Báo Cáo Báo cáo hóa học Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System

Báo cáo hóa học: Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System

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Research Article
Existence of Four Solutions of Some Nonlinear
Hamiltonian System
Tacksun Jung1 and Q-Heung Choi
2
1
Department of Mathematics, Kunsan National University, Kunsan 573-701, South Korea
2
Department of Mathematics Education, Inha University, Incheon 402-751, South Korea
Correspondence should be addressed to Q-Heung Choi, <a class="__cf_email__" href="http://www.cloudflare.com/email-protection" data-cfemail="f2839a97879c95b29b9c9a93dc9391dc9980">[email protected]<script type="text/javascript">
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Received 25 August 2007; Accepted 3 December 2007
Recommended by Kanishka Perera
We showthe existence of four 2π-periodic solutions of the nonlinearHamiltonian systemwith some
conditions. We prove this problem by investigating the geometry of the sublevels of the functional
and two pairs of sphere-torus variational linking inequalities of the functional and applying the
critical point theory induced from the limit relative category.
Copyright q 2008 T. Jung and Q.-H. Choi. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction and statements of main results
Let Ht, z be a C2
function defined on R1
× R2n which is 2π-periodic with respect to the first
variable t.In this paper, we investigate the number of 2π-periodic nontrivial solutions of the
following nonlinear Hamiltonian system
˙ z  J

Hzt, zt

, 1.1
where z : R → R2n,˙ z  dz/dt,
J 

0 ưIn
In 0

, 1.2
In is the identity matrix on Rn, H : R1
× R2n → R,and Hz is the gradient of H.Let z p, q,
p z1, .,zn,q zn1, .,z2n ∈ Rn. Then 1.1 can be rewritten as
˙ p  ưHqt, p, q,
˙ q  Hpt, p, q.
1.3
We assume that H ∈ C2
R1
× R2n,R1
 satisfies the following conditions.