# A Note on the Minimal Period Problem for Second Order Hamiltonian Systems

Authors: Huafeng Xiao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study periodic solutions of second order Hamiltonian systems with even potential. By making use of generalized Nehari manifold, some sufficient conditions are obtained to guarantee the multiplicity and minimality of periodic solutions for second order Hamiltonian systems. Our results generalize the outcome in the literature.


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## Body

## 1. Introduction

Denote byN,Z,R*,R the sets of all natural numbers, integers, nonnegative real numbers, and real numbers, respectively. For N∈N, denote by RN the N-dimensional Euclidean space with the usual inner product (·,·) and norm |·|.Consider the second order Hamiltonian systems(1)x¨+Ax+V′(x)=0,∀x∈RN.
Assume that(A1)
A is a symmetric, negative semidefinite matrix;(V1)
V∈C1(RN,R) and V(x)≥0 for all x∈RN;(V2)
V(x)=o(|x|2) as x→0;(V3)
|x|-2V(x)→+∞as|x|→∞;(V4)
for anyx∈RN with |x|=1, λ→(1/λ)(V′(λx),x) is strictly increasing on (0,∞);(V5)
there exists>2 and a1,a2>0 such that |V(x)|≤a1+a2|x|s;(V6)
V is even; that is, V(-x)=V(x) for all x∈RN.WhenA=0, in his pioneering work [1] of 1978, Rabinowitz established the existence of periodic solutions of (1) when V satisfies (V1), (V2), and the well-known AR-condition:(V7)
(2)∃μ>2,r0>0suchthat0<μV(x)≤(x,V′(x)),∃μ>2,r0>0suchthat0<μV(xoi)≤∀|x|≥r0.He conjectured that (1) possesses a nonconstant solution with any prescribed minimal period under the same assumptions. Since then, many authors devoted themselves to the study of periodic solutions with prescribed minimal period of (1).WhenA=0, in 1993, Long (cf. [2]) proved the existence of T-periodic solutions with minimal period T or T/3 of (1) under the assumptions that V∈C2(RN,R*) satisfies (V2), (V6), and (V7). Releasing assumption (V6), Long showed that (1) possesses periodic solutions with minimal periodic T/k for some integer k satisfying 1≤k≤n+2 (cf. [3]) or 1≤k≤n+1 (cf. [4]). In 2001, Fei et al. (cf. [5]) proved that (1) has a nonconstant periodic solution with any prescribed minimal period when V∈C2(RN,R*) satisfies (V2), (V6), and (V7), and V′′(x) is positive semidefinite.WhenA is positive semidefinite, V∈C2(RN,R*) satisfies (V2) and (V7); in 1997, Fei and Wang (cf. [6]) showed that (1) possesses nonconstant T-periodic solutions with minimal periodic T/k for some odd interger k satisfying 1≤k≤2(iT(A)+vT(A))+3, where iT(A)=∑m=0∞‍M-(Tm(A)), vT(A)=∑m=0∞‍M0(Tm(A)), and Tm(A)=(2πm/T)2I-A. If V is even, replacing (V7) by (V3), and the following condition
(3)∃β∈[0,2],d≥0suchthatμV(x)-(V′(x),x)≤d|x|β,∀|x|≥r0,
Fei and Wang (cf. [7]) estimated that the minimal period is not smaller than T/(iT(A)+vT(A)+2).The above results on minimal period problem were obtained by making use of index theory. Another method being used to study such a problem is Nehari manifold. As is well known, the Nehari manifold, introduced by Nehari (cf. [8, 9]), has been used widely to study the existence of ground state solutions of partial differential equations (cf. [10–14]) and that of (1) (cf. [13, 15, 16]). A ground state solution of a system is a solution which possesses the minimal energy of all solutions of the system. However, such kind of solutions may not have prescribed minimal periods. In 1981, Ambrosetti and Mancini made use of Nehari manifold to study the existence of periodic solutions with any prescribed minimal period of first order Hamiltonian systems with convex potential (cf. [17]). In 2010, Xiao (cf. [18]) proved that (1) possesses a periodic solution with any prescribed minimal period and when A=0, V satisfies (V6) and the following assumption:(V8)
(4)V∈C2(RN,R),0<ν(V′(x),x)≤(V′′(x)x,x),0<ν(V′(x),x)≤(V′′(x)wexi,x)∀x∈RN∖{0},where ν>1. For more results on this direction, we refer to [19, 20].Motivated by [13, 17, 18], in this paper, we consider the multiplicity and minimality of periodic solutions of (1) under the assumption (A1), (V1)–(V6). Our main result reads as follows.Theorem 1.
Assume that (A1), (V1)–(V6) hold. Then, for anyT>0, (1) possesses a periodic solution with minimal period T.Corollary 2.
IfA=0 and V satisfies (V6) and (V8), then, for any T>0, (1) possesses a periodic solution with minimal period T.Theorem 3.
Assume that (A1), (V1)–(V6) hold. Then, for anyT>0, (1) possesses infinitely many pairs of T-periodic solutions.The rest of this paper is divided into two parts. In Section2, we establish the variational functional and state some useful lemmas. In Section 3, we introduce Nehari Manifold and prove our main results.


