Rotating periodic solutions for super-linear second order Hamiltonian systems

Accepted Manuscript Rotating periodic solutions for super-linear second order Hamiltonian systems

Guanggang Liu, Yong Li, Xue Yang

PII: DOI: Reference:

S0893-9659(17)30361-0 https://doi.org/10.1016/j.aml.2017.11.024 AML 5386

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Applied Mathematics Letters

Received date : 9 September 2017 Revised date : 29 November 2017 Accepted date : 29 November 2017 Please cite this article as: G. Liu, Y. Li, X. Yang, Rotating periodic solutions for super-linear second order Hamiltonian systems, Appl. Math. Lett. (2017), https://doi.org/10.1016/j.aml.2017.11.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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ROTATING PERIODIC SOLUTIONS FOR SUPER-LINEAR SECOND ORDER HAMILTONIAN SYSTEMS GUANGGANG LIU, YONG LI∗ , XUE YANG Abstract. In this paper we consider a class of super-linear second order Hamiltonian systems. We use Morse theory to obtain the existence and multiplicity of rotating periodic solutions, which might be periodic, subharmonic or quasi-periodic ones.

1. Introduction Consider the following second order Hamiltonian system − x00 = Vx (t, x),

(1.1)

where Vx (t, x) ∈ C (R × R , R) with V (t + T, x) = V (t, Q x) for some orthogonal matrix Q ∈ O(n). The aim of this paper is to show that (1.1) admits solutions of the form x(t + T ) = Qx(t), ∀t ∈ R . Usually, we call this type of solutions the rotating periodic solutions of (1.1). This kind of solutions might be periodic if Q = In , where In denotes the identity matrix in Rn , subharmonic if Qk = In for some k ∈ Z+ with k ≥ 2, and quasi-periodic if Qk 6= In for any k ∈ Z+ with k ≥ 1. The existence of periodic solutions for second order Hamiltonian systems has been extensively studied in the past thirty years (see [2, 9, 10, 12, 13] and references therein). Recently, the rotating periodic solutions for nonlinear differential equations has become a very interesting topic. In [3, 4], Chang and Li studied the second order dynamical systems, by using the coincidence degree they obtained some existence results of rotating periodic solutions. In [6], Hu et al build up some important stability criteria for rotating periodic solutions of Hamiltonian systems using the Maslov index theory. In [7], by using Morse theory and the technique of penalized functional, we obtained one nontrivial rotating periodic solutions for a class of asymptotically linear second order Hamiltonian systems with resonance at infinity. In this paper, we shall study the existence and multiplicity of rotating periodic solutions for (1.1) via Morse theory when Vx (t, x) is super-linear at infinity. We make the following assumptions: (H0) V (t, x) ∈ C 2 (R × Rn , R) with V (t, 0) = 0 and V (t + T, x) = V (t, Q−1 x) for some orthogonal matrix Q ∈ O(n); 1

n

−1

2010 Mathematics Subject Classification. 34C25, 37J45, 37B30 . Key words and phrases. Second order Hamiltonian systems; rotating periodic solutions; Morse theory. The first author is supported by Liaocheng University Doctoral Fund. The second author is supported by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065) and NSFC (grant No. 11171132). The third author is supported by NSFC (grant No. 11201173). ∗ Corresponding author. 1

2

(H1) Vx (t, x) = A(t)x + Gx (t, x), where A(t) is an n × n continuous symmetric matrix and Gx (t, x) = o(|x|) as |x| → 0; (H2) lim|x|→∞ G(t,x) |x|2 = +∞ uniformly for t ∈ [0, T ], where | · | is the usual norm in Rn ; (H3) Gx (t, sx)·sx−2G(t, sx) is increasing with respect to s for every x ∈ Rn \{0} and sufficiently large s > 0, where · is the usual inner product in Rn ; (H4) G(t, −x) = G(t, x).

