- Research Article
- Open Access
Existence of Four Solutions of Some Nonlinear Hamiltonian System
© The Author(s) 2008
- Received: 25 August 2007
- Accepted: 3 December 2007
- Published: 14 January 2008
We show the existence of four -periodic solutions of the nonlinear Hamiltonian system with 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.
- Weak Solution
- Bilinear Form
- Orthogonal Projection
- Nontrivial Solution
- Smooth Manifold
Our main results are the following.
Chang proved in  that, under conditions – ,system (1.1) has at least two nontrivial -periodic solutions. He proved this result by using the finite dimensional variational reduction method. He first investigate the critical points of the functional on the finite dimensional subspace and the condition of the reduced functional and find one critical point of the mountain pass type. He also found another critical point by the shape of graph of the reduced functional.
For the proofs of Theorems 1.1 and ,1.2 we first separate the whole space into the four mutually disjoint four subspaces , , , which are introduced in Section 3 and then we investigate two pairs of sphere-torus variational linking inequalities of the reduced functional and of on the submanifold with boundary and , respectively, and translate these two pairs of sphere-torus variational links of and into the two pairs of torus-sphere variational links of and , where and are the restricted functionals of to the manifold with boundary and , respectively. Since and are strongly indefinite functinals, we use the notion of the condition and the limit relative category instead of the notion of condition and the relative category, which are the useful tools for the proofs of the main theorems. We also investigate the limit relative category of torus in (torus, boundary of torus) on and , respectively. By the critical point theory induced from the limit relative category theory we obtain two nontrivial -periodic solutions in each subspace and , so we obtain at least four nontrivial -periodic solutions of (1.1).
In Section 2, we introduce some notations and some notions of condition and the limit relative category and recall the critical point theory on the manifold with boundary. We also prove some propositions. In Section 3, we prove Theorem 1.1 and in Section 4, we prove Theorem 1.2.
We need the following facts which are proved in .
Now, we consider the critical point theory on the manifold with boundary induced from the limit relative category. Let be a Hilbert space and be the closure of an open subset of such that can be endowed with the structure of manifold with boundary. Let be a functional, where is an open set containing . The condition and the limit relative category (see ) are useful tools for the proof of the main theorem.
Let be a sequence of a closed finite dimensional subspace of with the following assumptions: where , for all ( and are subspaces of ), , , are dense in . Let , for any , be the closure of an open subset of and has the structure of a manifold with boundary in . We assume that for any there exists a retraction . For a given , we will write . Let be a closed subspace of .
We have the following multiplicity theorem (for the proof, see ).
Now, we state the following multiplicity result (for the proof, see [4, Theorem 4.6]) which will be used in the proofs of our main theorems.
We have the following two pairs of the sphere-torus variational linking inequalities.
Lemma 3.1. (First Sphere-Torus Variational Linking).
So we proved the first case.
Lemma 3.5. (Second Sphere-Torus Variational Linking).
Proof of Theorem 1.1.
Proof of Theorem 1.2.
This research is supported in part by Inha University research grant.
- Chang K-C: Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications. Volume 6. Birkhäuser, Boston, Mass, USA; 1993:x+312.View ArticleGoogle Scholar
- Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Providence, RI, USA; 1986:viii+100.View ArticleGoogle Scholar
- Fournier G, Lupo D, Ramos M, Willem M: Limit relative category and critical point theory. In Dynamics Reported. Volume 3. Springer, Berlin, Germany; 1994:1-24. 10.1007/978-3-642-78234-3_1View ArticleGoogle Scholar
- Micheletti AM, Saccon C: Multiple nontrivial solutions for a floating beam equation via critical point theory. Journal of Differential Equations 2001,170(1):157-179. 10.1006/jdeq.2000.3820MathSciNetView ArticleMATHGoogle Scholar
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