b > 0, ? ? R. It will be shown that when p ? 2, the structure of the spectrum in the general periodic case (that is, with q ? 0 and periodic boundary conditions), can be completely different from those of the following known cases: (i) the general periodic case with p = 2, (ii) the periodic case with p ? 2 and q = 0, and (iii) the general separated case with any p > 1. © 2006.