Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO SECOND-ODE NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBOSETTI-ABINOWITZ CONDITIONS ONG YUAN, ZIHENG ZHANG Abstract. This article studies the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system q Ltq + W qt, q = 0, where L C, n is a symmetric and positive definite matrix for all t. The function W C 1 n, is not assumed to satisfy the global Ambrosetti-abinowitz condition. Assuming reasonable conditions on L and W, we prove the existence of at least one nontrivial homoclinic solution, and for W t, q even in q, we prove the existence of infinitely many homoclinic solutions. 1. Introduction The purpose of this work is to study the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system q Ltq + W q t, q = 0, 1.1 where L C, n is a symmetric and definite matrix for all t, W C 1 n,. We say that a solution qt of 1.1 is homoclinic to 0 if q C, n, qt 0 and qt 0 as t ±. If qt 0, qt is called a nontrivial homoclinic solution. The existence of homoclinic solutions is one of the most important problems in the history of Hamiltonian systems, and has been studied intensively by many mathematicians. Assuming that Lt and W t, q are independent of t, or T -periodic in t, many authors have studied the existence of homoclinic solutions for Hamiltonian system 1.1 via critical point theory and variational methods, see for instance [, 4, 6, 8, 3, 15, 17] and the references therein. More general cases are considered in the recent papers [9, 19]. In this case, the existence of homoclinic solutions can be obtained by going to the limit of periodic solutions of approximating problems. In this paper we are interested in the case where 1.1 is non-autonomous. If Lt and W t, q are neither autonomous nor periodic in t, this problem is quite 000 Mathematics Subject Classification. 34C37, 35A15, 37J45. Key words and phrases. Homoclinic solutions; critical point; variational methods; mountain pass theorem. c 010 Texas State University - San Marcos. Submitted January 14, 010. Published February 5, 010. Supported by National Natural Science Foundation of China and FDP. 1
. YUAN, Z. ZHANG EJDE-010/9 different from the ones just described, because of the lack of compactness of the Sobolev embedding, see for instance [10, 14, 18] and the references therein. In [18], the authors considered 1.1 without periodicity assumptions on L and W and showed that 1.1 possesses one homoclinic solution by using a variant of the Mountain Pass Theorem without the PS condition. In [14], under the same assumptions of [18], the authors, by employing a new compact embedding theorem, obtained the existence and multiplicity of homoclinic solution of 1.1. The authors in [10] removed the technical coercivity condition on L and proved the existence of homoclinic solutions when L and W are even in t by the method of approximating the homoclinic orbits from solutions of boundary problems. By the way, a concentration-compactness principle by Lions [11, 1] has also been used to find one homoclinic solution of 1.1, for example [1, 5]. Here, we must point that all the results in the works mentioned above are obtained under the assumption that W satisfies the global Ambrosetti-abinowitz condition on q; i.e., there is a constant θ > such that, for every t and q n \{0}, 0 < θw t, q W q t, q, q, 1. where, : n n denotes the standard inner product in n and is the induced norm. More recently, in [13], the authors discussed the existence of even homoclinic solutions under a class of superquadratic conditions on W which is weaker than the global Ambrosetti-abinowitz condition see Corollary in [13]. More precisely, they assumed that W and L are even in t and satisfy: W1 there exist constants d 1 > 0 and ϑ such that W t, q d 1 q ϑ for all t and q n ; W W t, q = o q as q 0 uniformly with respect to t ; W3 W q t, q 0 as q 0 uniformly with respect to t ; W4 there exist constants d > 0 and δ > ϑ and τ L 1, + such that Wq t, q, q W t, q d q δ τt for all t and q n ; and some other reasonable assumptions on L and W. They also gave an example which satisfies their conditions but do not satisfy the global Ambrosetti-abinowitz condition [13, emark ]. On the other hand, compared with the case that W t, q is superquadratic as q +, the literature is smaller which is available for the case that W t, q is subquadratic as q +. As far as the authors are aware, only the papers [7, 0, 1] dealt with this case. Motivated by the works mentioned above, in this paper we give some more general conditions on L and W to guarantee that 1.1 has nontrivial homoclinic solutions. Explicitly, assuming that W t, q satisfies some superquadratic conditions which are weaker than the global Ambrosetti-abinowitz condition and different from in [13], and some other reasonable assumptions on L and W, we give a new result to guarantee that 1.1 has at least one nontrivial homoclinic solution and infinitely many homoclinic solutions if W t, q is even in q, which generalizes and improves the previous results in the literature. To state our main result, we state the basic hypotheses on L and W : H1 L C, n is a symmetric and positive definite matrix for all t ; there is a continuous function α : such that αt > 0 for all t and Ltq, q αt q and αt + as t +.
EJDE-010/9 HOMOCLINIC SOLUTIONS 3 From this condition, we see that there is a positive constant β > 0 such that Ltq, q β q for all t, q n. 1.3 H W t, q 0 for all t, q n and there exist constants M > 0 and 1 > 0 such that W t, q M q for all t, q n, q 1, where M < β, with β defined in 1.3; H3 there exist α 0 t > 0 and constants α 1 >, > 0 such that W t, q α 0 t q α1 for all t, q n, q ; H4 there exist constants µ > and α with 0 α < µ / such that µw t, q W q t, q, q α Ltq, q for all t, q n ; H5 W q t, q = O q as q 0 uniformly with respect to t ; H6 there exists W C n, such that for every t and q n. Our main result reads as follows. W q t, q W q Theorem 1.1. Suppose that the conditions H1-H6 are satisfied, then 1.1 possesses one nontrivial homoclinic solution. Moreover, if we assume that W t, q is even in q, i.e., H7 W t, q = W t, q for all t and q n, then 1.1 has infinitely many homoclinic solutions. emark 1.. Note that if 1. holds, so does H4, however the reverse is not true. Now, we give an example of W which satisfies H-H6 but does not satisfy the conditions 1., W1 and W. This does not say anything about W4. Example 1.3. W t, q = at q exp q γ, 1.4 where γ > 0 is a constant and at is a positive, continuous, bounded function with inf t at > 0. Then we have W t, q sup at exp γ 1 q := M q for all t, q n, q 1, t where 1 > 0 is any given constant, which implies that H holds if sup t at is small enough. Moreover, it is easy to check that W t, q at q +γ, W q t, q = at exp q γ q + γat exp q γ q γ q, Wq t, q, q = at q exp q γ + γat q γ+ exp q γ. 1.5 So, for any constant µ >, we have which yields µw t, q W q t, q, q = at q exp q γ µ γ q γ, 0 < µw t, q W q t, q, q µ sup t at exp µ q γ 1.6
4. YUAN, Z. ZHANG EJDE-010/9 for all t, q n and 0 < q µ 1/γ; γ i.e., 1. does not hold for every t and q n \ {0}; and µw t, q W q t, q, q 0 for all t, q n, q µ γ 1/γ, which, combining with 1.6, implies that, for some µ >, if sup t at is small enough note that 1.3, H4 holds. On the other hand, by 1.4, we have and, by 1.5, we have and W t, q lim q 0 q = at inf at > 0, t W q t, q = at exp q γ q + γat exp q γ q γ q sup at exp q γ + γ q γ q t W q t, q inf at lim t q 0 q = at sup at. t The remainder of the paper is organized as follows. In section, some preliminary results are given. In section 3, we give the proof of Theorem 1.1.. Preliminary esults To establish our result via the critical point theory, we firstly describe some properties of the space on which the variational framework associated with 1.1 is defined. Let E = { q H 1, n [ : qt + Ltqt, qt ] dt < + }, then the space E is a Hilbert space with the inner product [ẋt, ] x, y = ẏt + Ltxt, yt dt and the corresponding norm x = x, x. Note that E H 1, n L p, n for all p [, + ] with the embedding being continuous. In particular, for p = +, there exists a constant C > 0 such that q C q, q E..1 Here L p, n p < + and H 1, n denote the Banach spaces of functions on with values in n under the norms 1/p q p := qt dt p and q H 1 := q + q 1/ respectively. L, n is the Banach space of essentially bounded functions from into n equipped with the norm q := ess sup{ qt : t }. Lemma.1 [14, Lemma 1]. Suppose that L satisfies H1. Then the embedding of E in L, n is compact.
EJDE-010/9 HOMOCLINIC SOLUTIONS 5 Similar to Lemma of [14], we can get the following result. For the reader s convenience, we give the details of its proof. Lemma.. Suppose that H1, H5, H6 are satisfied. If q k q 0 weakly in E, then W q t, q k W q t, q 0 in L, n. Proof. Assume that q k q 0 in E. Then, by Banach-Steinhaus Theorem and.1, there is a constant b 1 > 0 such that sup q k b 1. k N Assumptions H5 and H6 imply the existence of b > 0 such that for all k N and t. Hence W q t, q k t b q k t W q t, q k t W q t, q 0 t b q k t + q 0 t b q k t q 0 t + q 0 t. Since, by Lemma.1, q k q 0 in L, passing to a subsequence if necessary, it can be assumed that q k q 0 < +, k=1 which implies that q k t q 0 t for almost every t and q k t q 0 t = ζt L, n. k=1 Therefore, W q t, q k t W q t, q 0 t b ζt + q 0 t. Then, using the Lebesgue s Convergence Theorem, the lemma is readily proved. Now we introduce more notations and some necessary definitions. Let B be a real Banach space, I C 1 B,, which means that I is a continuously Fréchetdifferentiable functional defined on B. Definition.3 [16]. I C 1 B, is said to satisfy the PS condition if any sequence {u j } j N B, for which {Iu j } j N is bounded and I u j 0 as j +, possesses a convergent subsequence in B. Moreover, let B r be the open ball in B with the radius r and centered at 0 and B r denote its boundary. We obtain the existence and multiplicity of homoclinic solutions of 1.1 by use of the following well-known Mountain Pass Theorems, see [16]. Lemma.4 [16, Theorem.]. Let B be a real Banach space and I C 1 B, satisfying the PS condition. Suppose that I0 = 0 and that A1 there exist constants ρ, α > 0 such that I Bρ α, A there exists e B \ B ρ such that Ie 0. Then I possesses a critical value c α given by where c = inf g Γ max s [0,1] Igs, Γ = {g C[0, 1], B : g0 = 0, g1 = e}.
6. YUAN, Z. ZHANG EJDE-010/9 Lemma.5 [16, Theorem 9.1]. Let B be an infinite dimensional real Banach space and I C 1 B, be even satisfying the PS condition and I0 = 0. If B = V X, where V is finite dimensional, and I satisfies: A3 there exist constants ρ, α > 0 such that I Bρ X α and A4 for each finite dimensional subspace Ẽ B, there is an = Ẽ such that I 0 on Ẽ\B Ẽ. Then I has an unbounded sequence of critical values. 3. Proof of Theorem 1.1 Now we establish the corresponding variational framework to obtain homoclinic solutions of 1.1. Define the functional I : B = E as follows [1 Iq = qt + 1 ] Ltqt, qt W t, qt dt = 1 3.1 q W t, qtdt. Lemma 3.1. Under the conditions of Theorem 1.1, we have I [ qv = qt, vt + Ltqt, vt Wq t, qt, vt ] dt, 3. for all q, v E, which yields that I qq = q Wq t, qt, qt dt. 3.3 Moreover, I is a continuously Fréchet-differentiable functional defined on E, i.e., I C 1 E, and any critical point of I on E is a classical solution of 1.1 with q± = 0 = q±. Proof. We firstly show that I : E. By H, there exist constants M > 0 and 1 > 0 such that W t, q M q for all t, q n, q 1. 3.4 Let q E, then q C 0, n the space of continuous functions q on such that qt 0 as t +. Therefore there is a constant > 0 such that t implies that qt 1. Hence, by 3.4, we have W t, qtdt W t, qtdt + M qt dt < +. 3.5 t Combining 3.1 and 3.5, we show that I : E. Next we prove that I C 1 E,. ewrite I as I = I 1 I, where [1 I 1 := qt + 1 ] Ltqt, qt dt and I := W t, qtdt. It is easy to check that I 1 C 1 E, and I 1qv [ ] = qt, vt + Ltqt, vt dt. 3.6 Therefore it is sufficient to show this is the case for I. In the process we will see that I qv = Wq t, qt, vt dt, 3.7
EJDE-010/9 HOMOCLINIC SOLUTIONS 7 which is defined for all q, v E. For any given q E, let us define Jq : E as Jqv = Wq t, qt, vt dt, v E. It is obvious that Jq is linear. Now we show that Jq is bounded. Indeed, for any given q E, there exists a constant M 1 > 0 such that q M 1 and, by.1, q CM 1. So according to H5 and H6, there is a constant b 3 > 0 such that W q t, qt b 3 qt, for all t, which yields that, by 1.3 and the Hölder inequality, Jqv = Wq t, qt, vt dt b 3 q v b 3 q v. 3.8 β Moreover, for q and v E, by the Mean Value Theorem, we have W t, qt + vtdt W t, qtdt = W q t, qt + htvt, vt dt, where ht 0, 1. Therefore, by Lemma. and the Hölder inequality, we have Wq t, qt + htvt, vt dt Wq t, qt, vt dt 3.9 = W q t, qt + htvt W q t, qt, vt dt 0 as v 0. Combining 3.8 and 3.9, we see that 3.7 holds. It remains to prove that I is continuous. Suppose that q q 0 in E and note that sup I qv I q 0 v = sup Wq t, qt W q t, q 0 t, vt dt v =1 v =1 W q, q W q, q 0 v v β W q, q W q, q 0. By Lemma., we obtain I qv I q 0 v 0 as q q 0 uniformly with respect to v, which implies the continuity of I and we show that I C 1 E,. Lastly, we check that critical points of I are classical solutions of 1.1 satisfying qt 0 and qt 0 as t +. We have known that E C 0, n. Moreover if q is one critical point of I, by 3., we have Ltq W q t, q is the weak derivative of q. Since L C, n, W C 1 n, and E C 0, n, which yields that q C, n ; i.e., q is a classical solution of 1.1. Moreover, it is easy to check that qt 0 as t +. Lemma 3.. Under the conditions H1, H4, H5, H6, I satisfies the PS condition. Proof. Assume that {u j } j N E is a sequence such that {Iu j } j N is bounded and I u j 0 as j +. Then there exists a constant C 1 > 0 such that for every j N. Iu j C 1, I u j E C 1 3.10
8. YUAN, Z. ZHANG EJDE-010/9 We firstly prove that {u j } j N is bounded in E. By 3.1, 3.3 and H4, we have µ 1 u j = µiu j I u j u j + µw t, uj t W q t, u j t, u j t dt 3.11 µiu j I u j u j + α Ltuj t, u j t dt. Let us define ηq = [ µ qt + µ α Ltqt, qt ] dt, then we have µ 1 q ηq µ q, 3.1 where µ 1 = µ α and µ = µ. Thus, combining 3.10, 3.11 with 3.1, we obtain µ 1 u j ηu j µiu j I u j u j µc 1 + C 1 u j. 3.13 Since µ 1 > 0, the inequality 3.13 shows that {u j } j N is bounded in E. By Lemma.1, the sequence {u j } j N has a subsequence, again denoted by {u j } j N, and there exists u E such that u j u, weakly in E, u j u, strongly in L, n. Hence I u j I u u j u 0, and by Lemma. and the Hölder inequality, we have W q t, u j t W q t, ut, u j t ut dt 0 as j +. On the other hand, an easy computation shows that I u j I u, u j u = u j u W q t, u j t W q t, ut, u j t ut dt. Consequently, u j u 0 as j +. Now, we present the proof of Theorem 1.1, divided into several steps. Proof of Theorem 1.1. Step 1. It is clear that I0 = 0 by H and I C 1 E, satisfies the PS condition by Lemmas 3.1 and 3.. Step. We now show that there exist constants ρ > 0 and α > 0 such that I satisfies the condition A1 of Lemma.4. In fact, assume that q E and 0 < q 1. Then, by 1.3 and H, we have W t, qtdt M qt dt M q M β q, and in consequence, combining this with 3.1, we obtain Iq 1 q M β q = 1 M 1 q. 3.14 β
EJDE-010/9 HOMOCLINIC SOLUTIONS 9 Note that H implies 1 M β > 0. Set ρ = 1 C, α = β M 1 βc > 0. 3.15 By.1, if q = ρ, then 0 < q 1 and 3.14 gives that I Bρ α. Step 3. It remains to prove that there exists e E such that e > ρ and Ie 0, where ρ is defined in 3.15. By 3.1, we have, for every m \{0} and q E\{0}, Im q = m q W t, m qtdt. Take some Q E such that Q = 1. Then there exists a subset Ω of positive measure of such that Qt 0 for t Ω. Take m > 0 such that m Qt for t Ω. Then, by H and H3, we obtain that Im Q m mα1 Ω α 0 t Qt α1 dt. 3.16 Since α 0 t > 0 and α 1 >, 3.16 implies that Im Q < 0 for some m > 0 such that m Qt for t Ω and m Q > ρ, where ρ is defined in 3.15. By Lemma.4, I possesses a critical value c α > 0 given by where Hence there is q E such that c = inf g Γ max s [0,1] Igs, Γ = {g C[0, 1], E : g0 = 0, g1 = e}. Iq = c, I q = 0. Step 4. Now suppose that W t, q is even in q; i.e., H7 holds, which implies that I is even. Moreover, we already know that I0 = 0 and I C 1 E, satisfies the PS condition in Step 1. To apply Lemma.5, it suffices to prove that I satisfies the conditions A3 and A4 of Lemma.5. A3 is identically the same as in Step, so it is already proved. Now we prove that A4 holds. Let Ẽ E be a finite dimensional subspace. From Step 3 we know that, for any Q Ẽ E such that Q = 1, there is m Q > 0 such that Im Q < 0, for every m m Q > 0. Since Ẽ E is a finite dimensional subspace, we can choose an = Ẽ > 0 such that Iq < 0, q Ẽ\B. Hence, by Lemma.5, I possesses an unbounded sequence of critical values {c j } j N with c j +. Let q j be the critical point of I corresponding to c j, then 1.1 has infinitely many homoclinic solutions. eferences [1] C. O. Alves, P. C. Carrião and O. H. Miyagaki; Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 003, no. 5, 639-64. [] A. Ambrosetti and V. Coti Zelati; Multiple homoclinic orbits for a class of conservative systems, end. Sem. Mat. Univ. Padova., 89 1993, 177-194.
10. YUAN, Z. ZHANG EJDE-010/9 [3] F. Antonacci; Periodic and homoclinic solutions to a class of Hamiltonian systems with indefinite potential in sign, Boll. Un. Mat. Ital., B 7 10 1996, no., 303-34. [4] P. Caldiroli and P. Montecchiari; Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1 1994, no., 97-19. [5] P. C. Carrião and O. H. Miyagaki; Existence of homoclinic solutions for a class of timedependent Hamiltonian systems, J. Math. Anal. Appl., 30 1999, no. 1, 157-17. [6] V. Coti Zelati and P. H. abinowitz; Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 1991, no. 4, 693-77. [7] Y. Ding; Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 5 1995, 1095-1113. [8] Y. Ding and M. Girardi; Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl., 1993, no. 1, 131-145. [9] M. Izydorek and J. Janczewska; Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 19 005, no., 375-389. [10] P. Korman and A. C. Lazer; Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, 1994 1994, no. 01, 1-10. [11] P. L. Lions; The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 1984, no., 109-145. [1] P. L. Lions; The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 1984, no. 4, 3-83. [13] Y. Lv and C. Tang; Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 007, no. 7, 189-198. [14] W. Omana and M. Willem; Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 199, no. 5, 1115-110. [15] E. Paturel; Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Val. Partial Differential Equations, 1 001, no., 117-143. [16] P. H. abinowitz; Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS eg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, I, 1986. [17] P. H. abinowitz; Homoclinic orbits for a class of Hamiltonian systems, Proc. oy. Soc. Edinburgh Sect. A., 114 1990, no. 1-, 33-38. [18] P. H. abinowitz and K. Tanaka; Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 06 1991, no. 3, 473-499. [19] X. Tang and L. Xiao; Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351 009 586-594. [0] Q. Zhang and C. Liu; Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 7 010, 894-903. [1] Z. Zhang and. Yuan; Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems, Nonlinear Anal. 71 009, 415-4130. ong Yuan School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China E-mail address: ryuan@bnu.edu.cn Ziheng Zhang School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China E-mail address: zhzh@mail.bnu.edu.cn