A multiplicity result for nonlinear second order periodic equations with nonsmooth potential
type: Item
instance of: scholarly article
title: A multiplicity result for nonlinear second order periodic equations with nonsmooth potential
zbMATH Open document ID: 1056.47056
author: Nikolaos S. Papageorgiou
author: Leszek Gasiński
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-01-22T00:00:00Z
review text: This article deals with the following periodic problem: \[ (| x'(t)| ^{p-2}x'(t))' \in \partial j(t,x(t)) \;\text{ almost everywhere on } [0,b], \quad x(0) = x(b), \;\;\;x'(0) = x'(b) \] with \(2 \leq p < \infty\) and a potential function \(j: [0,b] \times {\mathbb R} \to {\mathbb R}\) measurable in the first variable and locally Lipschitz in the second one (\(\partial\) represent the Clarke subdifferential). This problem is equivalent to analysis of critical points for the functional \[ \phi(x) = \frac1p \, \| x'\| _p^p + \int_0^b j(t,x(t))\, dt \] defined on \(W^{1,p}([0,b])\) and satisfying the nonsmooth \(PS_c\)-condition for any \(c \neq \int_0^b j_\pm(t) \, dt\) (\(j_\pm(t) = \lim_{\zeta \to \pm \infty} j(t,\zeta)\)) and bounded below under some natural conditions. The main result is the existence at least three distinct solutions. Some examples are presented.
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 2031082
zbMATH Keywords: first eigenvalue
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Clarke subdifferential
zbMATH Keywords: generalized variational derivatives
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: monotone and pseudomonotone operators
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2031082