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
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.
and is endowed with the norm
Then, from , is a sparable and reflexive Banach space.
Hence, a critical point of φ gives us a weak solution of BVP (1.1).
In the following, let H be a Banach space, let φ be continuously differentiable, and we state (C) condition .
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.
Proof The proof of this lemma is similar to that of . For the sake of completeness, we give a simple proof here.
Whence, by the fundamental lemma,
3 Existence of constant-sign solutions
Lemma 3.1Assume that
Remark 3.1 Let
which is equivalent to the condition:
Hence, by Fatou’s lemma,
Theorem 3.1Assume that (A1)-(A3) and
hold, then, BVP (1.1) has at least one positive solution.
Theorem 3.2Assume (A1)-(A5) and
hold, then, BVP (1.1) has two positive solutions.
Let with and consider the functional , we can apply Ekeland’s variational principle  and obtain such that
With the similar discussion above, we have the following result.
Theorem 3.3Assume (A1), (A3)-(A6) and
hold, then, BVP (1.1) has at least two positive solutions.
Theorem 3.4Assume that (A5) and
hold, then, BVP (1.1) has at least two negative solutions.
Theorem 3.5Assume that (B1), (B3)-(B5), (A5) and
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).
George, PK, Nandakumaran, AK, Arapostathis, A: A note on controllability of impulsive systems. J. Math. Anal. Appl.. 241, 276–283 (2000). Publisher Full Text
Akhmetov, MU, Zafer, A: Controllability of the Vallée-Poussin problem for impulsive differential systems. J. Optim. Theory Appl.. 102, 263–276 (1999). Publisher Full Text
Nenov, S: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. TMA. 36, 881–890 (1999). Publisher Full Text
Georescu, P, Morosanu, G: Pest regulation by means of impulsive controls. Appl. Math. Comput.. 190, 790–803 (2007). Publisher Full Text
Liu, X, Willms, AR: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng.. 2, 277–299 (1996). Publisher Full Text
Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second-order impulsive differential equations. Appl. Math. Comput.. 114, 51–59 (2000). Publisher Full Text
Lee, EK, Lee, YH: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput.. 158, 745–759 (2004). Publisher Full Text
Liu, Y: Positive solutions of periodic boundary value problems for nonlinear first order impulsive differential equations. Nonlinear Anal. TMA. 70, 2106–2122 (2009). Publisher Full Text
Yang, A, Wang, H, Wang, D: The unique solution for periodic differential equations with upper and lower solutions in reverse order. Bound. Value Probl.. 2013, (2013) Article ID 88. http://www.boundaryvalueproblems.com/content/2013/1/88
Papageorgiou, NS, Rocha, EM: Pairs of positive solutions for p-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition. Nonlinear Anal. (2008). Publisher Full Text
Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text
Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary problems with impulsives. Nonlinear Anal., Real World Appl. (2008). Publisher Full Text
Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text
Tian, Y, Ge, W: Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal.. 72, 277–287 (2010). Publisher Full Text