In this paper, we consider the existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Infinitely many solutions are obtained by using a version of the symmetric mountain-pass theorem, and this sequence of solutions converge to zero. Some recent results are extended.
MSC: 34B37, 35B38.
Keywords:impulsive effects; variational methods; Dirichlet boundary value problem; critical points
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  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:
(I1) ( ) are odd and satisfy
(I2) There exist , such that
(H1) , for all ;
(H2) uniformly for ;
(H3) uniformly for .
The main result of this paper is as follows.
Theorem 1.1Suppose 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.
(1) We do not make the nonlinearity f satisfy the well-known Ambrosetti-Rabinowitz condition ;
(2) 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.
(3) We can obtain a sequence of nontrivial solutions and as .
Remark 1.3 There exist many functions and satisfying conditions (I1)-(I2) and (H1)-(H3), respectively. For example, when , and .
2 Preliminary lemmas
In this section, we first introduce some notations and some necessary definitions.
Definition 2.1 Let E be a Banach space and . J is said to be sequentially weakly lower semi-continuous if as in E.
Definition 2.2 Let E be a real Banach space. For any sequence , if is bounded and as possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (denoted by (PS) condition for short).
In the Sobolev space , consider the inner product
which induces the norm
It is a consequence of Poincaré’s inequality that
Here, is the first eigenvalue of the Dirichlet problem
In this paper, we will assume that . We can also define the inner product
which induces the equivalent norm
If , then the norm and the norm are equivalent.
There exists such that if , then
For , we have that u and are both absolutely continuous, and , hence, for any . If , then u is absolutely continuous and . In this case, the one-side derivatives and may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that satisfies the Dirichlet condition . Assume that, for every , and . Let .
Taking and multiplying the two sides of the equality
by v and integrating between 0 and T, we have
Moreover, since , one has
Combining (2.3), we get
Lemma 2.3A weak solution of (1.1) is a function such that
for any .
Consider defined by
where . Using the continuity of f and , , we obtain the continuity and differentiability of J and . For any , one has
Thus, the solutions of problem (1.1) are the corresponding critical points of J.
Lemma 2.4If is a weak solution of problem (1.1), thenuis a classical solution of problem (1.1).
Proof Obviously, we have since . By the definition of weak solution, for any , one has
For , choose with for every . Then
By the definition of weak derivative, the equality above implies that
Hence and u satisfies the equation in (1.1) a.e. on . By integrating (2.7), we have
Combining this fact with (2.8), we get
Hence, for every , and the impulsive condition in (1.1) is satisfied. This completes the proof. □
Lemma 2.5If , then the functionalJis sequentially weakly lower semi-continuous.
Proof Let be a weakly convergent sequence to u in , then
We have that converges uniformly to u on . Then
This completes the proof. □
Under assumptions (H1) and (H2), we have
which means that for all , there exist such that
Hence, for every positive constant k, we have
Lemma 2.6Suppose that (I1)-(I2) and (H1)-(H3) hold, then satisfies the (PS) condition.
Proof Let be a sequence in such that is bounded and as . First, we prove that is bounded. By (2.5), (2.6) and (2.11), one has
By condition (I1), we can deduce that
Setting , we get
where and M is a positive constant. On the other hand, by (I1), (2.5) and (2.10), we have
Thus, (2.12) and (2.13) imply that is bounded in . Going if necessary to a subsequence, we can assume that there exists such that
as . Hence,
as . Moreover, one has
Therefore, as . That is converges strongly to u in . That is J satisfies the (PS) condition. □
3 Existence of a sequence of arbitrarily small solutions
In this section, we prove the existence of infinitely many solutions of (1.1), which tend to zero. Let X be a Banach space and denote
For , we define genus as
If there is no mapping φ as above for any , then . We list some properties of the genus (see ).
Proposition 3.1LetAandBbe closed symmetric subsets ofX, which do not contain the origin. Then the following hold.
(1) If there exists an odd continuous mapping fromAtoB, then ;
(2) If there is an odd homeomorphism fromAtoB, then ;
(3) If , then ;
(4) Thenn-dimensional sphere has a genus of by the Borsuk-Ulam theorem;
(5) IfAis compact, then and there exists such that and , where .
Let denote the family of closed symmetric subsets A of X such that and . The following version of the symmetric mountain-pass lemma is due to Kajikiya .
Lemma 3.1LetEbe an infinite-dimensional space and , and suppose the following conditions hold.
(C1) is even, bounded from below, and satisfies the Palais-Smale condition;
(C2) For each , there exists an such that .
Then either (R1) or (R2) below holds.
(R1) There exists a sequence such that , and converges to zero;
(R2) There exist two sequences and such that , , , , , , , and converges to a nonzero limit.
Remark 3.1 From Lemma 3.1, we have a sequence of critical points such that , and .
In order to get infinitely many solutions, we need some lemmas. Under the assumptions of Theorem 1.1, let , we have
Let . As attains a local but not a global minimum (P is not bounded below), we have to perform some sort of truncation. To this end, let , be such that , where m is the local minimum of , and M is the local maximum and . For these values and , we can choose a smooth function defined as follows
Then it is easy to see and is . Let and consider the perturbation of :
From the arguments above, we have the following.
Lemma 3.2Let is defined as in (3.1). Then
(i) andGis even and bounded from below;
(ii) If , then , consequently, and ;
(iii) Suppose that (I1)-(I2) and (H1)-(H3) hold, then satisfies the (PS) condition.
Proof It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 2.6. □
Lemma 3.3Assume that (I2) and (H3) hold. Then for any , there exists such that .
Proof Firstly, by (H3) of Theorem 1.1, for any fixed , , we have
Secondly, from Lemma 5 of , we have that for any finite dimensional subspace of and any , there exists a constant such that
Therefore, for any with and ρ small enough, we have
since . That is,
This completes the proof. □
Now, we give the proof of Theorem 1.1 as following.
Proof of Theorem 1.1 Recall that
By Lemma 3.2(i) and Lemma 3.3, we know that . Therefore, assumptions (C1) and (C2) of Lemma 3.1 are satisfied. This means that G has a sequence of solutions converging to zero. Hence, Theorem 1.1 follows by Lemma 3.2(ii). □
The authors declare that they have no competing interests.
CZ carried out the theoretical studies, and participated in the sequence alignment and drafted the manuscript. FM participated in the design of the study and performed the statistical analysis. SL conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
The authors are supported by the Research Foundation during the 12th Five-Year Plan Period of Department of Education of Jilin Province, China (Grant  No. 252), the China Postdoctoral Science Foundation (Grant No. 2012M520665), the Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), the open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03), the Natural Science Foundation of Changchun Normal University.
Sun, J, Chen, H: Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems. Nonlinear Anal., Real World Appl.. 11, 4062–4071 (2010)