Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

Abstract

This paper is concerned with the following nonlinear second-order nonautonomous problem: , where , , and , are not periodic in and is a continuous function and with . The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.

1. Introduction

Consider fast homoclinic solutions of the following problem: where , , and are not periodic in and is a continuous function and with

When , problem (1) reduces to the following second-order Hamiltonian system:

When , where is a positive definite symmetric matrix-valued function for all , then problem (1) reduces to the following damped vibration problem:

If and , problem (1) reduces to the following second-order Hamiltonian system:

It is well known that the existence of homoclinic orbits plays a very important role in the study of the behavior of dynamical systems. The first work about homoclinic orbits was done by Poincaré [1].

In the past years, many researchers paid attention to the existence and multiplicity of homoclinic solutions for systems (3) and (5) by critical point theory. For example, see [2–13] and references cited therein. However, there is only a few researches about the existence of homoclinic solutions for damped vibration problems (4) when . Wu and Zhou [14] established some results for a class of damped vibration problems with obstacles. Wu et al. [15] obtained some results for (4) with some boundary value conditions by variational methods. Zhang and Yuan [16] studied the existence of homoclinic solutions for (4) when is a constant. Later, Chen et al. [17] investigated fast homoclinic solutions for (4) and obtained the following Theorem A under more relaxed assumptions on , which resolved some open problems in [16]. Zhang et al. [18] considered a special nonautonomous problem and obtained some results of fast homoclinic orbits. Zhang [19] investigated a class of damped vibration problems with impulsive effects and established some existence and multiplicity results of fast homoclinic solutions. For the applications of weighted Sobolev space in elliptic equations, please see [20, 21] and references cited therein.

Theorem A (see [17]). Assume that , , and satisfy (2) and the following assumptions:(L) is a -valued continuous function of and there exists constant such that (W1), , and there exists constant such that   uniformly in .(W2) There is constant such that (W3) and there exists a constant such that Then problem (4) has at least one nontrivial fast homoclinic solution.

Otherwise, Chen et al. [17] obtained the multiplicity of fast homoclinic solutions for (4) if the following condition holds:(W4), .

Motivated by the abovementioned works, we will establish some results for (1). In order to introduce the concept of the fast homoclinic solutions for (1), we first state some properties of the weighted Sobolev space on which the certain variational associated with (1) is defined and the fast homoclinic solutions are the critical points of the certain functional.

Let where is defined in (2) and for , where the inner product is given by Then is a Hilbert space with the norm given by It is obvious that with the embedding being continuous. Here denotes the Banach spaces of functions on with values in under the norm Similar to [16–19], we have the following definition of fast homoclinic solutions.

Definition 1. If (2) holds, a solution of (1) is called a fast homoclinic solution if .

Now, we state our main results.

Theorem 2. Suppose that , satisfy (2) and (W1)–(W3) and satisfies the following conditions: (K1), and there exist two positive constants and such that   where satisfies (L).(K2) There exists a constant such that (K3) There is a constant such that Then problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 3. Suppose that , , and satisfy (2), (K1)–(K3), (W2), and the following conditions:(W1)′, , and   uniformly in .(W3)′ and there exists a constant such that Then problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 4. Suppose that , , and satisfy (2), (K1)–(K3), (W1)–(W3), and the following condition:(W4)′, , .Then problem (1) has an unbounded sequence of fast homoclinic solutions.

Theorem 5. Suppose that , , and satisfy (2), (K1)–(K3), (W1)′, (W2), (W3)′, and (W4)′. Then problem (1) has an unbounded sequence of fast homoclinic solutions.

Remark 6. We point out that (K1) is used in [13]. There are functions which can not be written in the form ; then the results we obtain here are different. It is also remarked that the function is not necessarily homogeneous of degree 2 with respect to and so may not be a norm in general.

The rest of this paper is organized as follows: in Section 2, some preliminaries are presented. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2. Preliminaries

The functional corresponding to (1) on is given by Clearly, it follows from (K1), (W1), or (W1)′ that and one can easily check that Furthermore, the critical points of in are classical solutions of (1) with .

Let and be given in Section 1. The following lemma is important.

Lemma 7 (see [17]). For any , where , .

The following two lemmas are Mountain Pass Theorem and Symmetric Mountain Pass Theorem, which are useful in the proofs of our theorems.

Lemma 8 (see [22]). Let be a real Banach space and satisfying (PS)-condition. Suppose and(i)there exist constants , such that ;(ii)there exists such that .Then possesses critical value which can be characterized as , where and is an open ball in of radius centered at .

Lemma 9 (see [22]). Let be a real Banach space and with being even. Assume that and satisfies (PS)-condition, (i) of Lemma 8, and the following condition:(iii)For each finite dimensional subspace , there is such that for , where is an open ball in of radius centered at .Then possesses an unbounded sequence of critical values.

Remark 10. Since it is very difficult to check condition (iii) of Lemma 9, only a few results about infinitely many homoclinic solutions can be seen in the literature by using Lemma 9, let alone fast homoclinic solutions. Motivated by the idea of Tang and Lin [10], we use Lemma 9 to obtain infinitely many fast homoclinic solutions for problem (1).

