Electron. J. Differential Equations, Vol. 2022 (2022), No. 13, pp. 1-12.

Remarks on the second Neumann eigenvalue

Jose C. Sabina de Lis

Abstract:
This work reviews some basic features on the second (first nontrivial) eigenvalue $\lambda_2$ to the Neumann problem

where $\Omega$ is a bounded Lipschitz domain of $\mathbb{R}^N$, $\nu$ is the outer unit normal, and $\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator. We are mainly concerned with the variational characterization of $\lambda_2$ and place emphasis on the range 1 < p < 2, where the nonlinearity $|u|^{p-2}u$ becomes non smooth. We also address the corresponding result for the p-Laplacian in graphs.

Submitted August 16, 2021. Published February 20, 2022.
Math Subject Classifications: 35J70, 35J92, 35P30.
Key Words: p-Laplacian operator; eigenvalues; Neumann conditions.
DOI: https://doi.org/10.58997/ejde.2022.13

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José C. Sabina de Lis
Departamento de Análisis Matemático and IUEA
Universidad de La Laguna
C. Astrofísico Francisco Sánchez s/n, 38203
La Laguna, Spain
email: josabina@ull.edu.es

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