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This article has been retracted since the limits of the three sequences included in the manuscript are likely to be the same, which cannot be ruled out in the proof.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
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.
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.
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 [1–15]). For instance, Yang et al.  have shown the existence of infinitely many homoclinic solutions for (1) by using fountain theorem.
Theorem A (see ). Assume thatandsatisfy the following conditions: (H1)and(H2)is a symmetric and positive definite matrix for alland there is a continuous functionsuch thatfor alland
as(H3)consider the following
whereis a positive continuous function such thatand, , and are constants.
Then (1) possesses infinitely many homoclinic solutions.
Moreover, Tang and Xiao  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 ). Assume thatandsatisfy the following conditions: (H4)and , where is T-periodic with respect to , and ; (H5), as uniformly with respect to ; (H6) there is a constantsuch 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  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  are listed as follows: (H11) there exist, , , , , and such that (H12) there exist, , , such that
Existence and multiplicity of solutions for -Laplacian boundary value problem have been studied extensively in the literature (see [23–26]). 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 .
In order to receive the homoclinic solution of , similar to  we consider a sequence of system of differential equations as follows: whereis a -periodic extension of restriction ofto the interval, . We will prove the existence of three homoclinic solutions of as the limit of the -periodic solutions of as in . However, many technical details in our paper are different from [10, 12].
For each, letdenote the Sobolev space of -periodic functions onwith values inunder the norm which is equivalent to the usual one. We define the norm inas .
Considerdefined 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.
Proof. Assume thatis a critical point of; then for all, one has It follows that By the definition of weak derivative, (18) implies that Thusandsatisfies 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. Letbe a Banach space and.is said to be sequentially weakly lower semi-continuous if as in.
Definition 4. Supposeis a real Banach space. For, we say that satisfies PS condition if any sequencefor whichis bounded andaspossesses a convergent subsequence.
Lemma 5 (see ). For, one then has, where
Lemma 6 (see ). Letbe a nonempty set, andare two real functions on. Assume that there are, such that Then, for eachsatisfying one has
Lemma 7 (see ). Letbe a separable and reflexive real Banach space,an interval, anda function satisfying the following conditions: (i)for each, the functionis continuous and concave; (ii)for each, the functionis sequentially weakly lower semicontinuous and Dâteaux differentiable, and; (iii)there exists a continuous concave functionsuch that Then, there exist an open intervaland a positive real number, such that, for each, the equation has at least two solutions inwhose norms are less than. If, in addition, the functionis (strongly) continuous in, and, for each, the functionisand 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) overand 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 constantsand, with, and such that , where (V2) there exist constantand functionswith such that (V3)and are continuous functions.
Remark 9. If there exist constantand functionswithsuch that
then (V2) holds.
In fact, (32) implies that there existssuch that which combining the continuity of onyields that there exists constantsuch that
Lemma 10. Assume that (V1) holds; then, for each, there exists a continuous concave functionsuch that
Proof. We define
It is clear that. It follows from
Let, . It is clear thathas the following properties: strictly increases for and has unique solutionfor 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 ofFrom the definition ofand, we have. Thus, implies, which combining (41) yields that
Since, we obtain It follows from that, which combiningyields 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,andsatisfies thecondition.
Proof. Letbe a sequence insuch thatandis 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 thatandis bounded in. Next, we will prove thatconverges strongly to somein. The proof is similar to . Sinceis bounded in, there exists a subsequence offor simplicity denoted again by such thatconverges weakly to somein. Thenconverges uniformly toon(see ). Therefore, as, for each. In view that andconverges weakly to some, we get as , for each. Then, from (15), one has for each. By , for each, there existsuch 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 haveas, for each.
Therefore,converges strongly toin, for each. Thus, for each, satisfies thecondition.
Proof. Letbe a weakly convergent sequence to in; thenconverges uniformly sequence toon. The continuity and convexity of imply thatis sequentially weakly lower continuous [28, Lemma 1.2], for each, which combining the continuity ofyields that
Hence,is sequentially weakly lower semi-continuous, for each.
It is obvious thatis continuous and concave for each. In view of Lemmas 10 and 11, it follows from Lemma 7 that there exist an open intervaland a positive real number, such that, for each and, has at least three critical points inwhose norms are less than. Therefore, we can reach our conclusion by using Lemma 2.
Lemma 13. Assume that (V3) holds. Letbe one of the three -periodic solutions of system obtained by Lemma 12 for each. Then there exists a subsequenceofconvergent to a certainin.
Proof. From Lemma 12, we have
which combining Lemma 5 yields that there exists a positive constantindependent ofsuch that
Thus, we obtain that is a uniformly bounded sequence. Next, we will show thatandare also uniformly bounded sequences. Sinceis 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 choosesuch that
it follows that for, Consequently,
Now we prove that the sequencesand are uniformly bounded and equicontinuous. In fact, for everyand, we have by (67) Similarly, from (64), we have Then, by application of the Arzelà-Ascoli Theorem, we obtain the existence of a subsequenceof and a functionsuch that
Thus, Lemma 13 is proved.
Lemma 14. Letbe determined by Lemma 13. Thenis a nontrivial homoclinic solution of system .
Proof. The first step is to show thatis a solution of system . By Lemma 13, one has
for, . Takewith. There existssuch that for allone has
Integrating (72) fromto, we obtain
for. Since (70) shows thatuniformly onanduniformly onas. Letin (73), we get
for. Sinceandare arbitrary, (74) yields thatis a solution of system . It is easy to see thatis not a solution of system forand so.
Secondly, we will prove thatas. By (59), we have For every, there existssuch that for Letin the above and use (70), and it follows that for each, Letin the above, and we get Thus Combining the above with (V3) we have By (26), we obtain Combining (80) with (81), we getas.
Finally, we show that From (60) and (70), one has From this and (64), we have If (82) does not hold, then there existand a sequencesuch that which yield that for It follows that which contradicts to (78) and so (82) holds. The proof is completed.
Example 1. Consider the following -Laplacian problem: where,, and