Periodic and subharmonic solutions for a class of second-order p-Laplacian Hamiltonian systems
© Lian et al.; licensee Springer 2014
Received: 28 August 2014
Accepted: 3 December 2014
Published: 30 December 2014
In this paper, the periodic and subharmonic solutions are investigated for a class of second-order non-autonomous ordinary differential equations with a p-Laplacian. With the perturbation technique and the dual least action principle, some existence results are given of solutions to the convex p-Laplacian systems.
Since F is T-periodic in t, it is natural to seek T-periodic solutions of (1) and (2). Also, F is kT-periodic for , one can search the kT-periodic solutions, which is called subharmonic solutions. By a subharmonic solution, it means a kT periodic solution with an integer, that is, the minimal period is strictly greater than T. When , it is a periodic solution or harmonic. Clearly, a solution x of (1) over verifying and can be extended by kT-periodicity over ℝ to give a kT-periodic solution. So, it is fine to study the periodic boundary value of problem of (1) over to present conditions for the existence of the periodic and subharmonic solutions of (1). For the study of the subharmonic solutions, we refer to – for a few examples.
The dual action principle was firstly introduced by Clarke  and developed by Clarke and Ekeland –, which is from the spirit of optimal control theory and convex analysis; see . Following this use and the direct variational method, the periodic solutions are obtained for the Hamiltonian system of (2) in , , ,  and the references therein. Mawhin and Willem  presented the existence results of solutions to the more general systems by using such a principle and the perturbation technique argument. In 2007, Tian and Ge  generalized the existence results to the p-Laplacian system (1). By transforming the variable, they found a first-order Hamiltonian system equivalent to the second-order p-Laplacian system (1). Then by applying the Clarke duality, the perturbation technique, and the least action principle, they obtained the existence result of the periodic solution. The authors  also discussed the multi-point boundary value problem of a second-order differential equation with a p-Laplacian.
Motivated by the work listed above, we aim to discuss the periodic and subharmonic solutions to the p-Laplacian systems of (1). The proofs are depending on the dual least action principle and the perturbation arguments. To the best of our knowledge, there is no paper discussing the subharmonics solutions of p-Laplacian system by using the dual least action principle. This paper is a first try. The existence of the periodic solutions obtained in this paper slightly improves the result in . New arguments are considered for the posterior estimates of the periodic solutions and the existence of subharmonic solutions are presented, which extend those in , .
The paper is organized as follows. In Section 2, we establish the variational structure of (1) and transfer the existence of the solution into the existence of a critical point of the corresponding functional. The dual action is mainly discussed here. In Section 3, sufficient conditions are presented to guarantee the existence of the periodic solution of (1). We also estimate the prior bounds of all the periodic solutions of (1). The existence of the subharmonic solutions are given in the last section.
2 Preliminary and dual action
with the norm and , respectively. Here q is a constant such that . It is easy to verify that X and Y are reflexive Banach spaces and .
We easily find the following inequality.
where is the symplectic matrix. Obviously, and for all .
where is the inner product in . Because the first part of ψ is indefinite, the dual least process is applied to discuss the existence of the periodic solutions of (4).
Suppose the following condition holds.
holds for a.e. and.
From the definition of and Lemma 2.2, we have the following result.
is the functional we needed since the critical points of on Y coincide with the solutions of (4). Because , it suffices to find a critical point in . Similarly to the discussion of the related lemmas in reference , we have the following results.
Suppose the condition (A0) holds. Ifis a critical point of, then the functionis the kT-periodic solution of (4) in X.
3 Periodic solutions
In this section, we discuss the kT-periodic solution of (4). Here we note that the is T-periodic in t for each .
Suppose the following conditions are satisfied.
From the discussion in Section 2, we can see that if the problem (4) has one solution , then is for the kT-periodic solutions of (1). Now we need to prove the problem (4) has at least one kT-periodic solution in X. The proof is divided into three parts.
Step 1: Existence of a solution for the perturbed problem.
with , . Thus every minimizing sequence of on is bounded by (13) and Lemma 2.1. From the continuity of and the definition of , we can see that the second term of is weakly lower semi-continuous on . Meanwhile the first part of is weakly continuous. So is weakly lower semi-continuous. This implies that has a minimum at some . So (11) has a solution .
Step 2: Estimation of .
we have .
Step 3: Existence of a solution for the problem (4).
that is, u is a solution of (4) in X.
So for all . The proof is complete. □
Theorem 3.1 still holds if condition (A1) is changed to:
where m is an integer such that .
When the parameter α is smaller, we can obtain the prior bound for all the solutions of the p-Laplacian system (1).
which completes the proof. □
4 Subharmonic solutions
and such that the minimal periodoftends to ∞ when.
for all and .
is incompatible with (21) when n is sufficiently large. Thus when .
This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Beijing Higher Education Young Elite Teacher Project.
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