By (H2), G(t, x) is asymptotically superquadratic at infinity. Note that our superquadratic conditions are different from the usual Ambrosetti-Rabinowitz condition, i.e., there is a µ > 2 such that 0 < µG(t, x) ≤ Gx (t, x) · x for all x ∈ Rn \ {0}. 2 2 For example, G(t, x) = (2 + sin 2π T t)|x| ln(1 + |x| ) satisfies (H2) and (H3) but it does not satisfy Ambrosetti-Rabinowitz condition. Let λ1 < λ2 < · · ·λi < · · · be distinct eigenvalues of the linear Hamiltonian systems  −x00 − A(t)x = λx, (1.2) x(T ) = Qx(0), x0 (T ) = Qx0 (0), with the corresponding eigenspaces E(λ1 ), E(λ2 ), · · ·, E(λi ), · · ·. Denote Ej = Lj i=1 E(λi ), `j = dim E(j). By (H1), x = 0 is a solution of (1.1) which is usually called trivial solution. Our aim is to find nontrivial rotating periodic solutions of (1.1). Now we state the main result of this paper.

Theorem 1.1. Assume that (H0), (H1), (H2), (H3) hold, and λk < 0 < λk+1 for some k ≥ 1, then (1.1) has at least one nontrivial rotating periodic solution. Moreover, if (H4) also holds, then (1.1) has infinitely many rotating periodic solutions. It should be pointed out that the study of quasi-periodic solutions is generally very difficult duo to the presence of small divisor. Our result shows that one can deal with the existence of quasi-periodic solutions via variational methods when systems under consideration possess certain rotation invariance usually corresponding to rigid rotation. The paper is organized as follows. In Section 2, we give some preliminary results about Morse theory. In Section 3, we will give the proof of Theorem 1.1 using Morse theory. 2. Preliminaries Let H be a real Hilbert space and J ∈ C 2 (H, R) be a functional satisfying the Cerami condition, i.e., any sequence {un } ⊂ H for which J(un ) is bounded and (1 + kun k)J 0 (un ) → 0 as n → ∞ possesses a convergent subsequence. Denote by Hq (A, B) the q − th singular relative homology group of the topological pair with coefficients in a field F. Let u be an isolated critical point of J with J(u) = c. The group Cq (J, u) := Hq (J c , J c \ {u}), q ∈ Z is called the q − th critical group of J at u, where J c = {u ∈ H | J(u) ≤ c}. Denote K(J) = {u ∈ H | J 0 (u) = 0}. Suppose that J(K) is bounded from below by a ∈ R. The critical groups of J at infinity are defined by Cq (J, ∞) := Hq (H, J a ), q ∈ Z.

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The Morse index of the critical point u is defined by the dimension of the negative space corresponding to the spectral decomposing of J 00 (u) and denoted by m− (J 00 (u)). The following results can be found in [1] and [2]. Proposition 2.1. Suppose J satisfies the Cerami condition. If K(J) = ∅, then Cq (J, ∞) ∼ = 0, q = 0, 1, 2···. If K = {u0 }, then Cq (J, ∞) ∼ = Cq (J, u0 ), q = 0, 1, 2···.

Proposition 2.2. Suppose that J ∈ C 2 (H, R) and u is a nondegenerate critical point of J with Morse index j, then Cq (J, u) = δq,j F.

Theorem 2.1. Suppose that J ∈ C 2 (H, R) satisfies the Cerami condition, and K = {u1 , . . . uk }, then ∞ ∞ X X Mq tq = βq tq + (1 + t)Q(t), q=0

q=0

where Q(t) is a formal series with nonnegative coefficients, Mq = and βq = rankCq (J, ∞), q = 0, 1, 2, · · ·.

