In this paper, we study the following nonlinear impulsive differential equations with Dirichlet boundary conditions
where , , , is continuous, , N is a positive integer, , , are continuous. With the help of the symmetric mountain-pass lemma due to Kajikiya , we prove that there are infinitely many small weak solutions for equations (1.1) with the general nonlinearities .
In recent years, a great deal of works have been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph . For some general and recent works on the theory of impulsive differential equations, we refer the reader to [3–13]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory , the method of upper and lower solutions with a monotone iterative technique , and some fixed point theorems in cones [16, 17].
On the other hand, in the last few years, many authors have used a variational method to study the existence and multiplicity of solutions for boundary value problems without impulsive effects [18–21]. For related basic information, we refer the reader to [22, 23].
For a second order differential equation , one usually considers impulses in the position u and the velocity . However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in the position [24–27].
A new approach via critical point and variational methods is proved to be very effective in studying the boundary problem for differential equations. For some general and recent works on the theory of critical point theory and variational methods, we refer the reader to [28–37].
More precisely, in [28
] the authors studied the following equations with Dirichlet boundary conditions:
They obtained the existence of solutions for problems by using the variational method. Zhang and Yuan  extended the results in . They obtained the existence of solutions for problem (1.2) with a perturbation term. Also, they obtained infinitely many solutions for problem (1.2) under the assumption that the nonlinearity f is a superlinear case. Soon after that, Zhou and Li  extended problem (1.2). In all the above-mentioned works, the information on the sequence of solutions was not given.
Motivated by the fact above, the aim of this paper is to show the existence of infinitely many solutions for problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions, converging to zero, by using a new version of the symmetric mountain-pass lemma due to Kajikiya . Our main results extend the existing study.
Throughout this paper, we assume that is continuous, and satisfies the following conditions:
) are odd and satisfy
) There exist
(H1) , for all ;
(H2) uniformly for ;
(H3) uniformly for .
The main result of this paper is as follows.
Theorem 1.1 Suppose that (I1)-(I2) and (H1)-(H3) hold. Then problem (1.1) has a sequence of nontrivial solutions and as .
Remark 1.1 Without the symmetry condition (i.e., and ), we can obtain at least one nontrivial solution by the same method in this paper.
We should point out that Theorem 1.1 is different from the previous results of [28
] in three main directions:
We do not make the nonlinearity f satisfy the well-known Ambrosetti-Rabinowitz condition ;
We try to use Lusternik-Schnirelman’s theory for -invariant functional. But since the functional is not bounded from below, we could not use the theory directly. So, we follow  to consider a truncated functional.
Remark 1.3 There exist many functions and satisfying conditions (I1)-(I2) and (H1)-(H3), respectively. For example, when , and .