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Periodic and subharmonic solutions for a class of second-order p-Laplacian Hamiltonian systems
Boundary Value Problems volume 2014, Article number: 260 (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.
In this paper, we consider the second-order ordinary differential equations with a p-Laplacian
where , , . Here stands the Euclidean norm in . In what follows we always suppose that , is measurable and T-periodic in t for every , continuously differentiable and convex in x for a.e. . When , (1) reduces to the second-order Hamiltonian system
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
Let be any integer and a constant. The Sobolev space is the space of functions with having a weak derivative and . The norm over is defined by
Here, we recall that
Consider the spaces X and Y defined by
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 .
For any , the mean value is defined by
Similarly, we can define and . We denote by the subspace of Y by
We easily find the following inequality.
For everyand, we have
For everyand, we have
Let , , then the second-order p-Laplacian system (1) can be changed to the first-order ones,
where is a parameter. Define , , and by
for . Then the system (3) can be written by
where is the symplectic matrix. Obviously, and for all .
By the kT-periodic property, we have the Hamiltonian action of (4) on X given by
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).
Let the set of all convex lower semi-continuous functions from to whose effective domain (not equal to ∞) is nonempty. Then the Fenchel transform of is defined by
Similarly, we also can define
Suppose the following condition holds.
(A0):There exist positive constants α, δ, and positive functionssuch that
holds for a.e. and.
Furthermore, we haveand
From the definition of and Lemma 2.2, we have the following result.
Easily we find . So when F satisfies the condition (A0), is continuously differentiable in v for a.e. and
Let and by duality, we have
So the dual action can be defined on Y by
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. Thenis continuously differential on. For any, we have
Suppose the condition (A0) holds. Ifis a critical point of, then the functionis the kT-periodic solution of (4) in X.
For every, we have
For every, we have
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.
(A1):There existssuch that for alland a.e. , one has
(A2):There exists, such that for alland a.e. , one has
minimizes the dual action
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.
Choose such that
and for any , let
Clearly, is strictly convex and continuously differentiable in for a.e. . From (A1) and (A2), we have
Because the function , , attains its minimum at , we have
Let . Easily we find , where . From Lemma 2.4 and Lemma 2.5, we find that the perturbed dual action
is continuously differentiable on and if is a critical point of , the function defined by
is a solution of
where we rewrite . Meanwhile, from Lemma 2.2, we have
which together with (10) and Lemma 2.6 implies that
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 .
It follows from a similar discussion to Lemma 2.2 that
So for any , . Then the function
is continuously differentiable. By (A3), has a minimum at some point such that
So, let , and then has a unique solution . By duality, we also have
and . From the inequality , we have , and from (13), we obtain
By Lemma 2.1, we have , where and are constants independent of ϵ, as well as the following constants , . Furthermore, from , we have
and from , i.e.
we have .
Meanwhile, by the convexity of , we have
This together with condition (A3) implies that is bounded. Consequently,
Step 3: Existence of a solution for the problem (4).
Since is bounded, there is a sequence () with and a function such that
Moreover, from , we have
So converges weakly to . From (12), we have the integrated form
Because converges weakly to u in X, converges uniformly to u in (see Proposition 1.2 in ). So, let and we have
that is, u is a solution of (4) in X.
Finally, we show minimizes the dual action on . Because is a minimum of and , we have
By the duality between and and the definition of , we have
Moreover, as converges weakly to v in , converges weakly to Jv. Letting we obtain, by ,
So for all . The proof is complete. □
Theorem 3.1 still holds if condition (A1) is changed to:
(A4): There exists such that for all and a.e. , one has
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).
If there exist, , , andsuch that
for a.e. and, then each solution of (1) satisfies the inequalities
We still set , , and the equalities (3) hold. Easily we find
It follows from
Integrating over and using Lemma 2.6, we have
Meanwhile, by the convexity of F, we have
which completes the proof. □
4 Subharmonic solutions
Assume thatis continuous. Suppose further that
asuniformly in. Then for each, the system (1) has a kT-periodic solutionsuch that
and such that the minimal periodoftends to ∞ when.
Let . Then condition (16) implies that there exists a such that
for all and with . By the convexity of F, we have
for all and with . Furthermore, the continuity of F implies that there exist positive constants δ, β such that
for all and .
By the condition (17), there exists such that
minimizes the dual action
Next we estimate the upper bound of . For any , we have
Let with , , and . Define the function
Obviously, and . Meanwhile, from (18), we can see that when with , we have
So it follows from (20) that
If does not hold when . Then there exist a subsequence and a constant such that
From (1), we have
and so for some constants . This implies that
On the other hand, the inequality
is incompatible with (21) when n is sufficiently large. Thus when .
It remains to prove that the minimal period of tends to +∞ as . If not, there exist and a subsequence such that the minimal period of is less than R. Meanwhile, by (18), (19), and Theorem 3.2, we have
We notice that
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This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Beijing Higher Education Young Elite Teacher Project.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.