## 2. Preliminary

Denote byST=R/(TZ). H=W1,2(ST,RN) is a Hilbert space equipping with the usual norm and inner product
(5)∥x∥H2=∫0T‍(|x˙|2+|x|2)dt,〈x,y〉H=∫0T‍[(x˙,y˙)+(x,y)]dt.
The variational functional defined on H, corresponding to (1), is
(6)φ(x)=∫0T‍[12|x˙(t)|2-12(Ax(t),x(t))-V(x(t))12]dt.
If (A1), (V1)–(V5) hold, then φ is continuous differentiable on H and
(7)〈φ′(x),y〉H=∫0T‍[(V′(x(t)),y(t))(x˙(t),y˙(t))-(Ax(t),y(t))-(V′(x(t)),y(t))]dt,∀x,y∈H.
Moreover, ϕ(x)=∫0T‍V(x(t))dt is weakly continuous and ϕ′:H→H* is compact.Define a subspace ofH as follows:
(8)E={x∈H∣x(-t)=-x(t)}.
Then E is a closed subspace of H. If x∈E, it has a Fourier expansion
(9)x(t)=∑n=1∞‍ansin2πntT.
Obviously, E∩RN={0}. We can define an equivalent inner product on E(10)〈x,y〉1=∫0T‍(x˙(t),y˙(t))dt.Let us state a useful lemma and omit the proof. One can find the details in [21].Lemma 4.
Ifx is a critical point of φ on E, then x is a critical point of φ on H.Define an operator onE by extending bilinear form
(11)〈Lx,y〉1=∫0T‍[(x˙(t),y˙(t))-(Ax(t),y(t))]dt.
It is easy to verify that L is a linear self-adjoint operator. Since A is a negative semidefinite matrix, then L has a sequence of eigenvalues
(12)0<λ1≤λ2≤⋯≤λn≤⋯
with λn→+∞ as n→∞. Thus for any x∈E∖{0}, 〈Lx,x〉1≥λ1∥x∥12, where ∥·∥1 denotes the norm induced by 〈·,·〉1. Define another equivalent inner product on E(13)〈x,y〉=〈Lx,y〉1.
Then, for all x∈E, there exist C1,C2>0 such that C1∥x∥≤∥x∥1≤C2∥x∥, where ∥·∥ denotes the norm induced by 〈·,·〉.Functional (6) can be rewritten as
(14)φ(x)=12∥x∥2-∫0T‍V(x(t))dt.In the end of this section, we introduce two useful lemmas.Lemma 5 (see [22]).
Ifx∈H and ∫0T‍x(t)dt=0, then
(15)∫0T‍|x(t)|2dt≤T24π2∫0T‍|x˙(t)|2dt,∥x∥∞2≤T12∫0T‍|x˙|2dt,
where ∥·∥∞=max⁡t∈[0,T]|x(t)|.Ifx∈E, ∫0T‍x(t)dt=0. Thus elements of E satisfy the above two inequalities.LetX be a Banach space such that the unit sphere S1 in X is a submanifold of class (at least) C1 and let J∈C1(S1,R). A sequence {xn}⊂S1 is called (PS) sequence for J if it satisfies {J(xn)} is bounded and J′(xn)→0 as n→∞. We say that J satisfies the (PS) condition if every (PS) sequence contains a convergent subsequence.Lemma 6 (see [13]).
IfX is infinite-dimensional and  J∈C1(S1,R) is bounded below and satisfies the (PS) condition on S1, then J has infinitely many pairs of critical points.


