- Open Access
Existence of homoclinic solutions for a class of second order p-Laplacian systems with impulsive effects
© Li and Chen; licensee Springer 2014
- Received: 28 July 2014
- Accepted: 16 September 2014
- Published: 2 October 2014
This paper is concerned with the existence of homoclinic solutions for a class of second order p-Laplacian systems with impulsive effects. A new result is obtained under more relaxed conditions by using the mountain pass theorem, a weak convergence argument, and a weak version of Lieb’s lemma.
MSC: 34C37, 35A15, 37J45, 47J30.
- homoclinic solutions
- variational methods
- critical point
- impulsive effects
where , is of class , , , , and as . ℤ denotes the sets of integers, and () are impulsive points. Moreover, there exist a positive integer m and a positive constant T such that , , , . and represent the right and left limits of at , respectively.
By using the mountain pass theorem, a weak convergence argument, and a weak version of Lieb’s methods, Fang and Duan  investigated homoclinic solutions of problem (1.2) and obtained the following main result.
Assume that the following conditions hold:
(V2): uniformly for;
Then problem (1.2) possesses a nontrivial weak homoclinic orbit.
For , problem (1.1) involves impulsive effects. It is well known that impulsive differential equations are used in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books , .
Recently, the existence and multiplicity of solutions for impulsive differential equations via variational methods have been investigated by some researchers. See for example – and references therein. However, there are few papers , – concerning homoclinic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of homoclinic solutions for impulsive differential equations.
Motivated by the above papers, we will establish a new result for (1.1).
Here and in subsequence, and denote the inner product and norm in ℝ, respectively. () denote different positive constants. Now, we state our main result.
Suppose that a, I, and V satisfy (V1) and the following conditions:
(A): , , andas;
Then problem (1.1) has a nontrivial homoclinic solution.
is nondecreasing on ;
is nonincreasing on .
where, from (A).
Ifis bounded in E anddoes not converge to 0 in measure, then there exist a sequenceand a subsequenceofsuch thatin E.
Furthermore, the critical points of φ in E are classical solutions of (1.1) with.
The proof is complete. □
Proof of Theorem 1.1
Since and , it follows from (3.7) that there exists such that and . Set , then , and .
where , . Furthermore, does not converge to 0 in measure.
This is a contradiction. Hence, (3.8) holds and does not converge to 0 in measure.
Therefore, and u is a nontrivial homoclinic solution of φ. □
This work is supported by the Scientific Research Foundation of Guangxi Education Office of China (No. 2013LX171) and the Scientific Research Foundation of Guilin University of Aerospace Technology (No. YJ1301).
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