Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition
© Lu and Lu; licensee Springer. 2014
Received: 8 April 2014
Accepted: 2 September 2014
Published: 24 September 2014
By means of critical point theory and some analysis methods, the existence of homoclinic solutions for the p-Laplacian system with delay, , is investigated. Some new results are obtained. The interesting thing is that the function is only required to satisfy a local condition. Furthermore, the results are all explicitly related to the value of delay τ.
MSC:34C37, 58E05, 70H05.
where is a constant. The following theorem was obtained.
Theorem 1.1 (See )
Assume that F and f satisfy the following conditions:
(B1) is T-periodic with respect to t, is a constant;
(B3) is a continuous and bounded function such that .
Then (1.1) possesses a homoclinic solution.
From the proof of Theorem 1.1 in , we see that assumption (B2) is crucial for obtaining the existence of a homoclinic solution for (1.1).
where is a constant, , , and being τ-periodic in t. Under
and some other conditions, they obtained the result that (1.2) possesses a nontrivial homoclinic orbit.
where , , and is T-periodic in t, , , , and are constants. The interesting thing of this paper is that the period T of function with respect to the variable t may not be equal to τ, and the main results are all expressively related to the value of delay τ. Furthermore, even if for the case of , we do not require condition (B2) for guaranteeing the coercive potential condition.
As is well known, a solution of (1.3) is named homoclinic (to 0) if and as . In addition, if , then u is called a nontrivial homoclinic solution.
where , is a -periodic extension of restriction of e to the interval . We obtain the following theorem.
Theorem 1.2 Assume that G and e satisfy the following conditions:
(C1) for all ;
where ρ is a constant with ;
(C3) is a continuous and bounded function such that .
So by using Theorem 1.2, we have the following result.
which implies condition (B2) for guaranteeing the coercive potential does not hold for (1.3). Moreover, we do not require that is τ-periodic function with respect to t, which is required by , and local condition (C2) is essentially different from assumption [H2] in .
If μ in assumption (C2) is the case , then we have further the following result.
Theorem 1.3 Assume that assumption (C1) in Theorem 1.2 is satisfied together with the following conditions:
(D3) is a continuous and bounded function such that .
So by using Theorem 1.3, we have the following result.
where is determined in (1.12).
Let with and , where , , and σ are positive constants. Then as .
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
In order to investigate the existence of homoclinic solutions to (1.3), we should study the existence of -periodic solutions to (1.4) for each in the first case.
where and are constants independent of k.
The proof is complete. □
where is a constant determined by (D2) and (1.9). Clearly, ρ is independent of k.
The proof is complete. □
Lemma 2.7 
Let be the -periodic solution to (1.4) that satisfies (2.1) for each . Then there exists a subsequence of convergent to a in .
3 Proof of main result
Since for and is continuous differential of on for every , it follows that on . In view of being arbitrary with , , , that is, , is a solution to (1.5).
Below, we will prove and as .
By using assumption (C1), we get for all . So it follows from (3.5) that , which contradicts the fact that in assumption (C3). The proof is complete. □
Since the proof of Theorem 1.3 works almost exactly as the proof of Theorem 1.2, we omit the proof of Theorem 1.3 here.
where , , and is a constant.
Thus, by using Corollary 1.1, we see that (3.6) has a nontrivial homoclinic solution for small enough.
Clearly, we can choose such that all the conditions of Corollary 1.1 are satisfied. So (3.7) has a nontrivial homoclinic solution.
which implies that the crucial assumption (B2) for guaranteeing the coercive condition in  (see Theorem 1.1 in Section 1) does not hold. So the results in present paper are essentially new.
Research supported by the NSF of China (No. 11271197).
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