Pk

i=0

rankCq (J, uk )

Remark 2.2. The original form of Proposition 2.1 and Theorem 2.1 are under the usual Palais-Smale condition, but since the deformation lemma also holds under the Cerami condition, we can replace the Palais-Smale condition by the Cerami condition. 3. Proof of the main result In [7], we have shown that the rotating periodic solutions of (1.1) is equivalent to the solutions of the following Q-boundary value problem  −x00 = Vx (t, x), (3.1) x(T ) = Qx(0), x0 (T ) = Qx0 (0). This leads us to consider the Hilbert space E defined by

E = {x ∈ H 1 ([0, T ]) | x(T ) = Qx(0), x0 (T ) = Qx0 (0)}

and the functional I on E defined by Z Z T 1 T 0 2 I(x) = |x (t)| dt − V (t, x)dt. 2 0 0

(3.2)

By (H0), the functional I is a C 2 functional and rotating periodic solutions of (1.1) correspond to critical points of the functional I, see [7] for details. This allows us to apply the critical point theory to the existence of rotating periodic solutions of (1.1). Notice that λk < 0 < λk+1 , then corresponding to the eigenvalue λk , E is split as E = E − ⊕ E + with E − = Ek and E + = Ek⊥ . Throughout this paper, for any x ∈ E, we always denote by x− , x+ the vectors in E with x = x− + x+ , x− ∈ E − and x+ ∈ E + . For simplicity, we equip E with the following equivalent norm Z T Z T kxk2 = (|(x+ )0 |2 − A(t)x+ · x+ )dt − (|(x− )0 |2 − A(t)x− · x− )dt. 0

0

Then we can rewrite the functional I by I(x) =

1 (kx+ k2 − kx− k2 ) − 2

Z

0

T

G(t, x)dt.

4

Lemma 3.1. Assume (H0)-(H3) hold, then I satisfies Cerami condition. Proof. Let {xj } ⊂ E be a sequence such that

|I(xj )| ≤ C, (1 + kxj k)I 0 (xj ) → 0

(3.3)

as j → ∞. By a standard argument (see [2]), it suffices to show {xj } is bounded. Conversely, x if {xj } is unbounded, let wj = kxjj k , then wj ∈ E with kwj k = 1. Thus there exist w ∈ E and a sequence of {wj }, for simplicity still denoted by {wj }, such that wj * w in E and wj → w in C([0, T ]). Let I1 = {x ∈ [0, T ] | w(x) 6= 0}, then |xj (t)| → ∞ for t ∈ I1 . By (H2), G(t, xj (t)) |wj (t)|2 → ∞ |xj (t)|2

(3.4)

for t ∈ I1 . By (H2), there exists a constant C2 > 0 such that G(t, x) + C2 ≥ 0.

(3.5)

We claim that w = 0 a.e. in [0, T ], i.e. |I1 | = 0, where |I1 | is the Lebesgue measure of I1 . In fact, if |I1 | > 0, then by (3.4), (3.5) and Fatou’s lemma, we have Z G(t, xj (t)) +∞ = lim inf |wj (t)|2 dt j→∞ |xj (t)|2 I1 Z G(t, xj (t)) + C2 ≤ lim inf dt j→∞ kxj k2 I1 Z G(t, xj (t)) + C2 ≤ lim inf dt j→∞ kxj k2 I1 RT − 2 1 2 kxj k + 0 G(t, xj )dt ≤ lim inf j→∞ kxj k2 ≤

lim inf j→∞

+ 2 1 2 kxj k

− I(xj ) 1 ≤ , kxj k2 2

(3.6)

a contradiction. We choose a sequence {sj } ⊂ [0, 1] such that I(sj xj ) = max I(sxj ). s∈[0,1]

(3.7)

For any c > 0, let vj = cwj . Since wj → 0 in C([0, T ]), then vj → 0 in C([0, T ]), thus by (H2), Z T lim G(t, vj (t))dt = 0. (3.8) j→∞

Since kxj k → ∞, we have 0 <

0 c kxj k < −

have vj− * 0 in E − , note that E

1 for j large enough. From vj * 0 in E, we

is finite dimensional, thus we have kvj− k → 0

(3.9)

kvj+ k2 = kvj k2 − kvj− k2 = c2 − kvj− k2 → c2 .

