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At least three solutions for the Hamiltonian system and reduction method
Journal of Inequalities and Applications volume 2013, Article number: 91 (2013)
We investigate the multiplicity of solutions for the Hamiltonian system with some asymptotically linear conditions. We get a theorem which shows the existence of at least three 2π-periodic solutions for the asymptotically linear Hamiltonian system. We obtain this result by the variational reduction method which reduces the infinite dimensional problem to the finite dimensional one. We also use the critical point theory and the variational method.
1 Introduction and statement of the main result
Let be a function defined on which is 2π-periodic with respect to the first variable t. In this paper we investigate the number of 2π-periodic solutions of the following Hamiltonian system:
where , . Let J be the standard symplectic structure on , i.e.,
where is the identity matrix. Then (1.1) can be rewritten as
where and is the gradient of G. We assume that satisfies the following asymptotically linear conditions:
(G1) as , , , where .
(G2) There exist constants α, β (without loss of generality, we may assume ) such that
(G3) Let be an integer within such that
(G4) exists and there exists which satisfies
(G5) G is 2π-periodic with respect to t.
We are looking for the weak solutions of (1.1) with conditions (G1)-(G5). The 2π-periodic weak solution of (1.1) satisfies
where E is introduced in Section 2. By Lemma 2.1 in Section 2, the weak solutions of (1.1) coincide with the critical points of the functional
(G3)′ Let , , … , and be all integers within (without loss of generality, we may assume ) such that . Suppose that there exist and such that and
(G4)′ and such that
then (1.1) has at least two nontrivial 2π-periodic weak solutions. Jung and Choi proved in  that if G satisfies the following conditions:
(G1) is with .
(G2) There exists such that
(G3) There exists such that
(G4) There exists an integer Γ such that ,
then (1.1) has at least m weak solutions, which are geometrically distinct and nonconstant.
Our main result is the following:
Theorem 1.1 Assume that G satisfies conditions (G1)-(G5). Then system (1.1) has at least three 2π-periodic solutions.
Theorem 1.1 will be proved by the finite dimensional reduction method, the critical point theory and the variational method for the perturbed operator . The finite dimensional reduction method combined with the critical point theory and the variational method reduces the critical point results of the functional on the infinite dimensional space to those of the corresponding functional on the finite dimensional subspace.
The outline of this paper is organized as follows. In Section 2, we introduce the Hilbert normed space E, show that the corresponding functional of (1.1) is in , Fréchet differentiable and prove the reduction lemma for the perturbed operator . In Section 3, we show that the reduced functional satisfies condition and is the strict local point of minimum of and prove Theorem 1.1 by the shape of graph of the reduced functional.
2 The perturbed operator
Let denote the set of 2n-tuples of the square integrable 2π-periodic functions and choose . Then it has a Fourier expansion , with , and . Let
with the domain
where ϵ is a positive small number. Then A is a self-adjoint operator. Let be the spectral resolution of A, and let
For each , we have the composition
where , , . According to A, there exists a small number such that . Let us define the space E as follows:
with the scalar product
and the norm
The space E endowed with this norm is a real Hilbert space continuously embedded in . The scalar product in naturally extends as the duality pairing between E and . We note that the operator is a compact linear operator from to E such that
Then and for , z has the decomposition , where
Thus we have
and that , , are isomorphic to , , , respectively. Let us define the functional on as follows:
where , and , . Let
By and (G2), . Let
The system (1.1) is equal to
The Euler equation of the functional is the system
Thus is a solution of (2.2) if and only if is a critical point of f. System (2.3)-(2.5) is reduced to
By (G2), there exists a such that
We note that
Let us set
Now we will prove a reduction lemma which reduces the problem on the infinite dimensional space E to that of the finite dimensional subspace.
Let be fixed and consider the function defined by
The function h has continuous partial Fréchet derivatives and with respect to its first and second variables given by
for and , . Let .
Lemma 2.1 Assume that G satisfies the conditions (G1)-(G5).
For given , there exists a unique satisfying the equation(2.11)
There exists such that if and are in and , then(2.12)
There exists such that if and are in and , then(2.13)
For given , if we put the unique solution of (2.11) as , then is continuous on and satisfies a uniform Lipschitz condition in with respect to norm (also norm ) and , . Moreover,
If is defined by
then has a continuous Fréchet derivative with respect to v, and
is a critical point of if and only if is a critical point of I.
Proof (i) Let . If , then equation (2.11) is equivalent to the equation
The operator is a self-adjoint, compact and linear map from into itself and its norm is . We note that
We claim that the right-hand side of (2.14) is a Lipschitz mapping of into itself with a Lipschitz constant . In fact, let v be a fixed element in and , be any elements in E. Then we have
Since the operator norm of is less than or equal to and
since . Therefore, by the implicit function theorem, for given , there exists a unique solution which satisfies (2.15).
For all ,(2.16)
For all ,
If , and are in , and , then
Since and (2.15) holds, we see that if and are in and , then
Similarly, using the fact that and (2.17) holds, we see that if and are in and , then
If denotes the unique which solves (2.11), then . In fact, if , and , , then we have
Thus we have
Thus θ is continuous. Since , . Since is finite and all topologies on are equivalent, we have
Let . If , then from (2.11) we have
Since , we have
for all .
Since the functional I has a continuous Fréchet derivative DI, has a continuous Fréchet derivative with respect to v.
Suppose that there exists such that . From for all , for all . Since for all , it follows that . Thus is a solution of (1.1). Conversely if u is a solution of (1.1) and , then . □
3 Proof of Theorem 1.1
Lemma 3.1 Assume that G satisfies the conditions (G1)-(G5). Then is bounded below and satisfies condition.
Proof Let . By the finite dimensional reduction,
where , , , , . Let . Then we have
Moreover, we have
By (G4), we have chosen a number γ such that . Thus we have
Thus is bounded from below and satisfies condition. □
Lemma 3.2 Assume that G satisfies conditions (G1)-(G5). Then is a strict local point of minimum of with .
Thus we have
Thus is a strict local point of minimum of . Since , . □
Proof of Theorem 1.1 By Lemma 2.1(v), is continuous and Fréchet differentiable in . By Lemma 3.1, is bounded above, satisfies the condition and as . By Lemma 3.2, is a strict local point of minimum of with a critical value . We note that is another critical value of . By the shape of the graph of the functional on the one-dimensional subspace , there exists the third critical point of . Thus (1.1) has at least three solutions, one of which is a trivial solution . □
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The authors appreciate very much the referees for their kind corrections. This work (Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2011-0026920).
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Jung, T., Choi, QH. At least three solutions for the Hamiltonian system and reduction method. J Inequal Appl 2013, 91 (2013). https://doi.org/10.1186/1029-242X-2013-91
- Hamiltonian system
- asymptotical linearity
- variational reduction method
- critical point theory
- variational method