Advances in Differential Equations

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

Raúl Manásevich and Jean Mawhin

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $p-1.$

Article information

Source
Adv. Differential Equations Volume 5, Number 10-12 (2000), 1289-1318.

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651224

Mathematical Reviews number (MathSciNet)
MR1785676

Zentralblatt MATH identifier
0992.34063

Subjects
Primary: 34B15: Nonlinear boundary value problems
Secondary: 34L30: Nonlinear ordinary differential operators 35J60: Nonlinear elliptic equations 35P05: General topics in linear spectral theory

Citation

Manásevich, Raúl; Mawhin, Jean. The spectrum of $p$-Laplacian systems with various boundary conditions and applications. Adv. Differential Equations 5 (2000), no. 10-12, 1289--1318. https://projecteuclid.org/euclid.ade/1356651224.


Export citation