-
Previous Article
The exponential behavior of Navier-Stokes equations with time delay external force
- DCDS Home
- This Issue
-
Next Article
Smoothing effect of the generalized BBM equation for localized solutions moving to the right
Existence and location of periodic solutions to convex and non coercive Hamiltonian systems
| 1. | Department of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy |
$J\dot u(t)+\nabla H(t,u(t))=0$ a.e. on $[0,T] $
$u(0)=u(T)$
where the function $H:[0,T]\times \mathbb R^{2N}\rightarrow \mathbb R$ is called Hamiltonian. Our attention will
be focused upon the case in which the Hamiltonian H, besides being measurable on $t\in[0,T]$, is convex and continuously
differentiable with respect to $u\in \mathbb R^{2N}$. Our basic assumption is that the Hamiltonian $H$ satisfies the
following growth condition:
Let $1 < p < 2$ and $q=\frac{p}{p-1}$. There exist positive constants $\alpha,\delta$ and functions $\beta,\gamma \in
L^q(0,T;\mathbb R^+)$ such that
$\delta|u|-\beta(t)\leq H(t,u)\leq\frac{\alpha}{q}|u|^q+\gamma(t),$
for all $u\in \mathbb R^{2N}$ and a.e. $t\in[0,T]$. Our main result assures that under suitable bounds on $\alpha,\delta$ and the functions $\beta,\gamma$, the problem above has at least a solution that belongs to $W_T^{1,p}$. Such a solution corresponds, in the duality, to a function that minimizes the dual action restricted to a subset of $\tilde{W}_T^{1,p}=${$v\in W_T^{1,p}: \int_0^{ T} v(t) dt=0$}.
| [1] |
Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939 |
| [2] |
Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069 |
| [3] |
Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059 |
| [4] |
Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75 |
| [5] |
Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 |
| [6] |
Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017 |
| [7] |
Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 |
| [8] |
Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 |
| [9] |
Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 |
| [10] |
Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222 |
| [11] |
Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 |
| [12] |
Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure & Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166 |
| [13] |
V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 |
| [14] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
| [15] |
Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268 |
| [16] |
Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 |
| [17] |
Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761 |
| [18] |
Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005 |
| [19] |
Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 |
| [20] |
Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]