Lemma 11. Assume that (W2) and (W3) or (W3)′ hold. Then for every , (i) is nondecreasing on ;(ii) is nonincreasing on .

The proof of Lemma 11 is routine and we omit it.

3. Proofs of Theorems

Proof of Theorem 2. Consider the following:
Step 1. The functional satisfies the (PS)-condition. Let satisfying be bounded and as . Then there exists a constant such that From (20), (21), (25), (K2), (W2), and (W3), we have The above inequalities imply that there exists a constant such that Now we prove that in . Passing to a subsequence if necessary, it can be assumed that in . For any given number , by (W1), we can choose such that Since as , we can choose such that It follows from (23), (27), and (29) that Similarly, by (24), (27), and (29), we have Since in , it is easy to verify that converges to pointwise for all . Hence, it follows from (30) and (31) that Since on , the operator defined by is a linear continuous map. So in . Sobolev theorem implies that uniformly on , so there is such that From (L), (27), (28), (30), (31), and (32), we have It follows from (33) and (34) that From (21) and (K3), as , we have It follows from (35) and (36) that Hence, in by (37). This shows that satisfies (PS)-condition.
Step 2. From (W1), there exists such that By (38), we have Let Set and ; it follows from Lemma 7 that for . From Lemma 11(i), (L), and (40), we have By (L), (K1), (W3), (39), and (41), we have Therefore, we can choose constant depending on such that for any with .
Step 3. From Lemma 11(ii) and (22), we have for any where and . Take such that and for . For , from Lemma 11(i) and (44), we get where . From (W3), (20), (43), (44), and (45), we get for Since and , it follows from (46) that there exists such that and . Set ; then , , and . It is easy to see that . By Lemma 8, has critical value given by where Hence, there exists such that The function is a desired solution of problem (1). Since , is a nontrivial fast homoclinic solution. The proof is complete.

Proof of Theorem 3. In the proof of Theorem 2, the condition in (W3) is only used in the proofs of (27) and Step 2. Therefore, we only need to prove that (27) and Step 2 still hold if we use (W1)′ and (W3)′ instead of (W1) and (W3), respectively. We first prove that (27) holds. From (K2), (W2), (W3)′, (20), (21), and (25), we have which implies that there exists a constant such that (27) holds. Next, we prove that Step 2 still holds. From (W1)′, there exists such that By (51), we have Let ; it follows from Lemma 7 that . It follows from (L), (K1), (20), and (52) that Therefore, we can choose constant depending on such that for any with . The proof of Theorem 3 is complete.

Proof of Theorem 4. Condition (W4)′ shows that is even. In view of the proof of Theorem 2, we know that and satisfies (PS)-condition and assumption (i) of Lemma 8. Now, we prove (iii) of Lemma 9. Let be a finite dimensional subspace of . Since all norms of a finite dimensional space are equivalent, there exists such that Assume that dim and is a base of such that For any , there exists , , such that Let It is easy to see that is a norm of . Hence, there exists a constant such that . Since , by Lemma 7, we can choose such that where is given in (52). Let Hence, for , let such that Then by (54)–(57), (59), and (60), we have This shows that and there exists such that , which, together with (58), implies that . Let and Since for all and and , it follows that . For any , from Lemma 7 and Lemma 11(i), we have where and . Since , , it follows that there exists such that Then for with and , it follows from (56), (59), (60), (61), and (64) that On the other hand, since for , then Therefore, from (62), (65), and (66), we have By (58) and (59), we have By (L), (K1), (20), (39), (63), (67), (68), and Lemma 11, we have for and Since , we deduce that there exists such that It follows that which shows that (iii) of Lemma 9 holds. By Lemma 9, possesses unbounded sequence of critical values with , where is such that for . If is bounded, then there exists such that By a similar fashion for the proof of (30) and (31), for the given in (39), there exists such that Hence, by (L), (20), (22), (39), (72), and (73), we have On the other hand, by (K1), we have It follows from (74) and (75) that This contradicts the fact that is unbounded, and so is unbounded. The proof is complete.

Proof of Theorem 5. In view of the proofs of Theorems 3 and 4, the conclusion of Theorem 5 holds. The proof is complete.

4. Examples

Example 1. Consider the following system: where , , . Let where and , , , . By an easy computation, we have Then satisfies (K1), (K2), (K3), and (W4)′. Let Then it is easy to check that all the conditions of Theorem 4 are satisfied with and . Hence, problem (77) has an unbounded sequence of fast homoclinic solutions.

Example 2. Consider the following system: where , , and . Let and be the same in Example 1 and where , , , and , . Let Then it is easy to check that all the conditions of Theorem 5 are satisfied with and . Hence, by Theorem 5, problem (81) has an unbounded sequence of fast homoclinic solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the NNSF of China (no. 11301108), Guangxi Natural Science Foundation (no. 2013GXNSFBA019004 and no. 2013GXNSFDA019001), the Scientific Research Foundation of Guangxi Education Office (no. 201203YB093), and the Program Sponsor Terms for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ().