The spectrum of the periodic p-Laplacian

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