(3.10)

and

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Then by (3.7), (3.8), (3.9) and (3.10), we have c I(sj xj ) ≥ I( xj ) = I(vj ) kxj k

Z T 1 + 2 − 2 (kv k − kvj k ) − G(t, vj (t))dt = 2 j 0 c2 + o(1). = 2 By the arbitrariness of c, we have that I(tj xj ) → ∞. Note that I(0) = 0 and |I(xj )| ≤ C, for j large enough we have that sj ∈ (0, 1) and 0 dI(sxj ) |s=sj = hI (sj xj ), sj xj i. ds Since sj ∈ (0, 1), by (H3) there exists a constant C1 > 0 such that

0 = sj

Gx (t, sj xj ) · sj xj − 2G(t, sj xj ) ≤ Gx (t, xj ) · xj − 2G(t, xj ) + C1 .

(3.11)

(3.12)

Then by (3.3), (3.11), (3.12) and I(tj xj ) → ∞, we have that Z 1 T 1 0 [Gx (t, xj ) · xj − 2G(t, xj )]dt C + o(1) ≥ I(xj ) − hI (xj ), xj i = 2 2 0 Z 1 T ≥ [Gx (t, sj xj )sj xj − 2G(t, sj xj ) − C1 ]dt 2 0 1 0 1 = I(sj xj ) − hI (sj xj ), sj xj i − C1 T 2 2 1 = I(sj xj ) − C1 T → ∞, 2 thus we get a contradiction and therefore {xj } is bounded.  Now we compute the critical groups of I at infinity. We follow the idea of [11, 8] on the study of p-Laplacian equations. Lemma 3.2. Assume that (H0)-(H3) hold, then the critical group of I at infinity is Cq (I, ∞) = 0, q = 0, 1, 2 · · · . Proof. By (H1) and (H2), for any M > 0, there exists CM > 0 such that G(t, x) ≥ M |x|2 − CM .

Then for x ∈ S with S = {x ∈ E | kxk = 1}, we have Z T s2 I(sx) = (kx+ k2 − kx− k2 ) − G(t, sx)dt 2 0 Z T s2 kxk2 − (M |sx|2 − CM )dt ≤ 2 0 s2 ≤ − M s2 kxk2L2 + CM T. 2 Choose M sufficiently large such that M kxk2L2 > 21 , then I(sx) → −∞ as s → +∞. By (H3), there exists M1 > 0 such that Gx (t, x) · x − 2G(t, x) ≥ −M1 .

6

Choose α sufficiently small such that α < min{−M1 T, inf I(x)}. kxk≤1

For any x ∈ S, there exists s > 1 such that I(sx) ≤ α. If I(sx) ≤ α, we have Z T dI(sx) = s(kx+ k2 − kx− k2 ) − Gx (t, sx) · xdt ds 0 Z T 1 2 1 2 [ s (kx+ k2 − kx− k2 ) − Gx (t, sx) · sxdt] = s 2 2 0 Z T 2 1 2 1 + 2 − 2 ≤ G(t, sx) · sxdt + M1 T ] [ s (kx k − kx k ) − s 2 2 0 2 1 = [I(sx) + M1 T ] < 0. s 2 Hence, by the Implicit Function Theorem, there exists a unique T ∈ C(S, R+ ) such that I(T (x)x) = α. Next we define a continuous mapping η : [0, 1] × (E \ {0}) → E \ {0} by  x x (1 − s)x + sT ( kxk ) kxk , x ∈ E \ (I α ∪ {0}), η(s, x) = x, x ∈ I α .

It is clear that η is a strong deformation retract from E \ {0} to I α , thus we have Cq (I, ∞) = Hq (E, I α ) ∼ = Hq (E, E \ {0}) = 0, q = 0, 1, 2 · · · .



