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RETRACTED
This article has been retracted since the limits of the three sequences included in the manuscript are likely to be the same, which cannot be ruled out in the proof. So, the three nontrivial homoclinic solutions in Theorem 15 are likely to be the same one.
Abstract and Applied Analysis

Volume 2013 (2013), Article ID 183585, 10 pages

http://dx.doi.org/10.1155/2013/183585
Research Article

Three Homoclinic Solutions for Second-Order -Laplacian Differential System

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China

Received 13 May 2013; Revised 23 July 2013; Accepted 31 July 2013

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2013 Jia Guo and Bin-Xiang Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider second-order -Laplacian differential system. By using three critical points theorem, we establish the new criterion to guarantee that this -Laplacian differential system has at least three homoclinic solutions. An example is presented to illustrate the main result.

1. Introduction

Let us consider the following second-order -Laplacian differential system: where , , with and , is continuous, , and . As usual, we say that a solution of is nontrivial homoclinic (to 0) if , and as .

In the past two decades, many authors have studied homoclinic orbits for the second-order Hamiltonian systems and the existence and multiplicity of homoclinic solutions for (1) have been extensively investigated via critical point theory (see [115]). For instance, Yang et al. [5] have shown the existence of infinitely many homoclinic solutions for (1) by using fountain theorem.

Theorem A (see [5]). Assume that and satisfy the following conditions: (H1) and (H2) is a symmetric and positive definite matrix for all and there is a continuous function such that for all and as (H3)consider the following where is a positive continuous function such that and , , and are constants.

Then (1) possesses infinitely many homoclinic solutions.

Moreover, Tang and Xiao [10] prove the existence of homoclinic solution of (1) as a limit of the -periodic solutions of the following extension of system (1): and they established the following theorem.

Theorem B (see [10]). Assume that and satisfy the following conditions: (H4) and , where is T-periodic with respect to , and ; (H5) , as uniformly with respect to ; (H6) there is a constant such that (H7) is a continuous and bounded function;(H8) there exist constants and such that (H9) there is a constant such that (H10)consider the following

Then system (1) possesses a nontrivial homoclinic solution.

For -Laplacian problem, Tian and Ge [16] obtained sufficient conditions that guarantee the existence of at least two positive solutions of -Laplacian boundary value problem with impulsive effects. Two key conditions of the main results of [16] are listed as follows: (H11) there exist , , , , , and such that (H12) there exist , , , such that

In [17], Ricceri established a three critical points theorem. After that, several authors used it to obtain some interesting results (see [1822]).

Existence and multiplicity of solutions for -Laplacian boundary value problem have been studied extensively in the literature (see [2326]). However, to our best knowledge, the existence of at least three homoclinic solutions for -Laplacian differential system has attracted less attention.

Motivated by the aforementioned facts, in this paper we are devoted to study the multiplicity homoclinic solutions of via three critical points theorem obtained by Ricceri [17].

In order to receive the homoclinic solution of , similar to [10] we consider a sequence of system of differential equations as follows: where is a -periodic extension of restriction of to the interval , . We will prove the existence of three homoclinic solutions of as the limit of the -periodic solutions of as in [10]. However, many technical details in our paper are different from [10, 12].

2. Preliminaries

For each , let denote the Sobolev space of -periodic functions on with values in under the norm which is equivalent to the usual one. We define the norm in as .

Consider defined by where Using the continuity of , one has that is (strongly) continuous in , and for any ,

In order to prove our main result, we list some basic facts in this section.

Definition 1. A function is said to be a -periodic solution of if satisfies the equation in .

Lemma 2. If is a critical point of ; then is a -periodic solution of   .

Proof. Assume that is a critical point of ; then for all ,  one has It follows that By the definition of weak derivative, (18) implies that Thus and satisfies the . Therefore, is a solution of .

Lemma 2 motivates us to apply three critical points theorem to discuss the multiplicity of the -periodic solution of . Here, at the end of this section, let us recall some important facts.

Definition 3. Let be a Banach space and . is said to be sequentially weakly lower semi-continuous if as in .

Definition 4. Suppose is a real Banach space. For , we say that satisfies PS condition if any sequence for which is bounded and as possesses a convergent subsequence.

Lemma 5 (see [16]). For , one then has , where

Lemma 6 (see [27]). Let be a nonempty set, and are two real functions on . Assume that there are , such that Then, for each satisfying one has

Lemma 7 (see [17]). Let be a separable and reflexive real Banach space, an interval, and a function satisfying the following conditions: (i)for each , the function is continuous and concave; (ii)for each , the function is sequentially weakly lower semicontinuous and Dâteaux differentiable, and ; (iii)there exists a continuous concave function such that Then, there exist an open interval and a positive real number , such that, for each , the equation has at least two solutions in whose norms are less than . If, in addition, the function is (strongly) continuous in , and, for each , the function is and satisfies the PS condition, then the above conclusion holds with “three” instead of “two.”

