On the structure of the second eigenfunctions of the p-Laplacian on a ball

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 $\{{\tau }_n,{\mathrm{\Psi }}_n\}$ such that ${\mathrm{\Psi }}_n$ is nonradial and has exactly $2n$ nodal domains. A few related open problems are also stated.