Nonconstant Periodic Solutions of Discrete p-Laplacian System via Clark Duality and Computations of the Critical Groups

and Applied Analysis 3 by which the norm ‖ ⋅ ‖ can be induced by ‖u‖ := ‖u‖2 = ( T ∑ n=1 |u (n)| 2 ) 1/2

When  = 2, (1) reduces to the second order discrete periodic Hamiltonian system: Δ 2  ( − 1) + ∇ (,  ()) = 0,  ∈ Z. ( In 2003, Guo and Yu [1] introduced the critical point theory (see, e.g., [2]) to the study of the existence of -periodic solutions of (2).By using Rabinowitz's saddle point theorem, they proved existence of -periodic solutions,  ∈ N, 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  = 0 and  satisfies a superquadratic Ambrosetti-Rabinowitz condition [3].The growth condition of ∇ was later removed in Zhou et al. [4] by using the linking theorem.
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][15][16][17][18][19][20].Many interesting results have been proved on the existence and multiplicity of solutions to (1) subject to the Dirichlet boundary condition (0) = ( + 1) = 0 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  > 1, He and Chen [22] obtained a result on the existence of periodic 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).
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  ∈  1 (, R).In Morse theory, the local behavior of  near an isolated critical point  0 at the level  is described by the critical groups: where   = { ∈  : () ≤ },  is a neighbourhood of  0 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 [24].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  ̸ = 2 because there are few results on the nonlinear eigenvalue problem: =  ( + 1) . ( 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 (, ) =  1 (,  1 ) +  2 (,  2 ) with And nonzero solutions of ( 14) correspond to nontrivial critical points of defined on For , V ∈ ,  can be equipped with the inner product Abstract and Applied Analysis 3 by which the norm ‖ ⋅ ‖ can be induced by where |⋅| denotes the Euclidean norm in R 2 and (⋅, ⋅) denotes the usual inner product in R 2 .It is easy to know that  is a finite dimensional Hilbert space which can be identified with R 2 .The variational functional  can be rewritten as where see [11] for details.Very fortunately, the linear eigenvalue problem can be worked out with eigenvalues So, 0 lies in the spectrum of  1 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 [25]).To conquer this difficulty, motivated by [2,9,22], we introduce a dual action functional  in the form and 0 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 0 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).

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  ∈ R. Apparently,  = 0 is an eigenvalue of ( 25) with the eigenfunction Through a simple calculation, we see that ( 25) is equivalent to If  ̸ = 0, then ( 27) is equivalent to It has been proved that ( 28) has a nontrivial solution if and only if 25) has a nontrivial solution if and only if  =   = 2 sin(/) with  ∈ Z(− + 1,  − 1) \ {0}.The multiplicity of  0 = 0 is 2 and the multiplicities of   ̸ = 0 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  = ( − 1)/2 if  is odd, and  = /2 if  is even.
is a solution of ( 14), and  1 is a solution of (1).If V is a nontrivial critical point, then  is a nonconstant -periodic solution of ( 14), and  1 is a nonconstant -periodic solution of (1).

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  ∈  1 (, R).Denote   = { ∈  :  () ≤ } , for  ∈ R.
In [27], Cerami introduced a weak version of the (PS) condition as follows.
If  satisfies the (PS) condition or the () condition, then  also satisfies the following deformation condition which is essential in Morse theory [28,29].
satisfies the () condition if  satisfies the (  ) condition for all  ∈ R.
Let  0 be an isolated critical point of  with ( 0 ) =  ∈ R, and let  be a neighborhood of  0 ; the group is called the th critical group of  at  0 , where   (, ) denotes the th singular relative homology group of the pair (, ) over a field F, which is defined to be quotient   (, ) =   (, )/  (, ), where   (, ) is the th singular relative closed chain group and   (, ) is the th singular relative boundary chain group [30].Bartsch and Li [25] defined the th critical group of  at infinity as provided that (K) is bounded from below by  ∈ R with K = { ∈  :   () = 0} and  satisfies the (  ) condition for all  ≤ .Assume ♯K < ∞ 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   (, ∞) ≇ 0, that is,   ̸ = 0 for some  ∈ Z, then there must exist a critical point  of  with   (, ) ≇ 0. Furthermore, the following results hold.
We will use the following result to compute the critical groups of  at infinity.Proposition 13 (see [24]).Let the functional  :  → R be of the form where  :  →  is a self-adjoint linear operator such that 0 is not in the spectrum of ,  ± are invariant subspaces corresponding to the positive/negative of spectrum of , respectively,  ± := |  ± has a bounded inverse on  ± , and  ∈  Assume that  = dim  − 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. Lemma 14.Under the conditions of Theorem 1, the functional  defined by (59) satisfies () condition.
Proof.Let {V () } ⊂  be a Cerami sequence of .Since dim  is finite, we only need to show that {V () } is bounded.
Proof.Recall the dual action functional  :  → R being of the form For any ũ, V ∈ , if we define a bilinear function as (ũ, V) = ∑  =1 (Δũ( − 1), V()), then by (40) one has By Riese representation theorem [31], we can define the unique continuous self-adjoint linear operator  on  by ⟨V, V⟩ = ∑  =1 (ΔV( − 1), V()).If  is in the spectrum of , then the equation ΔV( − 1) = V() yields nontrivial solutions in  which turns out to be the same as (28) with  being replaced by V and the same invariant subspaces  + and  − which can be given by eigensubspace.It is obvious that 0 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
of Theorem 1. First, we prove that 0 is a local minimum of  and, hence, 2+  −/  Hence 0 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.