In this paper, we consider a kind of Sturm-Liouville boundary value problems with impulsive effects. By using the mountain pass theorem and Ekeland’s variational principle, the existence of two positive solutions and two negative solutions is established. Moreover, we do not assume that the nonlinearity satisfies the well-known AR-condition.
MSC: 34B15, 34B37, 58E30.
Keywords:p-Laplacian; boundary value problem; variational; impulsive
Impulsive effects exist widely in many evolution processes, in which their states are changed abruptly at a certain moment of time. Impulsive differential equations have become more important in recent years in mathematical models of real processes and phenomena studied in control theory [1,2], population dynamics and biotechnology [3,4], physics and mechanics problems . There has been a significant development in the area of impulsive differential equations with fixed moments. We refer the reader to [6,7] and the references therein. Fixed-point theorems in cones [8-10] and the method of lower and upper solutions with monotone iterative technique [11-13], have been used to study impulsive differential equations.
Moreover, the Sturm-Liouville boundary value problems (for short BVPs) have received a lot of attention. Many works have been carried out to discuss the existence of at least one solution, multiple solutions. The methods used therein mainly depend on the Leray-Schauder continuation theorem, Mawhin’s continuation theorem. Since it is very difficult to give the corresponding Euler functional for Sturm-Liouville BVPs and verify the existence of the critical points for the Euler functional, few people consider the existence of solutions for Sturm-Liouville BVPs by critical point theory, and many works considered the existence of solutions for Dirichlet BVPs . Recently, few researchers have used variational methods to study the existence of solutions for impulsive differential equations with Dirichlet boundary conditions [15,16]. In , by mountain pass theorem, Tian and Ge considered the existence of positive solutions of a kind of Sturm-Liouville boundary value problems with impulsive effects. The authors require that the nonlinearity and . They have not obtained the existence of both positive solutions and negative solutions.
Based on the knowledge mentioned above, in this paper, we consider the constant-sign solutions of the following BVP
where , , , , , . Here and denote the right and left limits, respectively. Assume that , is continuous, is continuous on R, , .
Ambrosetti and Rabinowitz  established the existence of nontrival solutions for Dirichlet problems under the well-known Ambrosetti-Rabinowitz condition: there exist some and such that
for all and . Since then, the AR-condition has been used extensively. By the usual AR-condition, it is easy to show that the Euler-Lagrange functional associated with the system has the mountain pass geometry, and the Palais-Smale sequence is bounded. For example, in [16,17], based on (1.2), the authors considered the boundary value problems with impulsive effects.
In this paper, we study the existence of constant-sign solutions of BVP (1.1) without the AR-condition. The paper is organized as follows. In the forthcoming section, we give the Euler functional of BVP (1.1) and some basic lemmas. The aim of Section 3 is to prove the existence of at least two positive solutions of BVP (1.1) based on the mountain pass theorem and Ekeland’s variational principle. At last, we give some results of the existence of at least two negative solutions.
The Sobolev space is defined by
and is endowed with the norm
Then, from , is a sparable and reflexive Banach space.
Definition 2.1 We say that x is a classical solution of BVP (1.1) if it satisfies the equation of BVP (1.1) a.e. on , the limits and , , exist and the Sturm-Liouville boundary conditions hold.
However, if , then x is absolutely continuous and . In this case, the one-sided derivatives , may not exist. As a consequence, we need to introduce a different concept of solution.
Definition 2.2 We say that is a weak solution of BVP (1.1) if it satisfies
Consider defined by
It is clear φ is continuously differentiable on and by computation, one has
Hence, a critical point of φ gives us a weak solution of BVP (1.1).
There exists a positive constant such that
for any , . Here, .
For , suppose that , .
Lemma 2.2If , then, .
For , let , then, the following properties hold:
(i) , ;
(iv) if uniformly converges toxin , then, uniformly converges to ;
(vi) , .
In the following, let H be a Banach space, let φ be continuously differentiable, and we state (C) condition .
(C) Every sequence such that the following conditions hold: has a subsequence, which converges strongly in H.
(i) is bounded,
This condition is weaker than the usual Palais-Smale condition, but can be used in place of it when constructing deformations of sublevel sets via negative pseudo-gradient flows, and, therefore, also in minimax theorems such as the mountain pass lemma and the saddle point theorem.
Lemma 2.4If is a weak solution of BVP (1.1), then is a classical solution of BVP (1.1).
Proof The proof of this lemma is similar to that of . For the sake of completeness, we give a simple proof here.
