Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 RESEARCH Open Access Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions Chenxing Zhou1 , Fenghua Miao1 and Sihua Liang1,2* * Correspondence: liangsihua@163.com College of Mathematics, Changchun Normal University, Changchun, Jilin 130032, P.R China Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R China Abstract 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 Introduction In this paper, we study the following nonlinear impulsive differential equations with Dirichlet boundary conditions ⎧ p– ⎪ ⎨ –u (t) + a(t)u(t) = μ|u| u(t) + f (t, u(t)), a.e t ∈ [, T], + – u (tj ) = u (tj ) – u (tj ) = Ij (u(tj )), j = , , , N, ⎪ ⎩ u() = u(T) = , (.) where p > , T > , μ > , f : [, T] × R → R is continuous, a ∈ L∞ [, T], N is a positive integer, = t < t < t < · · · < tN < tN+ = T, u (tj ) = u (tj+ ) – u (tj– ) = limt→tj+ u (t) – limt→tj– u (t), Ij : R → R 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 (.) with the general nonlinearities f (t, u) 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 [–] 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 [, ] © 2013 Zhou et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 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 [–] For related basic information, we refer the reader to [, ] For a second order differential equation u = f (t, u, u ), one usually considers impulses in the position u and the velocity u 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 [–] 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 [–] More precisely, in [] the authors studied the following equations with Dirichlet boundary conditions: ⎧ ă + u(t) = f (t, u(t)), a.e t ∈ [, T], ⎨ –u(t) ˙ j ) = Ij (u(tj )), j = , , , p, u(t ⎪ ⎩ u() = u(T) = (.) 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 (.) with a perturbation term Also, they obtained infinitely many solutions for problem (.) under the assumption that the nonlinearity f is a superlinear case Soon after that, Zhou and Li [] extended problem (.) 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 (.), 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 Ij : R → R is continuous, and f (t, u) satisfies the following conditions: (I ) Ij (j = , , , N ) are odd and satisfy u(tj ) Ij (s) ds – Ij u(tj ) u(tj ) ≥ , u(tj ) Ij (s) ds ≥ ; (I ) There exist δj > , j = , , , N such that u(tj ) Ij (s) ds ≤ δj |u| , for u ∈ R \ {}; (H ) f (t, u) ∈ C([, T] × R, R), f (t, –u) = –f (t, u) for all u ∈ R; f (t,u) (H ) lim|u|→∞ |u| p– = uniformly for t ∈ [, T]; (H ) lim|u|→+ f (t,u) u = ∞ uniformly for t ∈ [, T] The main result of this paper is as follows Theorem . Suppose that (I )-(I ) and (H )-(H ) hold Then problem (.) has a sequence of nontrivial solutions {un } and un → as n → ∞ Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 Remark . Without the symmetry condition (i.e., f (x, –u) = –f (x, u) and I(–s) = –I(s)), we can obtain at least one nontrivial solution by the same method in this paper Remark . We should point out that Theorem . is different from the previous results of [–] in three main directions: () We not make the nonlinearity f satisfy the well-known Ambrosetti-Rabinowitz condition []; () We try to use Lusternik-Schnirelman’s theory for Z -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 () We can obtain a sequence of nontrivial solutions {un } and un → as n → ∞ Remark . There exist many functions Ij and f (t, u) satisfying conditions (I )-(I ) and (H )-(H ), respectively For example, when p = , Ij (s) = s and f (t, u) = et u/ Preliminary lemmas In this section, we first introduce some notations and some necessary definitions Definition . Let E be a Banach space and J : E → R J is said to be sequentially weakly lower semi-continuous if limn→∞ inf J(un ) ≥ J(u) as un u in E Definition . Let E be a real Banach space For any sequence {un } ⊂ E, if {J(un )} is bounded and J (un ) → as n → ∞ possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (denoted by (PS) condition for short) In the Sobolev space H (, T), consider the inner product T u, v H (,T) = u (t)v (t) dt, which induces the norm T u H (,T) u (t) dt = It is a consequence of Poincaré’s inequality that T u(t) dt T ≤√ λ u (t) dt Here, λ = π /T is the first eigenvalue of the Dirichlet problem –u (t) = λu(t), t ∈ [, T], u() = u(T) = (.) In this paper, we will assume that inft∈[,T] a(t) = m > –λ We can also define the inner product T u, v = T u (t)v (t) dt + a(t)u(t)v(t) dt, Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 which induces the equivalent norm T T u (t) dt + u = a(t) u(t) dt Lemma . [] If ess inft∈[,T] a(t) = m > –λ , then the norm · and the norm · are equivalent H (,T) Lemma . [] There exists c∗ such that if u ∈ H (, T), then u ∞ where u ≤ c∗ u , (.) = maxt∈[,T] |u(t)| ∞ For u ∈ H (, T), we have that u and u are both absolutely continuous, and u ∈ L (, T), hence, u (tj ) = u (tj+ ) – u (tj– ) for any t ∈ [, T] If u ∈ H (, T), then u is absolutely continuous and u ∈ L (, T) In this case, the one-side derivatives u (tj+ ) and u (tj– ) may not exist As a consequence, we need to introduce a different concept of solution Suppose that u ∈ C[, T] satisfies the Dirichlet condition u() = u(T) = Assume that, for every j = , , , N , uj = u|(tj ,tj+ ) and uj ∈ H (tj , tj+ ) Let = t < t < t < · · · < tN < tN+ = T Taking v ∈ H (, T) and multiplying the two sides of the equality –u (t) + a(t)u(t) = μ|u|p– u(t) + f t, u(t) by v and integrating between and T, we have T –u (t) + a(t)u(t) – μ|u|p– u(t) – f t, u(t) v(t) dt = Moreover, since u() = u(T) = , one has T – u (t)v(t) dt N tj+ =– u (t)v(t) dt j= tj N T – tj+ u (t)v(t)|t+ + =– u (t)v (t) dt j j= N T u (tj )v(tj ) – u ()v() + u (T)v(T) + =– – N = T u (tj )v(tj ) + u (t)v (t) dt j= N T Ij u(tj ) v(tj ) + = j= u (t)v (t) dt j= u (t)v (t) dt (.) Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 Combining (.), we get T T u (t)v (t) dt + |u|p– u(t)v(t) dt N T – T a(t)u(t)v(t) dt – μ Ij u(tj ) v(tj ) = f t, u(t) v(t) dt + j= Lemma . A weak solution of (.) is a function u ∈ H (, T) such that T T u (t)v (t) dt + |u|p– u(t)v(t) dt N T Ij u(tj ) v(tj ) = f t, u(t) v(t) dt + – T a(t)u(t)v(t) dt – μ (.) j= for any v ∈ H (, T) Consider J : H (, T) → R defined by J(u) = u – μ p T N T |u|p dt – u(tj ) F t, u(t) dt + Ij (s) ds, j= (.) u where F(t, u) = f (t, s) ds Using the continuity of f and Ij , j = , , , N , we obtain the continuity and differentiability of J and J ∈ C (H (, T), R) For any v ∈ H (, T), one has T T J (u)v = u (t)v (t) dt + a(t)u(t)v(t) dt T –μ |u|p– u(t)v(t) dt N T – Ij u(tj ) v(tj ) f t, u(t) v(t) dt + (.) j= Thus, the solutions of problem (.) are the corresponding critical points of J Lemma . If u ∈ H (, T) is a weak solution of problem (.), then u is a classical solution of problem (.) Proof Obviously, we have u() = u(T) = since u ∈ H (, T) By the definition of weak solution, for any v ∈ H (, T), one has T T u (t)v (t) dt + Ij u(tj ) v(tj ) = f t, u(t) v(t) dt + |u|p– u(t)v(t) dt N T – T a(t)u(t)v(t) dt – μ j= (.) Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 For j ∈ {, , , , N}, choose v ∈ H (, T) with v(t) = for every t ∈ [, tj ] ∪ [tj+ , T] Then tj+ tj+ u (t)v (t) dt + a(t)u(t)v(t) dt tj tj tj+ =μ T |u|p– u(t)v(t) dt + f t, u(t) v(t) dt tj By the definition of weak derivative, the equality above implies that –u (t) + a(t)u(t) = μ|u|p– u(t) + f t, u(t) , a.e t ∈ (tj , tj+ ) (.) Hence uj ∈ H (tj , tj+ ) and u satisfies the equation in (.) a.e on [, T] By integrating (.), we have N N – Ij u(tj ) v(tj ) u (tj )v(tj ) + u (T)v(T) – u ()v() + j= j= T + –u (t) + a(t)u(t) – μ|u|p– u(t) – f t, u(t) v(t) dt = Combining this fact with (.), we get N N Ij u(tj ) v(tj ) for any v ∈ H (, T) u (tj )v(tj ) = j= j= Hence, u (tj ) = Ij (u(tj )) for every j = , , , N , and the impulsive condition in (.) is satisfied This completes the proof Lemma . If ess inft∈[,T] a(t) = m > –λ , then the functional J is sequentially weakly lower semi-continuous Proof Let {un } be a weakly convergent sequence to u in H (, T), then u ≤ lim inf un n→∞ We have that {un } converges uniformly to u on C[, T] Then lim inf J(un ) n→∞ = lim n→∞ ≥ u un – μ p – T = J(u) This completes the proof μ p T N T |un |p dt – u(tj ) F t, u(t) dt + Ij (s) ds j= N T |u|p dt – un (tj ) F t, un (t) dt + Ij (s) ds j= Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 Under assumptions (H ) and (H ), we have f (t, u)u = o |u|p , F(t, u) = o |u|p as |u| → ∞, which means that for all ε > , there exist a(ε), b(ε) > such that f (t, u)u ≤ a(ε) + ε|u|p , (.) F(x, u) ≤ b(ε) + ε|u|p (.) Hence, for every positive constant k, we have F(x, u) – kf (x, u)u ≤ c(ε) + ε|u|p , (.) where c(ε) > Lemma . Suppose that (I )-(I ) and (H )-(H ) hold, then J(u) satisfies the (PS) condition Proof Let {un } be a sequence in H (, T) such that {J(un )} is bounded and J (un ) → as n → ∞ First, we prove that {un } is bounded By (.), (.) and (.), one has J(un ) – J (un )un – μ p = N un (tj ) + j= T Ij (s) ds – T (p – )μ –ε p ≥ – T |un |p dt + f t, un (t) un (t) – F t, un (t) dt N Ij un (tj ) un (tj ) j= N un (tj ) |un |p dt – c(ε)T + Ij (s) ds j= N Ij un (tj ) un (tj ) j= By condition (I ), we can deduce that N un (tj ) Ij (s) ds – j= Setting ε = T (p–)μ , p N Ij un (tj ) un (tj ) ≥ j= we get |un |p dt ≤ M + o() un , (.) Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 where o() → and M is a positive constant On the other hand, by (I ), (.) and (.), we have ∞ > J(un ) = ≥ un un – – μ p μ +ε p T N T |un |p dt – T un (tj ) F t, un (t) dt + Ij (s) ds j= |un |p dt – b(ε)T (.) Thus, (.) and (.) imply that {un } is bounded in H (, T) Going if necessary to a subsequence, we can assume that there exists u ∈ H (, T) such that u un un → u weakly in H (, T), strongly in C [, T], R , as n → ∞ Hence, J (un ) – J (u) (un – u) → , T un (t) – u(t) dt → , f t, un (t) – f t, u(t) T |un |p– un (t) – |u|p– u(t) un (t) – u(t) dt → , N Ij un (tj ) – Ij u(tj ) un (tj ) – u(tj ) → , j= as n → ∞ Moreover, one has T J (un ) – J (u) (un – u) = un – u – |un |p– un (t) – |u|p– u(t) un (t) – u(t) dt T f t, un (t) – f t, u(t) – un (t) – u(t) dt N – Ij un (tj ) – Ij u(tj ) un (tj ) – u(tj ) j= Therefore, un – u → as n → +∞ That is {un } converges strongly to u in H (, T) That is J satisfies the (PS) condition Existence of a sequence of arbitrarily small solutions In this section, we prove the existence of infinitely many solutions of (.), which tend to zero Let X be a Banach space and denote := A ⊂ X \ {} : A is closed in X and symmetric with respect to the orgin For A ∈ , we define genus γ (A) as γ (A) := inf m ∈ N : ∃ϕ ∈ C A, Rm \ {}, –ϕ(x) = ϕ(–x) Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page of 13 If there is no mapping ϕ as above for any m ∈ N , then γ (A) = +∞ We list some properties of the genus (see []) Proposition . Let A and B be closed symmetric subsets of X, which not contain the origin Then the following hold () If there exists an odd continuous mapping from A to B, then γ (A) ≤ γ (B); () If there is an odd homeomorphism from A to B, then γ (A) = γ (B); () If γ (B) < ∞, then γ (A \ B) ≥ γ (A) – γ (B); () Then n-dimensional sphere Sn has a genus of n + by the Borsuk-Ulam theorem; () If A is compact, then γ (A) < +∞ and there exists δ > such that Uδ (A) ∈ and γ (Uδ (A)) = γ (A), where Uδ (A) = {x ∈ X : x – A ≤ δ} Let k denote the family of closed symmetric subsets A of X such that ∈/ A and γ (A) ≥ k The following version of the symmetric mountain-pass lemma is due to Kajikiya [] Lemma . Let E be an infinite-dimensional space and I ∈ C (E, R), and suppose the following conditions hold (C ) I(u) is even, bounded from below, I() = and I(u) satisfies the Palais-Smale condition; (C ) For each k ∈ N , there exists an Ak ∈ k such that supu∈Ak I(u) < Then either (R ) or (R ) below holds (R ) There exists a sequence {uk } such that I (uk ) = , I(uk ) < and {uk } converges to zero; (R ) There exist two sequences {uk } and {vk } such that I (uk ) = , I(uk ) < , uk = , limk→∞ uk = , I (vk ) = , I(vk ) < , limk→∞ vk = , and {vk } converges to a nonzero limit Remark . From Lemma ., we have a sequence {uk } of critical points such that I(uk ) ≤ , uk = and limk→∞ uk = In order to get infinitely many solutions, we need some lemmas Under the assumptions of Theorem ., let ε = μp , we have J(u) = ≥ u u ≥ u – – μ p T =A u –B u p p |u|p dt – b(ε)T –b p – b(ε)T μ T p – C, where A= , B= μ p c T, p ∗ Ij (s) ds j= μ p c T u p ∗ – u(tj ) F t, u(t) dt + μ + ε cp∗ T u – p N T |u|p dt – μ +ε p u = T C=b μ T p Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page 10 of 13 Let P(t) = At – Bt p – C As P(s) 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 R , R be such that m < R < M < R , where m is the local minimum of P(s), and M is the local maximum and P(R ) > P(m) For these values R and R , we can choose a smooth function χ(t) defined as follows ⎧ ⎪ ≤ t ≤ R , ⎨ , χ(t) = , t ≥ R , ⎪ ⎩ ∞ C , χ(t) ∈ [, ], R ≤ t ≤ R Then it is easy to see χ(t) ∈ [, ] and χ(t) is C ∞ Let ϕ(u) = χ( u ) and consider the perturbation of J(u): G(u) = u – ϕ(u)μ p T |u|p dt N T – ϕ(u) u(tj ) F t, u(t) dt + Ij (s) ds j= (.) Then G(u) ≥ A u – Bϕ(u) u p –C =P u , where P(t) = At – Bχ(t)t p – C and P(t) = P(t), m∗ , ≤ t ≤ ρ , t ≥ ρ From the arguments above, we have the following Lemma . Let G(u) is defined as in (.) Then (i) G ∈ C (H (, T), R) and G is even and bounded from below; (ii) If