Lemma 8. Let . Then for every , the following inequality holds:

Proof. Fix . For every , Integrating (27) over and using the Hölder inequality, we get

3. Main Result

In this section, our main result of this paper is presented. First, we introduce the following three conditions:(V1) there exist constants and ,  with ,   and such that    , where (V2) there exist constant and functions with such that (V3) and are continuous functions.

Remark 9. If there exist constant and functions with such that then (V2) holds.

In fact, (32) implies that there exists such that which combining the continuity of on yields that there exists constant such that

Lemma 10. Assume that (V1) holds; then, for each , there exists a continuous concave function such that

Proof. We define It is clear that . It follows from that

Let , . It is clear that has the following properties: strictly increases for and has unique solution for each .

In view of (29), (38), and (1), one has which yields that It follows from Lemma 5, , and that Let ; then is a solution of From the definition of and , we have . Thus, implies , which combining (41) yields that

Therefore,

Since , we obtain It follows from that , which combining yields that . Therefore, in view of (V1) and (44), we get . Thus, it follows from (38) and (40) that

From (43), (45), and (V1), we have It is obvious that . Owing to Lemma 6, choosing , we obtain which combining implies the conclusion.

Lemma 11. If (V2) holds, then for each , and satisfies the condition.

Proof. Let be a sequence in such that and is bounded, for each .

Lemma 5 implies that It follows from (V2) and (48) that which yields that for each . Noting that , the above inequality implies that and is bounded in . Next, we will prove that converges strongly to some in . The proof is similar to [22]. Since is bounded in , there exists a subsequence of for simplicity denoted again by such that converges weakly to some in . Then converges uniformly to on (see [28]). Therefore, as ,  for each . In view that and converges weakly to some , we get as , for each . Then, from (15), one has for each . By [29], for each , there exist such that

If , it follows from (51)–(54) that as .

If , by Holder’s inequality, we obtain for each . Similarly,

It follows from and (54)–(56) that In view of (51)–(53) and (57), we have as ,  for each .

Therefore, converges strongly to in , for each . Thus, for each , satisfies the condition.

Lemma 12. Assume that (V1) and (V2) hold; then there exist an open interval and a positive real number , such that, for each   and ,   has at least three -periodic solutions in whose norms are less than .

Proof. Let be a weakly convergent sequence to in ; then converges uniformly sequence to on . The continuity and convexity of imply that is sequentially weakly lower continuous [28, Lemma 1.2], for each , which combining the continuity of yields that Hence, is sequentially weakly lower semi-continuous, for each .

It is obvious that is continuous and concave for each . In view of Lemmas 10 and 11, it follows from Lemma 7 that there exist an open interval and a positive real number , such that, for each and , has at least three critical points in whose norms are less than . Therefore, we can reach our conclusion by using Lemma 2.

Lemma 13. Assume that (V3) holds. Let be one of the three -periodic solutions of system   obtained by Lemma 12 for each . Then there exists a subsequence of convergent to a certain in .

Proof. From Lemma 12, we have which combining Lemma 5 yields that there exists a positive constant independent of such that Thus, we obtain that is a uniformly bounded sequence. Next, we will show that and are also uniformly bounded sequences. Since is a -periodic solutions of system for every ,  we have

By (60), (61), and (V3), we get which yields that Then, from (63), (V3), and the definition of , we obtain For , by the continuity of , we can choose such that

it follows that for , Consequently,

Now we prove that the sequences and are uniformly bounded and equicontinuous. In fact, for every and , we have by (67) Similarly, from (64), we have Then, by application of the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence of and a function such that

Thus, Lemma 13 is proved.

Lemma 14. Let be determined by Lemma 13. Then is a nontrivial homoclinic solution of system .

Proof. The first step is to show that is a solution of system . By Lemma 13, one has for , . Take with . There exists such that for all one has Integrating (72) from to , we obtain for . Since (70) shows that uniformly on and uniformly on as . Let in (73), we get for . Since and are arbitrary, (74) yields that is a solution of system . It is easy to see that is not a solution of system for and so .

Secondly, we will prove that as . By (59), we have For every , there exists such that for Let in the above and use (70), and it follows that for each , Let in the above, and we get Thus Combining the above with (V3) we have By (26), we obtain Combining (80) with (81), we get as .

Finally, we show that From (60) and (70), one has From this and (64), we have If (82) does not hold, then there exist and a sequence such that which yield that for It follows that which contradicts to (78) and so (82) holds. The proof is completed.

Lemmas 13 and 14 imply that the limit of the -periodic solutions of system is a nontrivial homoclinic solution of system . Combining this with Lemma 10–Lemma 12, we can get the following.

Theorem 15. Assume that (V1), (V2), and (V3) hold. Then system possesses three nontrivial homoclinic solutions.

4. Example

Example 1. Consider the following -Laplacian problem: where , , and