  In this paper, we prove that the second eigenfunctions of the $p$-Laplacian,
$p>1$, are not radial on the unit ball in $\mathbb{R}^N,$ for any $N\ge 2.$ Our
proof relies on the variational characterization of the second eigenvalue and a
variant of the deformation lemma. We also construct an infinite sequence of
eigenpairs $\{\tau_n,\Psi_n\}$ such that $\Psi_n$ is nonradial and has exactly
$2n$ nodal domains. A few related open problems are also stated.
