Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 13, pp. 1-12.

Title: Remarks on the second Neumann eigenvalue
       
Author: Jose C. Sabina de Lis (Univ. de La Laguna, Spain)

Abstract:
This work reviews some basic features on the second (first nontrivial) eigenvalue
$\lambda_2$ to the Neumann problem
$$\displaylines{
-\Delta_p u = \lambda |u|^{p-2}u \quad x\in \Omega\cr
|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=0 \quad x\in \partial\Omega,
}$$
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.