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

Anoop, T. V.; Drábek, P.; Sasi, Sarath

doi:10.1090/proc/12902

On the structure of the second eigenfunctions of the $p$-Laplacian on a ball
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by T. V. Anoop, P. Drábek and Sarath Sasi

Proc. Amer. Math. Soc. 144 (2016), 2503-2512

DOI: https://doi.org/10.1090/proc/12902

Published electronically: October 19, 2015

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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.

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