Periodic solutions of Hamiltonian systems with superquadratic potential
type: Item
instance of: scholarly article
title: Periodic solutions of Hamiltonian systems with superquadratic potential
zbMATH Open document ID: 0632.34036
DOI: 10.1007/BF01769209
author: Donato Fortunato
author: Vieri Benci
author: Alberto Capozzi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: The authors consider the Hamiltonian H(t,p.q), \[ H(t,p.q)=\sum^{n}_{i,j=1}a_{ij}(t,q)p_ ip_ j+\sum^{n}_{i=1}b_ i(t,q)p_ i+V(t,q), \] with suitable growth conditions on the coefficients. The important ones are (i) V(t,q) grows more than \(| q|^{\alpha}\) \((\alpha >2)\) at infinity, (ii) some plausible growth rates of \(a_{ij}(t,q)\) with \(q_ j\) in addition to \(\sum_{i,j}a_{ij}(t,p,q)p_ ip_ j>\nu (q)| p|^ 2\) \((\nu (q)>0)\), and (iii) relative growths of \(| a_{ij}|\), \(b^ 2_ j\) and \(| \partial b_ j/\partial q_ kq_ k|^ 2\) with respect to V(t,q) as \(q\to \infty\). It is shown that, in addition to the above, (1) if \(\partial H/\partial t=0\), the system has infinitely many nonconstant T-periodic solutions for every prescribed periodic T, and (2) if H(t,p.q) is T-periodic in t and satisfies suitable bounds on the second order differential coefficients with respect to p,q, then the system has at least a nonconstant T-periodic solution. In order to establish these results, the authors first demonstrate an abstract theorem on critical points for strongly indefinite functionals. As a prelude to this, they recall some notions on the critical point theory and the index theory.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4027925
zbMATH Keywords: representation of \(S^ 1\). Hamiltonian
zbMATH Keywords: second order differential coefficients
zbMATH Keywords: critical points
zbMATH Keywords: critical point theory
zbMATH Keywords: index theory
MaRDI profile type: Publication
cites work: Saddle points and multiple solutions of differential equations
cites work: Q3899074
cites work: Periodic solutions of asymptotically linear Hamiltonian systems
cites work: Solutions of minimal period for a class of convex Hamiltonian systems
cites work: Dual variational methods in critical point theory and applications
cites work: Forced vibrations of superquadratic Hamiltonian systems
cites work: Q4748863
cites work: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity
cites work: Multiple periodic solutions for a semilinear wave equation with nonmonotone nonlinearity
cites work: A geometrical index for the groupS1 and some applications to the study of periodic solutions of ordinary differential equations
cites work: On Critical Point Theory for Indefinite Functionals in The Presence of Symmetries
cites work: Q4746860
cites work: On asymptotically quadratic Hamiltonian systems
cites work: The dual method in critical point theory. Multiplicity results for indefinite functional
cites work: On subquadratic Hamiltonian systems
cites work: Q3679526
cites work: Q3876715
cites work: Q5636090
cites work: Periodic solutions to Hamiltonian inclusions
cites work: Hamiltonian trajectories having prescribed minimal period
cites work: Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz
cites work: Borsuk-Ulam Theorems for Arbitrary S 1 Actions and Applications
cites work: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
cites work: On Strongly Indefinite Functionals with Applications
cites work: Q3847203
cites work: Existence of infinitely many periodic solutions for a perturbed Hamiltonian system
cites work: Periodic solutions of hamiltonian systems
cites work: Periodic Solutions of Hamiltonian Systems: A Survey
cites work: Periodic solutions of large norm of Hamiltonian systems
cites work: Q3337779
cites work: Q5734753
full work available at URL: https://doi.org/10.1007/bf01769209
OpenAlex ID: W2072834638
description: scientific article; zbMATH DE number 4027925