Multiple periodic solutions for Hamiltonian systems with not coercive potential
type: Item
instance of: scholarly article
title: Multiple periodic solutions for Hamiltonian systems with not coercive potential
zbMATH Open document ID: 1192.37084
author: Gabriele Bonanno
author: R. Livrea
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-22T00:00:00Z
review text: The authors consider the existence of \(T\)-periodic solutions for the system of ordinary differential equations \(-u''+A(t)u = \lambda b(t)\nabla G(u)\), where \(A\) is an \(N\times N\) symmetric positive definite matrix with entries \(a_{ij}: [0,T]\to\mathbb{R}\), \(b\) is a nonnegative function on \([0,T]\) (\(b\not\equiv 0\)) and \(\lambda\) is a positive parameter. The first main result asserts that if \(G\) satisfies a certain comparison condition, then, for \(\lambda\) in an explicitly given interval, there exist at least two nontrivial periodic solutions. The second main result asserts the existence of infinitely many periodic solutions provided \(\lambda\) is in an explicitly given interval and \(G(\xi)/|\xi|^2\) has oscillatory behaviour as \(|\xi|\to\infty\) (roughly, \(\liminf_{|\xi|\to\infty}G(\xi)/|\xi|^2 < L\limsup_{|\xi|\to\infty}G(\xi)/|\xi|^2\), \(L\) a suitable constant). The proofs use some abstract critical point theorems due to the first author and his co-authors.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5651218
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: periodic solution
zbMATH Keywords: multiple solutions
zbMATH Keywords: infinitely many solutions
zbMATH Keywords: critical point
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.09.025
OpenAlex ID: W1964861242
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DOI: 10.1016/J.JMAA.2009.09.025
description: scientific article; zbMATH DE number 5651218