# American Institute of Mathematical Sciences

November  2019, 12(7): 1841-1850. doi: 10.3934/dcdss.2019121

## 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

Received  April 2018 Revised  May 2018 Published  December 2018

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

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.

Citation: Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121
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