Remarks on the second Neumann eigenvalue
José C. Sabina de Lis · Electronic Journal of Differential Equations · 2022
his work reviews some basic features on the second (first nontrivial) eigenvalue \(\lambda_2\) to the Neumann problem $$\displaylines{ -\Delta_p u = \lambda |u|^{p-2}u \quad x\in \Omega\cr | abla u|^{p-2}\frac{\partial u}{\partial u}=0 \quad x\in \partial\Omega, }$$ where \(\Omega\) is a bounded Lipschitz domain of \(\mathbb{R}^N\), \( u\) is the outer unit normal, and \(\Delta_p u = \text{div}(| abla u|^{p-2} abla u)\) 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.