Choose with for every , then
Whence, by the fundamental lemma,
Hence, , that is, , exist, and x satisfies the equation of BVP (1.1) a.e. on . Moreover,
Now multiplying the equation by and integrating between 0 and 1, together with (2.1), we get
Assume that , then, and ( ). Let , we arrive x satisfies the impulsive condition and
by (2.5). Let , then, , that is, . Let , then, , that is, . Hence, x is a solution of BVP (1.1). □
3 Existence of constant-sign solutions
Assume that , , a.e. on , for a.e. and ; for a.e. and ; , for , for , . Define , , and
It is obvious that is continuously differentiable and , , .
Lemma 3.1Assume that
(A1) , , , , ;
(A2) there exits a constant such that for a.e. , , , , ;
(A3) for .
Then, satisfies (C) condition.
Remark 3.1 Let
Then, satisfies (A1)-(A3). However, it does not satisfy the AR-condition while x is large.
Remark 3.2 The condition of for , , a.e. , is weaker than the following condition:
there is such that is increasing in ,
which is equivalent to the condition:
is increasing in .
Proof Let be a sequence such that
In order to prove that is bounded in , there are several steps.
Step 1. is bounded.
From (3.2), for , one has
We know that is an absolutely continuous function on , and so, the fundamental theorem of calculus ensures the existence of a set such that and is differentiable on , then, let ,
Then, is bounded.
Step 2. is bounded.
Suppose that as . Set for all . Obviously, , that is, is a bounded sequence in . Going to a subsequence if necessary, we may assume that
It is clear that and from the inequality as , there exists a sequence , and as such that
From , , one has
Moreover, , , and from (A1), one has
Let , then, as for . By the hypothesis,
Let , then, for all . If , then,
Hence, by Fatou’s lemma,
Then, from (3.5), we reach a contradiction, that is, . Since , we conclude that for a.e. . Then, for .
Assume that be such that
Fix an integer and define
that is, . Since , there exists an integer , for , one has . Whence,
Since uniformly for , then, uniformly for , and for as . Hence,
Therefore, we have as . Since and are bounded, then, is bounded. Together with , one has for all . Then,
Since is bounded, there exists such that
Since , then,
Since is an arbitrary integer, let , we have a contradiction. This proves that is bounded.
From step 1 and step 2, we obtain that is bounded. Hence, we may assume that
Moreover, for , one has
Since is a Cauchy sequence in , , is bounded in , , as , one has as . Moreover, is continuous in x, is continuous, uniformly in , whence, , , and
If , from Lemma 2.1, there exists a positive constant such that
If , by Lemma 2.1, the Hölder inequality and the boundedness of in , one has
Then, we have as . Hence, , that is, is a Cauchy sequence in . By the completeness of , one has that is a convergence sequence. □
Theorem 3.1Assume that (A1)-(A3) and
(A4) , , ;
hold, then, BVP (1.1) has at least one positive solution.
Proof From (A4), one has and
Let , then, . Hence, there exists such that . Obviously, , then, . We infer that there exists an such that for all .
Moreover, choose , , , . For, , there exists such that for . Choose , one has . Hence, and
Since is arbitrary, we have , that is, . Hence, from the mountain pass theorem, we obtain , such that
It follows . If for a.e. , then, . Hence, a.e. , that is and . This implies that is a positive solution of BVP (1.1). □
Theorem 3.2Assume (A1)-(A5) and
(A6) , , , ,
hold, then, BVP (1.1) has two positive solutions.
Proof Assume . Obviously, and
If is small enough and x is positive, we have , then,
Let with and consider the functional , we can apply Ekeland’s variational principle  and obtain such that
From (3.13), we have . Define , then, is a minimizer of on . Therefore, for small and all with , we have
Define , then, , that is, is a maximum of on . Therefore, for small and all with , with the same discussion above, one has
Let and set . Then, and . Since satisfies (C) condition, we may assume that in . Hence,
which implies that and is a critical point of . Moreover,
so, . If a.e. , with the same discussion in Theorem 3.1, a.e. . Hence, and , which implies is another positive solution of BVP (1.1). □
With the similar discussion above, we have the following result.
Theorem 3.3Assume (A1), (A3)-(A6) and
( ) there exists such that for all , we have for a.e. , all , , ,
hold, then, BVP (1.1) has at least two positive solutions.
Theorem 3.4Assume that (A5) and
(B1) , , , , ;
(B2) , , , ;
(B3) for a.e. ;
(B4) , , ;
(B5) , , ,
hold, then, BVP (1.1) has at least two negative solutions.
Theorem 3.5Assume that (B1), (B3)-(B5), (A5) and
( ) there exists such that for all , we have for a.e. , all , , ,
hold, then, BVP (1.1) has at least two negative solutions.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
The authors are thankful to the referees for their useful suggestions, which helped to enrich the content and considerably improved the presentation of this paper. This research is supported by the Beijing Natural Science Foundation (No. 1122016), the Scientific Research Common Program of Beijing Municipal Commission of Education (No. KM201311417006) and the New Departure Plan of Beijing Union University (No. zk201203).
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