- Open Access
Singular potential Hamiltonian system
© Jung and Choi; licensee Springer. 2013
- Received: 24 April 2013
- Accepted: 8 October 2013
- Published: 19 November 2013
We investigate multiple solutions for the Hamiltonian system with singular potential nonlinearity and periodic condition. We get a theorem which shows the existence of the nontrivial weak periodic solution for the Hamiltonian system with singular potential nonlinearity. We obtain this result by using the variational method, critical point theory for indefinite functional.
- Hamiltonian system
- singular potential nonlinearity
- variational method
- critical point theory for indefinite functional
where . We assume that satisfies the following conditions:
where is the distance function to C and is a constant.
Several authors ([1–4], etc.) studied the Hamiltonian system with nonsingular potential nonlinearity. Jung and Choi [2, 3] considered (1.1) with nonsingular potential nonlinearity or jumping nonlinearity crossing one eigenvalue, or two eigenvalues, or several eigenvalues. Chang  proved that (1.1) has at least two nontrivial 2π-periodic weak solutions under some asymptotic nonlinearity. Jung and Choi  proved that (1.1) has at least m weak solutions, which are geometrically distinct and nonconstant under some jumping nonlinearity.
where E is introduced in Section 2.
Our main result is as follows.
Theorem 1.1 Assume that G satisfies conditions (G1)-(G2). Then system (1.1) has at least one 2π-periodic solution.
For the proof of Theorem 1.1, we introduce the perturbed operator , such that is a compact operator and the associated functional corresponding to the operator , and approach the variational method, the critical point theory. In Section 2, we investigate the Fréchet differentiability of the associated functional and recall the critical point theorem for indefinite functional. In Section 3, we show that the associated functional satisfies the geometrical assumptions of the critical point theorem for indefinite functional and prove Theorem 1.1.
and that , , are isomorphic to , , , respectively.
For the sake of simplicity, from now on we shall denote the subset of , satisfying the 2π-periodic condition, by .
By the following lemma, the weak solutions of (1.2) coincide with the critical points of the functional .
Moreover, . That is, .
Lemma 2.2 Assume that G satisfies conditions (G1)-(G2). Let and weakly in X with . Then .
so we prove the lemma. □
Now, we recall the critical point theorem for indefinite functional (cf. ).
Theorem 2.1 (Critical point theorem for indefinite functional)
, where and is bounded and self-adjoint, ,
is compact, and
- (I3)there exists a subspace and sets , and constants such that
Q is bounded and ,
S and ∂Q link.
Then I possesses a critical value .
We shall show that the functional satisfies the geometric assumptions of the critical point theorem for indefinite functional.
Lemma 3.1 (Palais-Smale condition)
Assume that G satisfies conditions (G1) and (G2). Then there exists a constant depending on the norm of the function G on such that satisfies the condition in X for .
we have , which is a contradiction. □
Q is bounded and ,
and ∂Q link.
for . Then there exist constants and such that if , then .
We can choose a constant such that if , then . Thus we prove the lemma. □
Proof of Theorem 1.1 By Lemma 2.1, is continuous and Fréchet differentiable in X and, moreover, . By Lemma 2.2, if and weakly in X with , then . By Lemma 3.1, satisfies the condition for . By Lemma 3.2, there exist sets with radius , and a constant such that , Q is bounded and , and and ∂Q link. By the critical point theorem, possesses a critical value . Thus (1.1) has at least one nontrivial weak solution. Thus we prove Theorem 1.1. □
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-2010-0023985).
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