Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

Abstract

In this paper, we investigate infinitely many solutions for the generalized periodic boundary value problem under the potential function without the evenness assumption and obtain two new existence results by the multiple critical point theorem. Meanwhile, we give two corollaries for the periodic solutions of second-order Hamiltonian systems and an example that illustrates our results.

1. Introduction and Main Results

We consider the existence of infinitely many solutions for the following generalized periodic boundary value problem: where , , with , is the unit matrix of order , denotes the gradient of for , and satisfies (H₀), that is, is continuously differentiable in for a.e. and measurable in for every , and there exist and such that for all and a.e. , where .

Note that if , is 1-periodic and is 1-periodic in ; then, the solutions of problem (1) are the 1-periodic solutions of second-order Hamiltonian systems.

In his pioneer paper [1] of 1978, Rabinowitz studied for the existence of periodic solutions for Hamiltonian systems via the critical point theory. From then on, with the aid of the critical point theory, the existence of infinitely many periodic solutions for Hamiltonian systems has been extensively investigated in some papers (see [2–17]) and the excellent books (see [18–20]).

For second-order Hamiltonian systems, under various conditions, the authors in [3–5, 7, 8, 14–17] obtained infinitely many periodic solutions under the evenness assumption of . Without the evenness assumption of , the authors in [2, 6, 9–13] also obtained infinitely many periodic solutions for first- (or second-) order Hamiltonian systems under the potential function . In this paper, we are interested in the potential function , and without the evenness assumption. We study the existence of infinitely many solutions of problem (1) via the multiple critical point theorem established in [21, 22] under .

Now, we use the index defined in [23] (see Section 2) to state our results.

Theorem 1. Assume that,is positive definite; i.e., there existssuch thatfor all. In addition, assume thatsatisfies(H₀) and (H₁) () and (H₂): for all , where , , and is a constant of the compact imbedding (or see (30) in Section 2).
Then, for each , problem (1) possesses infinitely many solutions, where and

Theorem 2. The conclusion of Theorem 1still holds if we replace(H₂) with (H₂): We postpone the proofs to the next section and turn to applications to second-order Hamiltonian systems. For systematic researches of second-order Hamiltonian systems, we refer to the excellent books (see [18–20]).

As the special case, we consider the periodic solution problem: where , , and and are 1-periodic. After a simple calculation, we have , , , , , and for all . Therefore, the following corollaries are immediately obtained from Theorems 1 and 2.

Corollary 3. Assume thatis positive definite andsatisfies (H₀), (H₁), and (H_2,1): for all , where .
Then, for each , problem (7) possesses infinitely many 1-periodic solutions, where are given by (34).

Corollary 4. The conclusion of Corollary 3still holds if we replace (H_2,1) with (H_2,2):

Next, an example of problem (7) is given below.

Example 5. Let and for every . Define the continuous function as follows: where and For , by it is easy to see that the conditions (H₀) and (H₁) are satisfied. Noticing that we have via a simple computation. This shows that (H_2,1) holds. By Corollary 3, there exist infinitely many 1-periodic solutions for problem (7), for each . In particular, by , we can see that the following second-order Hamiltonian systems in also have infinitely many 1-periodic solutions.

Remark 6. In Example 5, we discard the evenness assumption of potential function , which means that Example 5 does not satisfy the assumptions in [3–5, 7, 8, 14–17]. Noticing that the potential function in Example 5, we can see that Example 5 does also not satisfy the assumptions in [2, 6, 9–13]. Therefore, our result is also new even in the case of periodic solutions for second-order Hamiltonian systems.

2. Variational Setting and Proof of the Main Result

In this section, we first recall the multiple critical point theorem due to [21, 22] and some conclusions of index theory due to [23, 24], respectively.

Lemma 7 ([21], Theorem 2.1; [22], Theorem 2.5). Letbe a reflexive real Banach space,be a (strongly) continuous, coercive, sequentially weakly lower semicontinuous and Gâteaux differentiable functional, andbe a sequentially weakly upper semicontinuous and Gâteaux differentiable functional. For all, let

Then, (a)For every and every , the restriction of the functional to admits a global minimum, which is a critical point (local minimum) of in (b)If , then, for each , the following alternative holds: either (b₁) possesses a global minimum or(b₂)there is a sequence of critical points (local minima) of such that (c)If , then, for each , the following alternative holds: either (c₁)there is a global minimum of which is a local minimum of or(c₂)there is a sequence of pairwise distinct critical points (local minima) of , with which weakly converges to a global minimum of

Index theory in [23] deals with a classification of associated with the following system: where , , and .

Let . Set with . By Section 2.4 in [23], we see that is self-adjoint and . In particular, if , then , and if , then . By Corollary 1.21 in [25], we know that is a Hilbert space with the norm for each , and the embeddings , , and are compact.

For any , we define a bilinear form as follows:

Proposition 8 ([24], Proposition 7.2.1). For any, the spacehas a-orthogonal decomposition:such that is positive definite, null, and negative definite on , , and , respectively. Moreover, and are finite-dimensional.

Definition 9 ([23], Definition 2.4.1; [24], Definition 7.1.3). For any, we defineWe call and the nullity and index of with respect to the bilinear form , respectively.

Proposition 10 ([24], Proposition 7.2.2). For any, we haveand.

Proposition 11 ([23], Proposition 2.4.2 (1); [24], Corollary 7.2.2 (i)). For any, we have thatis the solution subspace of systems (18) and (19), and.

Remark 12 ([23], Example 2.4.3; [24], Remark 7.1.3). Let be the eigenvalues of a constant symmetric matrix . For with , we have where denotes the number of elements in set . In particular, formulae (23) and (24) when were given first by Mawhin and Willem in [19].

Next, we establish the variational setting for problem (1).

It is known that the operator is also self-adjoint and is bounded from below. Noticing that and , by Definition 9 and Proposition 10, we know that the operator has a sequence of eigenvalues: and the system of eigenfunctions corresponding to forming an orthogonal basis in . Hence, we can define another inner product: with the corresponding norm Clearly, is equivalent to . Put

Since is positive definite, there exist such that for all . So, we have for all .

Noticing the compactness of the embedding , from (29), we know that there is an embedded constant such that for all , where and is the norm of .

Now, we define

From the assumption (H₀) and Theorem 1.2 in [19], it is easy to verify that is weakly lower semicontinuous on and is weakly continuous with for all . If , we easily find that the critical points of correspond to the solutions of problem (1) and omit the details.

Finally, we give the proofs of Theorems 1 and 2.

For convenience, put

Since , we have . So, if , we put , and if , we put .

Proof of Theorem 1. Set that Obviously, is (strongly) continuous, coercive, and Gâteaux differentiable, is sequentially weakly upper semicontinuous and Gâteaux differentiable, and . On the other hand, the critical points of in are the solutions of problem (1).
Take that . Pick a sequence such that and . Set . By (30), we have for any with . From definition of and (3) of H₂, we obtain which shows that Next, we prove that in is unbounded from below.
Again, pick a sequence such that and Noticing that , by Proposition 8, we have If , put By (34), we know that there exists such that This implies that Choosing small enough , from (34) and , we have If , put From (38), there exists such that This shows that Noting that , from (b) of Lemma 7, we know that admits a sequence of critical points with . The proof is complete.

Furthermore, from (c) of Lemma 7, we can prove Theorem 2 similar to Theorem 1. Here, we omit it.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 11901248) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJB110006).