Nonconstant Periodic Solutions of Discrete -Laplacian System via Clark Duality and Computations of the Critical Groups
We study the existence of periodic solutions to a discrete -Laplacian system. By using the Clark duality method and computing the critical groups, we find a simple condition that is sufficient to ensure the existence of nonconstant periodic solutions to the system.
Let , , and denote the respective sets of natural numbers, integers, and real numbers. For with , write . Let . We are concerned with the existence of nonconstant -periodic solutions to the following discrete -Laplacian system: where is the forward difference operator defined by and is the -Laplace operator defined by (). Consider , and denotes the gradient of with respect to . We assume that is -periodic in the first variable ; that is, .
When , (1) reduces to the second order discrete -periodic Hamiltonian system: In 2003, Guo and Yu  introduced the critical point theory (see, e.g., ) to the study of the existence of -periodic solutions of (2). By using Rabinowitz's saddle point theorem, they proved existence of -periodic solutions, , when either is bounded and is coercive with respect to or satisfies a subquadratic Ambrosetti-Rabinowitz condition and a related coercivity condition. In the same year, they also proved existence of at least two nontrivial -periodic solutions of (2) when satisfies a superlinear condition near and satisfies a superquadratic Ambrosetti-Rabinowitz condition . The growth condition of was later removed in Zhou et al.  by using the linking theorem. For additional studies on the existence and multiplicity of solutions to (2) subject to various boundary value conditions through the use of critical point theory, we refer the reader to [4–13].
There have been tremendous efforts devoted to the study of the -Laplacian system (1) and the systems involving the -Laplace operator in recent years [14–20]. Many interesting results have been proved on the existence and multiplicity of solutions to (1) subject to the Dirichlet boundary condition by using the critical point theory [16, 17, 19, 21]. However, we have seen a very limited success in the application of this theory to the study of the existence of periodic solutions for (1). For the general case , He and Chen  obtained a result on the existence of periodic solutions for (1) with a convex by using Clark duality and the perturbation technique. For the case , Luo and Zhang  proved the existence of nonconstant periodic solutions for by making use of the linking theorem.
Our major goal in this paper is to prove the following theorem which gives a simple sufficient condition for the existence of nonconstant -periodic solutions to the -Laplacian system (1).
Theorem 1. Let . Assume that is strictly convex in for every and there exist and such that, for all and , one has where with Then (1) has at least one nonconstant -periodic solution.
Our assumptions differ from those in [22, 23] considerably and can be verified easily. For instance, let , and Then is strictly convex in for every , and (4) and (5) are satisfied with , , , and . For this function , Theorem 1 confirms the existence of at least one nonconstant -periodic solution of (1).
In the remainder of this section, we outline our approach based on the Clark duality and computation of the critical groups. Let be a real Hilbert space and . In Morse theory, the local behavior of near an isolated critical point at the level is described by the critical groups: where , is a neighbourhood of containing no other critical points, and denotes singular homology. The critical groups distinguish between different types of critical points and are extremely useful for obtaining the existence and multiplicity of solutions for variational problems .
Nonzero -periodic solutions of (1) are the nontrivial critical points of the variational functional defined on the finite dimensional space see [9, 23] for details. However, it is difficult to compute the critical groups for the case because there are few results on the nonlinear eigenvalue problem: As a result, there are no known eigenspaces to work with. As usual, if (10) has nonzero solutions, then we say that is an eigenvalue of the discrete -Laplacian with periodic boundary condition. To overcome this difficulty, we introduce to transform (1) into the following equivalent first order nonautonomous system: Denote System (12) can be rewritten in the compact form where with And nonzero solutions of (14) correspond to nontrivial critical points of defined on For , can be equipped with the inner product by which the norm can be induced by where denotes the Euclidean norm in and denotes the usual inner product in . It is easy to know that is a finite dimensional Hilbert space which can be identified with . The variational functional can be rewritten as where see  for details. Very fortunately, the linear eigenvalue problem can be worked out with eigenvalues So, 0 lies in the spectrum of which brings another difficulty in computing the critical groups of at infinity (e.g., to compute the critical groups of at infinity, the variational functional may be required to satisfy the angle condition proposed in ). To conquer this difficulty, motivated by [2, 9, 22], we introduce a dual action functional in the form and is not in the spectrum of . Furthermore, nontrivial critical points of correspond to nonconstant -periodic solutions of (1). To show that the dual action functional has at least one nontrivial critical point, firstly, we show that satisfies the condition () which guarantees that the critical groups make sense. Then we compute the critical groups . And finally, we show that is a local minimum of and hence the critical groups at infinity of are different from the critical groups at zero of which is sufficient for the existence of at least one nontrivial critical point of and hence the existence of at least one nonconstant -periodic solution of (1).
2. The Dual Action Functional and Related Lemmas
In this section, we present several technical lemmas to facilitate our proof of Theorem 1 in Section 3. In order to decompose the space appropriately, we consider the eigenvalue problem with . Apparently, is an eigenvalue of (25) with the eigenfunction Through a simple calculation, we see that (25) is equivalent to If , then (27) is equivalent to It has been proved that (28) has a nontrivial solution if and only if with [1, 3]. So in this case (25) has a nontrivial solution if and only if with . The multiplicity of is and the multiplicities of are of the same number . So, on the eigenvalue problem (25), the following results hold.
Proposition 2. For the eigenvalue problem (25), the eigenvalues are which can be arranged as with if is odd, and if is even.
To make an explicit decomposition of the Hilbert space , we also need to compute the eigenfunctions of (25) corresponding to each .
