Nonresonance conditions on the potential with respect to the Fučik spectrum for the periodic boundary value problem
type: Item
instance of: scholarly article
title: Nonresonance conditions on the potential with respect to the Fučik spectrum for the periodic boundary value problem
zbMATH Open document ID: 0852.34040
author: Pierpaolo Omari
author: Fabio Zanolin
author: Patrick Habets
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-12-11T00:00:00Z
review text: This article deals with the solvability of the periodic boundary value problem (1) \(u''+ g(u)= h(t)\), \(u(0)- u(T)= u'(0)- u'(T)= 0\), where \(g: \mathbb{R}\to \mathbb{R}\) is a continuous function. It is assumed that there are positive constants \(q_\pm\) and \(Q_\pm\) such that \[ q_\pm\leq \liminf_{s\to \pm \infty} {g(s)\over s}\leq \limsup_{s\to \pm\infty} {g(s)\over s}\leq Q_\pm\tag{2} \] and (3) \((q_-, q_+)\in {\mathcal C}_m\), \((Q_-, Q_+)\in {\mathcal C}_{m+ 1}\), where \({\mathcal C}_m\) and \({\mathcal C}_{m+ 1}\) are two consecutive branches of the Dancer-Fučik spectrum for the periodic problem (recall that \({\mathcal C}_k= \{(\mu, \nu): \mu^{-{1\over 2}}+ \nu^{-{1\over 2}}= 2/k\}\), for \(k= 1, 2,\dots)\). It is known that the hypotheses (2) and (3) alone are not sufficient to guarantee the existence of solutions to problem (1) for every \(h(t)\). Thus, in order to avoid resonance with the critical sets \({\mathcal C}_m\) and \({\mathcal C}_{m+ 1}\), the authors impose further conditions of the form \[ q_+< \limsup_{s\to + \infty} {2G(s)\over s^2}, \qquad \text{and} \qquad \liminf_{s\to+ \infty} {{2G(s)} \over s^2}< Q_+ \] (or similar ones at \(- \infty\), or, also, a choice of one condition at \(- \infty\) and one at \(+ \infty\), where \(G= \int g\)). Analogous results are obtained for the case ``below'' \({\mathcal C}_1\) and extended to the equation \(u''+ f(u) u'+ g(u)= h(t)\) as well, with no condition on \(f(x)\) (except continuity). The proofs are based on degree and variational arguments.
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 894653
zbMATH Keywords: solvability
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: Dancer-Fučik spectrum
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072148
OpenAlex ID: W1996437435
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DOI: 10.1216/RMJM/1181072148
description: scientific article; zbMATH DE number 894653