Nonlinear resonant periodic problems with concave terms
type: Item
instance of: scholarly article
title: Nonlinear resonant periodic problems with concave terms
zbMATH Open document ID: 1220.34028
author: Sergiu Aizicovici
author: Nikolaos S. Papageorgiou
author: Vasile Staicu
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-01-12T00:00:00Z
review text: The authors continue their study of periodic BVPs for quasilinear equations involving the one-dimensional \(p\)-Laplacian. The problem under consideration reads: \[ -(|u'(t)|^{p-2} u'(t))' = \lambda |u(t)|^{q-2} u(t) + f(t, u(t)) \text{ on } [0,b], \] \[ u(0) = u(b), \quad u'(0) = u'(b), \] where \(1 < q < p < \infty\), with parameter \(\lambda > 0\). \(f\) is Carathéodory and satisfies some technical conditions. In particular, if \((\lambda_m)_{m \geq 1}\) is the sequence of eigenvalues of the negative periodic \(p\)-Laplacian on \([0,b]\), it is required that \(f(t,x)/(|x|^{p-2}x)\) belongs asymptotically, as \(|x| \to \infty\), to the spectral interval \([\lambda_m, \lambda_{m+1}]\) for some \(m \geq 2\). More precisely, the problem is permitted to be resonant at \(\lambda_{m+1}\) and non-uniformly nonresonant at \(\lambda_m\). This together with the appearance of the `concave' term \(\lambda |u|^{q-2} u\) constitutes the new feature compared with earlier work. The authors establish the existence of three distinct non-trivial solutions provided \(\lambda\) is not too large. One is of mountain pass type, one is a local minimizer, and the third one is detected using the Morse inequalities and the calculation of the critical groups at zero and at infinity.
reviewed by: Hans-Peter Heinz
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5835305
zbMATH Keywords: p-Laplacian
zbMATH Keywords: concave term
zbMATH Keywords: resonance
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: critical groups
zbMATH Keywords: Morse relations
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.09.009
OpenAlex ID: W2016766959
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DOI: 10.1016/J.JMAA.2010.09.009
description: scientific article; zbMATH DE number 5835305