Proof of Theorem 1.1 . : Since λk < 0 < λk+1 , we can conclude that x = 0 is a non-degenerate critical point of I and its Morse index is m− (I 00 (0)) = dim Ek = `k , so by Proposition 2.2, Cq (I, 0) = δq,`k F, q = 0, 1, 2 · · · . By Lemma 3.2, Cq (I, ∞) 6= Cq (I, 0). Then from Lemma 3.1 and Proposition 2.1, I has at least one nontrivial critical point, and therefore the Hamiltonian systems (1.1) has at least one nontrivial rotating periodic solution. If (H4) also holds, then obviously I is an even functional. Set Nτ (A) = {x ∈ E | d(x, A) ≤ τ }. If conversely I only has finitely many critical points, by a symmetric Marino-Prodi perturbation technique (see [5]), for ε and τ small enough we can e C 2 ≤ ε; (ii)I(x) = I(x) e obtain an even functional Ie ∈ C 2 (E, R) such that: (i) kI − Ik e e if x ∈ E \ N2τ (K(I) \ {0}); (iii) K(I) ⊂ Nτ (K(I)), and the critical points of I are all non-degenerate and finite in number. Then Ie has finitely many non-degenerate e 0) = Cq (I, 0) = δq` , q = 0, 1, 2···, Cq (I, e ∞) = Cq (I, ∞) = critical points, and Cq (I, k e 0, q = 0, 1, 2 · · · . Suppose that K(I) = {0, x1 , −x1 , x2 , −x2 , · · · · ·, xm , −xm }. Since Ie ∈ C 2 (E, R) is an even functional, we have that m− (Ie00 (xi )) = m− (Ie00 (−xi )), 1 ≤ i ≤ n. Thus by Proposition 2.2 and Theorem 2.1, we have m X − e00 t `k + 2 tm (I (xi )) = (1 + t)Q(t). (3.13) i=1

Substutite t = 1 into (3.13), the left hand is an odd number and the right hand is an even number, thus we get a contradiction. Therefore, I has infinitely many critical points and the Hamiltonian systems (1.1) has infinitely many rotating periodic solutions. 

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References [1] T. Bartsch, S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Analysis 28(3) (1997), 419-441. [2] K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkh¨ auser Boston 1993. [3] X. Chang, Y. Li, Rotating periodic solutions of second order dissipative dynamical systems, Discrete Continuous Dynamical Systems 36(2) (2016), 643-652. [4] X. Chang, Y. Li, Rotating periodic solutions for second order dynamical systems with singularities of repulsive type. Mathematical Methods in the Applied Sciences 40(8) (2017), 3092-3099. [5] N. Ghoussoub, Duality and perturbation methods in critical point theory. Cambridge University Press, 1993. [6] X. Hu, P. Wang, Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems, Journal of Functional Analysis 261(11) (2011), 3247-3278. [7] G. G. Liu, Y. Li, X. Yang, Roatting periodic solutions for asymptotically linear secondorder Hamiltonian systems with resonance at infinity, Mathematical Methods in the Applied Sciences 40(18) (2017), 7139-7150. [8] S. Liu, S. Li, Existence of solutions for asymptotically linear p-laplacian equations. Bulletin of the London Mathematical Society 36(1) (2004), 81-87. [9] S. Ma, Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems, Journal of Differential Equations 248(10)(2010), 2435-2457. [10] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag New York 1989. [11] K. Perera, Nontrivial Critical Groups in p-Laplacian Problems via the Yang Index, Topological Methods in Nonlinear Analysis 21(21) (2003), 301-309. [12] J. Pipan, M. Schechter, Non-autonomous second order Hamiltonian systems, Journal of Differential Equations 257(2)(2014), 351-373. [13] C. L. Tang, X. P. Wu, Periodic solutions for a class of new superquadratic second order Hamiltonian systems, Applied Math. Letters 34(2)(2014),65-71. Guanggang Liu School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, P. R. China and School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, P. R. China E-mail address: [email protected] Yong Li School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, P. R. China and College of Mathematics, Jilin University, Changchun 130012, P. R. China E-mail address: [email protected] Xue Yang School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, P. R. China and College of Mathematics, Jilin University, Changchun 130012, P. R. China E-mail address: [email protected]