For each fixed , a solution of (28) can be written as where and are constant vectors in . By using the relation between and , that is, (27) with , we obtain Let , , denote the canonical basis of . If we choose and , then , and ; if we choose and , then , and . Therefore, the eigenfunctions of (25) corresponding to each can be given as Let Note that if is even and , then which shorten the dimension of the eigensubspace corresponding to to . Hence and has a eigensubspace decomposition as Thus, for any , can be expressed in the form where and . Obviously, .
Denote Then we have the following Wirtinger type inequalities: On the other hand, we can also define the norm on as follows: with . Since and are equivalent, there exist constants such that and
From [2, Theorem 2.2 and Proposition 2.4], we have the following lemma.
Lemma 3. Let, for every , be continuously differentiable and strictly convex in , and Then for every , is continuously differentiable in and
Remark 4. If, for , , can be split into two parts , then by (45), we have , , and .
From [2, Proposition 2.2], we have the following lemma.
Lemma 5. Let convex lower semicontinuous function be such that, for some , , , and , one has whenever . Then, if one has where .
Furthermore, on the we have the following.
Lemma 7. For all and every , one has
Proof. By Lemma 3, . Let be defined by
where . Then
Hence (4), (53), and the fact that if give (50).
To show that (51) holds, we can apply Lemma 5 to with , , and , and . Then (48) implies that (51) holds.
The variational functional of (14) defined on is see  for details. If then, by (45) and (54), we obtain where Now, we introduce a dual action functional where . Note that for with and ; hence it is sufficient to consider the functional on the subspace of . By Remark 6, the functional is continuously differentiable on . And, for any , we have If is a critical point of on the subspace , that is, for any , then That is, where . Setting , we get , and by relation (45) and (62), . Therefore, . And due to . Hence, we have the following.
Remark 8. If is a critical point of the dual action functional , then there exists a constant such that with , is a solution of (14), and is a solution of (1). If is a nontrivial critical point, then is a nonconstant -periodic solution of (14), and is a nonconstant -periodic solution of (1).
3. Proof of the Main Result
As our proof of Theorem 1 is mainly based on the computation of the critical groups in Morse theory, we recall several basic concepts about critical groups [2, 24]. Let be a real Hilbert space, and . Denote for .
Definition 9. The functional satisfies the Palais-Smale (PS) condition if any sequence such that is bounded and as has a convergent subsequence.
In , Cerami introduced a weak version of the (PS) condition as follows.
Definition 10. The functional satisfies the Cerami condition (the () condition for short) if any sequence such that is bounded and as has a convergent subsequence.
Definition 11 (deformation condition). The functional satisfies the () condition at the level if, for any and any neighborhood of , there are and a continuous deformation such that(1) for all ;(2) for all ;(3) is nonincreasing in for any ;(4). satisfies the () condition if satisfies the () condition for all .
Let be an isolated critical point of with , and let be a neighborhood of ; the group is called the th critical group of at , where denotes the th singular relative homology group of the pair over a field , which is defined to be quotient , where is the th singular relative closed chain group and is the th singular relative boundary chain group .
Bartsch and Li  defined the th critical group of at infinity as provided that is bounded from below by with and satisfies the () condition for all .
Assume and satisfies the () condition. The Morse-type numbers of the pair are defined by and the Betti numbers of the pair are see [2, 24]. Furthermore, the following relations hold: Thus, if , that is, for some , then there must exist a critical point of with . Furthermore, the following results hold.
Proposition 12 (see ). Let be a real Hilbert space and . Assume that and that satisfies the () condition. If there exists some such that(i), then must have a critical point with ,(ii), then must have a nontrivial critical point.
We will use the following result to compute the critical groups of at infinity.
Proposition 13 (see ). Let the functional be of the form where is a self-adjoint linear operator such that is not in the spectrum of , are invariant subspaces corresponding to the positive/negative of spectrum of , respectively, has a bounded inverse on , and has a compact differential with Assume that is finite and satisfies the deformation condition. Then
For the proof of Theorem 1, in what follows we may assume that has only finitely many critical points. Firstly, we show that satisfies the condition () which guarantees that the critical groups of at infinity make sense. Then, via computations of critical groups of at infinity and at zero, we complete the proof of Theorem 1.
Proof. Let be a Cerami sequence of . Since is finite, we only need to show that is bounded.
If is unbounded, up to a subsequence, still denoted by , we may assume that, for some , In particular, However, by (50) and (51), we have By relation (11) and , one has which implies that , and hence At the same time, note that so, And assumption (5) implies that so holds which gives a contradiction to (74). This completes the proof.
Lemma 15. Under the conditions of Theorem 1, with , .
Proof. Recall the dual action functional being of the form For any , if we define a bilinear function as , then by (40) one has By Riese representation theorem , we can define the unique continuous self-adjoint linear operator on by . If is in the spectrum of , then the equation yields nontrivial solutions in which turns out to be the same as (28) with being replaced by and the same invariant subspaces and which can be given by eigensubspace. It is obvious that is not in the spectrum of from the definition of the subspaces of and . Hence, by Proposition 13, we only need to prove that where Note that with . And By (51), , and the equivalence of and , we see that (84) holds. Hence by Proposition 13 and (36).
Proof of Theorem 1. First, we prove that is a local minimum of and, hence,
Firstly, by (40), for all , we have Hence, by (40), (42), and (50), for , we have Take Then, for , one has Hence is a local minimum of , and (88) must hold. By Lemma 15, (88), and Proposition 12, must have at least one nontrivial critical point. The proof is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This project is supported by the National Natural Science Foundation of China (no. 11301103), SRF of Guangzhou Education Bureau (no. 10A012), Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), and the Projects for Outstanding Young Teachers and High Level Talents of Guangdong Higher Education Institutes.
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