Abstract
Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.
1. Introduction
In this paper, we consider the following superlinear system with noninstantaneous impulsive effects: where , , , , , , the impulses start abruptly at points and keep the derivative constant on a finite time interval , . Here, is a given positive integer and . Denote where .
This paper is mainly motivated by two facts. For the past thirty years, impulsive differential equations have been studied extensively in the literature. There are many approaches to investigate solutions for impulsive differential equations, such as the method of lower and upper solutions, fixed point theory, coincidence degree theory, and geometric approach (see for example [1–8]). In addition, variational methods have been also dealt with impulsive differential equations since papers [9, 10] appeared. Meanwhile, many existence and multiplicity results of solutions for various impulsive differential equations have been obtained. We refer the reader to a large quantity of the references for papers [9, 10]. It is worth noting that in all these references, most of impulsive effects are instantaneous. However, in [11, 12], the authors considered a kind of noninstantaneous impulsive effects, which is motivated by paper [13]. On the other hand, in recent years, some new critical point theorems are used to study nonlinear differential equations with no impulsive effects (see for example, [14–21]). Based on these two facts, we investigate the existence of nontrivial solution for (1).
This paper is organized as follows. In Section 2, we recall a critical point theorem in [16] and introduce a definition of the weak solution for (1), which generalizes Definition 2.3 in [12] to an -dimensional case. In Section 3, we present our result and complete its proof via variational methods. The result is new in the field of studying noninstantaneous impulsive problem. In Section 4, we present an example in order to illustrate our result.
2. Preliminaries
The following critical point theorem is crucial in our arguments.
Theorem 1 (see [16], Theorem 3.2). Let be a real Banach space and let be two continuously Gâteaux differentiable functions such that is bounded from below and . Fix such that and assume that for each , the functional satisfies the (PS)-condition and it is unbounded from below. Then, for each , the functional admits at least two distinct critical points.
Consider the Hilbert space: with the inner product where is the inner product in . The corresponding norm is defined by
Meanwhile, for every , we define which is also an inner product in , whose corresponding norm is that
Poincare’s inequality [22] implies that where is the first eigenvalue of the Dirichlet problem
Hence,
That is, is equivalent to . It is well known that is compactly embedded in , where . Since the norms and are equivalent, there exists a positive number such that where .
Following the idea of the variational approach for impulsive differential equation of [11, 12] and combining with (1), one has that for every ,
In addition, combing with (1) again, one has
On the other hand,
Hence, (14) and (15), together with , imply that
Similar to Definition 2.3 in [12], we can introduce the following concept of a weak solution for problem (1).
Definition 2. Say that is a weak solution for problem (1) if (16) holds for any .
Now, we consider the functionals defined by putting
Since , and , and are two continuously Gâteaux differentiable with for every . Hence,
Namely, . Definition 2, together with (20)–(22), shows that a critical point of corresponds to a weak solution for (1).
3. Main Result and Its Proof
From now on, we refer to as the range of , unless specifically stated. Let be the usual norm in . The following theorem is our main result.
Theorem 3. Suppose that
() There exist and such that for ,
hold. Then, for every and for every
(1) admits at least one nontrivial weak solution.
Proof. We complete the proof in four steps.
Step 1. Clearly, is a real Banach space. are two continuously Gâteaux differentiable functional. In addition, (17), (18), and imply that is bounded from below and .
Step 2. satisfies the (PS)-condition. That is, every such that is bounded and as contains a convergent subsequence.
By (11), we have
Let , then . From and (25), we deduce that
which implies that is bounded in . Since is a reflexive Banach space, passing to a subsequence if necessary, we may assume that there is a such that in . Then, converges uniformly to on and in . In addition,
Hence, the second term on the right hand of (27) converges to as because of (28) and . Moreover, as implies
as . (27)–(29) show that as . By the completeness of , we see that contains a convergent subsequence in .
Step 3. is unbounded from below.
By (), there exist such that
for every .
Denote , , . Hence,
for every .
Let and with and is not a constant for a.e. . In view of (31), we have
Denote
Then,
Hence, by (17)–(19) and (32)–(34), we have
which shows that as because the coefficient of is negative.
Step 4. .
Let be such that
and put
By (18), one has
Now, we apply Theorem 1 to conclude that (1) admits at least one nontrivial weak solution.
4. Example
Let us consider the following problem: where .
In this case, , , , . Choose , one can verify that all the conditions in Theorem 3 are satisfied and (39) admits at least one nontrivial weak solution.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
This work is supported by scientific research project of department director of Zhenjiang College (grant number: 2117080045).