Antiperiodic Solutions for p-Laplacian Systems via Variational Approach

have acted as one of the mainstream research problems in the field of differential equation. Under various assumptions of the potential F(t, x), there have been lots of existence and multiplicity of results in the literatures by using the tool of nonlinear analysis, such as degree theory, minimax methods, and Morse theory. Here we do not even try to review the huge bibliography, but we only list some references for our purpose; for example, we refer the readers to see [1–6] and the references therein. Comparing problem (1) with problem (3), we observe that the only difference is the boundary conditions. In order to use the variational methods, one of the main difficulties is the variational principle. For this matter, we try to modify some work space such that the variational principle can be established. Thanks to the work of Tian and Henderson [7], we borrow their ideas to give the variational principle for problem (1). Note that the study of antiperiodic solutions for nonlinear differential systems of the form

In the last few decades, the following second order systems involving periodic boundary condition: − ü = ∇ (,  ()) a.e.[0, ] ,  (0) =  () , u (0) = u () , have acted as one of the mainstream research problems in the field of differential equation.Under various assumptions of the potential (, ), there have been lots of existence and multiplicity of results in the literatures by using the tool of nonlinear analysis, such as degree theory, minimax methods, and Morse theory.Here we do not even try to review the huge bibliography, but we only list some references for our purpose; for example, we refer the readers to see [1][2][3][4][5][6] and the references therein.
Comparing problem (1) with problem (3), we observe that the only difference is the boundary conditions.In order to use the variational methods, one of the main difficulties is the variational principle.For this matter, we try to modify some work space such that the variational principle can be established.Thanks to the work of Tian and Henderson [7], we borrow their ideas to give the variational principle for problem (1).
Note that the study of antiperiodic solutions for nonlinear differential systems of the form is closely related to the study of its periodic solutions.Indeed, if we assume that ( + , ) = (, ) and ∇(, −) = −∇(, ), then let and we get that V is a solution of systems (4) with conditions in , V can be extended to be 2-periodic over R, and hence V is a 2-periodic solution of systems (4).
In this paper, we will establish the existence of solutions for problem (1) by variational method.As far as we know, there are few papers studying the second order systems with antiperiodic boundary conditions by variational methods.
This paper is organized as follows.In Section 2, we introduce a variational principle for problem (1).In Section 3, we prove our main results.
where (⋅, ⋅) denotes the inner product on R  , then and there exists  ∈ R  such that Remark 2. From Lemma 1, we have the following facts.
(i) A function V satisfying ( 6) is called a weak derivative of .By a Fourier series argument, the weak derivative, if it exists, is unique.We denote by   the weak derivative of .(ii) We will identify the equivalence class  and its continuous representation (iii) Equations ( 7) and (8) imply that (0) = −() = .
For this matter, we only show that ()+ = 0. Indeed, using integration by parts, we have (iv) If   is continuous on [0, ], then by (9)   is the classical derivative of  = û.
(v) It follows from (9) and Rademacher theorem that   is the classical derivative of  a.e. on [0, ].
The Sobolev space Ŵ1,  is the space of functions  ∈   (0, ; R  ) having a weak derivative It is easy to see that Ŵ1,  is a reflexive Banach space and  ⊂ Ŵ1,  .
With the proof of Lemma 3.3 in [7] and Theorem 8.8 in [8], we have the following embedding theorem.
By Lemma 1, the norm ‖ ⋅ ‖ Under the assumption of (H), we have the following.for every V ∈ Ŵ1,  .Moreover, if   () = 0, then  is a solution of problem (1); that is,  ∈ Ŵ1,  satisfies the equation and antiperiodic condition in (1).
Proof.Similarly as the proof of Theorem 1.4 in [1], we obtain that  ∈  1 ( Ŵ1,  , R) and (28) holds.If for all V ∈ Ŵ1,  and hence for all V ∈ , by Lemma Accordingly, the conclusion is completed.
Next, we study the eigenvalue problem

Main Results and Proof
In this section, we will give some existence and multiplicity of results for problem (1).
Remark 11.By hypothesis (H), we see that (, ) is summable over  ∈ [0, ] in the neighborhood of zero, and thus the condition ( 0 ) can be weaken to hold for || large.
If  = , we may assume the function () in ( 0 ) satisfies that as the same proof of Theorem 10, and we can get the following theorem.
Theorem 12.Under the above assumptions of (, ), then problem (1) has at least one solution in Ŵ1,  .
Theorem 13.Assume that (, ) satisfies hypothesis (H) and the following conditions: Then problem (1) possesses at least one nontrivial solution in Ŵ1,  .
In order to prove Theorem 13, we need the following results.