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 RN, for any N ≥ 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 {τn,Ψn} such that Ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated.