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

TL;DR

This paper proves that second eigenfunctions of the p-Laplacian on a ball are nonradial for any dimension, using variational methods, and constructs eigenfunctions with specific nodal domain properties.

Contribution

It establishes the nonradiality of second eigenfunctions for the p-Laplacian on a ball and constructs an infinite sequence of eigenfunctions with prescribed nodal domains.

Findings

Second eigenfunctions are nonradial for all dimensions.

Constructed an infinite sequence of eigenfunctions with 2n nodal domains.

Provided insights into the structure of p-Laplacian eigenfunctions.

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.