New properties of the spectrum \(\sigma\) are determined for the boundary value problems {\parindent6.5mm \begin{itemize}\item[(i)] \(-\Delta_p u= (\lambda- q(x))|u|^{p-1}\text{sgn\,}u\), \(p> 1\), \(x\in(0,\pi_p)\) with periodic boundary conditions \item[(ii)] \(u(0)= u(\pi_p)\), \(u'(0)= u'(\pi_p)\), \(q(x)\in C^1[0,\pi_p]\), \(\lambda\in \mathbb R\), where \(\Delta_p\) is the \(p\)-Laplacian and \(2\pi_p\) is the period of the \(p\)-sine function. \end{itemize}} If \(p=2\), then \(\Delta_p u= u''\), \(\sin_p(x)= \sin x\), \(\pi_p= \pi\) (see the paper for definitions). Let \(\sigma(k)\), \(k= 0,1,2,\dots\) be the set of eigenvalues whose eigenfunctions have \(2k\) zeros. It is proved that, for every pair of integers \(k\), \(n\), there exists \(q= q(k,n)\) such that \(\sigma(k)\) contains at least \(n\) distinct eigenvalues. This extends a theorem by \textit{M. Zhang} [J. Lond. Math. Soc., II. Ser. 64, No.~1, 125--143 (2001; Zbl 1109.35372)].