Variational Approach to Impulsive Differential Equations with Singular Nonlinearities

Differential equations with impulsive effects appear naturally in the description of many evolution processes whose states experience sudden changes at certain times, called impulse moments. There is an extensive bibliography about the subject. For recent references, see [1]. Variational methods have been successfully employed to investigate regular second order differential equations with impulsive effects; See, for instance, [2–8]. In particular, the paper [8] considers the existence of ndistinct pairs of nontrivial solutions. However, very few papers have used variational methods to investigate the case of impulsive second order boundary value problems with singular nonlinearities. In fact, it seems that the work [9] is the first paper along this line. Singular boundary value problems without impulses have attracted the attention of many researchers; see [10] for details and references. This paper is devoted to the study of the existence and multiplicity of periodic solutions for impulsive second order differential equations with singular nonlinearities. More specifically, we consider the following impulsive problem:


Introduction
Differential equations with impulsive effects appear naturally in the description of many evolution processes whose states experience sudden changes at certain times, called impulse moments.There is an extensive bibliography about the subject.For recent references, see [1].

The Main Result
In this section we state and prove our main result.Theorem 2. Suppose that conditions (H1) and (H2) hold.Then for any  ∈ N, there exists   such that for  >   problem (1) has infinitely many distinct nontrivial solutions.Proof.To give the proof of the main result, we first modify problem (1) to another one which is not singular.
First, we show that   is bounded from below.Define a subset Ω of  1  as follows: noticing that For  ∈ Ω, we have If  ∉ Ω, we use the partition of the interval  =  1 ∪  2 ∪  3 , where By (H2) and (jjj), we have From (jjj),   (, ()) < 0 on  1 , which implies Since   is a Caratheodory function, it follows that   is bounded by some positive constant .Hence, This shows that   is bounded from below.
Journal of Applied Mathematics 5 Now, we apply Theorem 9.12 in [11] to conclude that   possesses infinitely many distinct pairs of nontrivial critical points.That is, problem (11) has infinitely many distinct pairs of distinct nontrivial solutions.
We consider  ≥  in the definition of   ().We have  (54) Applying our main result, we see that when  >   , for any  ∈ N * , problem (49) has infinitely many distinct nontrivial solutions.