The spectrum of $p$-Laplacian systems with various boundary conditions and applications

Abstract

Let $p > 1,$ and $\psi_p : {{\mathbb R}}^N \to {{\mathbb R}}, \; v \mapsto |v|^{p-2}v,$ with $|v|$ the Euclidian norm of $v.$ This paper is devoted to the study of the corresponding eigenvalue problem $$\left(\psi_p(u')\right)' + \lambda \psi_p(u) = 0,$$ under the Dirichlet, Neumann and periodic boundary conditions. The eigenvalues in the Dirichlet and Neumann cases are the same when $N = 1$ and $N \geq 2,$ but not in the periodic case, where the exact nature of the set of eigenvalues is still open. We provide some information about this set. Variational characterizations of the first positive eigenvalue are obtained in the case of all three boundary conditions, as well as the corresponding generalized Poincaré's or Wirtinger's inequalities. Applications are given to forced Liénard-type systems and to systems with growth of order $p-1.$