At least three solutions for the Hamiltonian system and reduction method
© Jung and Choi; licensee Springer 2013
Received: 20 August 2012
Accepted: 9 February 2013
Published: 5 March 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
where and is the gradient of G. We assume that satisfies the following asymptotically linear conditions:
(G1) as , , , where .
(G5) G is 2π-periodic with respect to t.
then (1.1) has at least two nontrivial 2π-periodic weak solutions. Jung and Choi proved in  that if G satisfies the following conditions:
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
Now we will prove a reduction lemma which reduces the problem on the infinite dimensional space E to that of the finite dimensional subspace.
for and , . Let .
- (i)For given , there exists a unique satisfying the equation(2.11)
- (ii)There exists such that if and are in and , then(2.12)
- (iii)There exists such that if and are in and , then(2.13)
- (iv)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,
- (v)If is defined by
is a critical point of if and only if is a critical point of I.
- (ii)For all ,(2.16)
- (iii)Similarly, using the fact that and (2.17) holds, we see that if and are in and , then
- (iv)If denotes the unique which solves (2.11), then . In fact, if , and , , then we have
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.
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 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 . □
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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