G(u) < m, then P( u ) < m, consequently, u < ρ and J(u) = G(u); (iii) Suppose that (I )-(I ) and (H )-(H ) hold, then G(u) satisfies the (PS) condition Proof It is easy to see (i) and (ii) (iii) are consequences of (ii) and Lemma . Lemma . Assume that (I ) and (H ) hold Then for any k ∈ N , there exists δ = δ(k) > such that γ ({u ∈ H (, T) : G(u) ≤ –δ(k)} \ {}) ≥ k Proof Firstly, by (H ) of Theorem ., for any fixed u ∈ H (, T), u = , we have F(x, ρu) ≥ M(ρ)(ρu) with M(ρ) → ∞ as ρ → Secondly, from Lemma of [], we have that for any finite dimensional subspace Ek of H (, T) and any u ∈ Ek , there exists a constant d > such that T |u|s = s u(t) dt s ≥d u , s ≥ , ∀u ∈ Ek Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 Page 11 of 13 Therefore, for any u ∈ Ek with u = and ρ small enough, we have μ G(ρu) = J(ρu) ≤ ρ – ρ p p ≤ + T T |u|p dt – M(ρ)ρ N |u| dt + ρ δj c∗ j= N δj c∗ – M(ρ)d ρ j= = –δ(k) < , since lim|ρ|→ M(ρ) = +∞ That is, {u ∈ Ek : u = ρ} ⊂ u ∈ H (, T) : G(u) ≤ –δ(k) \ {} This completes the proof Now, we give the proof of Theorem . as following Proof of Theorem . Recall that k = A ∈ H (, T) \ {} : A is closed and A = –A, γ (A) ≥ k and define ck = inf sup G(u) A∈ k u∈A By Lemma .(i) and Lemma ., we know that –∞ < ck < Therefore, assumptions (C ) and (C ) of Lemma . are satisfied This means that G has a sequence of solutions {un } converging to zero Hence, Theorem . follows by Lemma .(ii) Competing interests The authors declare that they have no competing interests Authors’ contributions 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 Author details College of Mathematics, Changchun Normal University, Changchun, Jilin 130032, P.R China Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R China Acknowledgements The authors are supported by the Research Foundation during the 12th Five-Year Plan Period of Department of Education of Jilin Province, China (Grant [2013] 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 Received: 17 June 2013 Accepted: 20 August 2013 Published: September 2013 Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 References Kajikiya, R: A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations J Funct Anal 225, 352-370 (2005) Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential 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with state-dependent delay Math Comput Model 49, 1920-1927 (2009) 37 Zhang, D, Bai, BX: Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions Math Comput Model 53, 1154-1161 (2011) 38 Garcia Azorero, J, Peral Alonso, I: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term Trans Am Math Soc 323, 877-895 (1991) Page 12 of 13 Zhou et al Boundary Value Problems 2013, 2013:200 http://www.boundaryvalueproblems.com/content/2013/1/200 doi:10.1186/1687-2770-2013-200 Cite this article as: Zhou et al.: Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions Boundary Value Problems 2013 2013:200 Page 13 of 13 ... article as: Zhou et al.: Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions Boundary Value Problems 2013 2013:200 Page 13 of 13 ... (2010) 31 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... study the existence and multiplicity of solutions for boundary value problems without impulsive effects [–] For related basic information, we refer the reader to [, ] For a second order differential