Subharmonic solutions for a class of Lagrangian systems

Anouar Bahrouni; Marek Izydorek; Joanna Janczewska

doi:10.3934/dcdss.2019121

Article Contents

2019, Volume 12, Issue 7: 1841-1850. Doi: 10.3934/dcdss.2019121 open access

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$ A_m $

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Subharmonic solutions for a class of Lagrangian systems

1.
Department of Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia
2.
Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

^* Corresponding author: Marek Izydorek
^* Corresponding author: Marek Izydorek

Received: March 31, 2018

Revised: April 30, 2018

Published: June 2019

M. Izydorek and J. Janczewska are supported by Grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081

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Abstract

We prove that second order Hamiltonian systems $ -\ddot{u} = V_{u}(t,u) $ with a potential $ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $ of class $ C^1 $, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [14]. Indeed, we weaken the latter condition in a neighbourhood of $ 0\in \mathbb{R} ^N $. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

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Mathematics Subject Classification: Primary: 37J45, 70H03; Secondary: 34C25, 34C37.

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References

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