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

Abstract

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.