Abstract
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.
MSC:35Q70.
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1 Introduction
Let D be an open subset in with a compact complement , . Let , , and . In this paper we investigate the number of solutions with singular potential nonlinearity and periodic condition
Let us set and
Then (1.1) can be rewritten as
where . We assume that satisfies the following conditions:
(G1) There exists such that
(G2) There is a neighborhood Z of C in such that
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 [1] proved that (1.1) has at least two nontrivial 2π-periodic weak solutions under some asymptotic nonlinearity. Jung and Choi [2] proved that (1.1) has at least m weak solutions, which are geometrically distinct and nonconstant under some jumping nonlinearity.
In this paper we are trying to find the weak solutions of system (1.2) such that
i.e.,
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.
2 Variational method
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 α be a positive number such that and contains only one element 0 of σA. Let
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
Let
Let
Then and for , z has the decomposition , where
Thus we have
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 .
Let us introduce an open set of the Hilbert space E as follows:
Let
Then and for , z has the decomposition , where
The associated functional of (1.2) on X is as follows:
where , . Let
By , . Let
System (1.2) can be rewritten as
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 I. System (2.3)-(2.5) is reduced to
By the condition and (G2),
By (G3) and (G4), there exists such that
By the following lemma, the weak solutions of (1.2) coincide with the critical points of the functional .
Lemma 2.1 Assume that G satisfies conditions (G1)-(G2). Then is continuous and Fréchet differentiable in X with the Fréchet derivative
Moreover, . That is, .
Proof First we prove that is continuous and Fréchet differentiable in X. For ,
We have
Thus we have
Next we shall prove that is Fréchet differentiable in X. For ,
By (2.10), we have
Thus
□
Lemma 2.2 Assume that G satisfies conditions (G1)-(G2). Let and weakly in X with . Then .
Proof To prove the conclusion, it suffices to prove that
Since is bounded from below, it suffices to prove that there is an interval such that
By definition, means that there exists such that . By G(2), there exists a constant such that
Thus we have
for all . By Schwarz’s inequality, we have
Thus we have
Since the embedding is compact, we have
Thus by Fatou’s lemma, we have
Thus
Thus if ,
so we prove the lemma. □
Now, we recall the critical point theorem for indefinite functional (cf. [5]).
Let
Theorem 2.1 (Critical point theorem for indefinite functional)
Let X be a real Hilbert space with and . Suppose that satisfies (PS) and
-
(I1)
, where and is bounded and self-adjoint, ,
-
(I2)
is compact, and
-
(I3)
there exists a subspace and sets , and constants such that
-
(i)
and ,
-
(ii)
Q is bounded and ,
-
(iii)
S and ∂Q link.
-
(i)
Then I possesses a critical value .
3 Proof of Theorem 1.1
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 .
Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence satisfying
and
where is a compact operator and . Since is bounded from below, (3.1) implies that there exists a constant such that
We claim that has a convergent subsequence. It suffices to prove that the sequence is bounded in X. By contradiction, we suppose that up to a subsequence, satisfies . Then, for large k, we have
It follows from (3.4) that
By (3.3) and (3.5),
Letting
we have , which is a contradiction. □
Let
Lemma 3.2 Assume that G satisfies conditions (G1) and (G2). Then there exist sets with radius , and a constant such that
-
(i)
and ,
-
(ii)
Q is bounded and ,
-
(iii)
and ∂Q link.
Proof (i) Let us choose . Then . By (G1), is bounded above and there exists a constant
for . Then there exist constants and such that if , then .
(ii) Let us choose . Let . Then , , . We note that
By (G2), is bounded from below. Thus, by Lemma 2.2, there exists a constant such that if , then we have
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. □
References
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Jung T, Choi QH: On the number of the periodic solutions of the nonlinear Hamiltonian system. Nonlinear Anal. TMA 2009, 71: e1100-e1108. 10.1016/j.na.2009.01.095
Jung T, Choi QH: Periodic solutions for the nonlinear Hamiltonian systems. Korean J. Math. 2009, 17: 331–340.
Rabinowitz PH CBMS. Regional Conf. Ser. Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.
Benci V, Rabinowitz PH: Critical point theorems for indefinite functionals. Invent. Math. 1979, 52: 241–273. 10.1007/BF01389883
Acknowledgements
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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Jung, T., Choi, QH. Singular potential Hamiltonian system. J Inequal Appl 2013, 545 (2013). https://doi.org/10.1186/1029-242X-2013-545
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DOI: https://doi.org/10.1186/1029-242X-2013-545