## 3. Proofs of the Main Results

Denote byh(x)=〈φ′(x),x〉. We define the Nehari manifold
(16)M={x∈E∖{0}∣h(x)=0}.Denote byS={x∈E∣∥x∥=1}. Fixing x∈S, define gx(r)=φ(rx) for r∈(0,∞). On the one hand, if x∈M, then y=x/∥x∥∈S and
(17)gy′(∥x∥)=〈φ′(∥x∥y),y〉=〈φ′(x),x〉∥x∥=h(x)∥x∥=0.
On the other hand, if gx′(r)=0, then
(18)h(rx)=〈φ′(rx),rx〉=r〈φ′(rx),x〉=rgx′(r)=0.
Thus, for any x∈S, gx′(r)=0 if and only if rx∈M. The following lemma shows that M≠∅.Lemma 7.
For anyx∈S, there exists a unique r0 dependent on x such that
(19)gx(r0)=sup⁡r∈R*gx(r).Proof.
By the definition ofgx, we have
(20)gx(r)=φ(rx)=12r2-∫0T‍V(rx(t))dt.
By (V2), for anyϵ>0, there exists δϵ>0 such that V(x)≤ϵ|x|2 for |x|≤δϵ. If x∈S, then Lemma 5 implies that ∥x∥L∞≤T/12C2 and |x(t)|≤T/12C2 for t∈[0,T]. Fixing ϵ1<3/(T2C22), there exists r1=δϵ1/(T/12C2) such that for all x∈S and 0<r<r1(21)φ(rx)=12r2-∫0T‍V(rx(t))dt≥12r2-ϵ1r2∫0T‍|x|2dt>r24.
Letα0=1/T∫0T‍|x(t)|dt. Then α0>0. For any 0<α<α0, denote
(22)Aα={t∈[0,T]∣|x(t)|≥α}.
Then meas(Aα)>0, where meas(Aα) denotes the measure of Aα. Thus (V1) and (V3) yield that
(23)gx(r)r2=12-∫0T‍V(rx(t))r2dt≤12-∫Aα‍V(rx(t))|rx(t)|2|x(t)|2dt⟶-∞,12-∫Aα‍V(rx(t))|rx(t)|2|x(t)|2asr⟶∞.
Consequently, there exists r2>0 dependent on x such that gx(r)<0 for r≥r2. By the mean value theorem, there exists r0>0 dependent on x such that gx(r0)=sup⁡r∈R*gx(r) and gx′(r0)=0.
By the definition,gx is continuous differentiable on (0,+∞). Differentiating gx, we have
(24)gx′(r)r=1-1r∫0T‍(V′(rx),x)dt=1-∫0T‍|x|2(1r|x|V′(r|x|y),y)dt,
where y=x/|x|. Because of (V4), gx′(r)/r is strictly decreasing on (0,+∞). Thus gx′(r)/r and hence gx′(r) has a unique zero point on (0,+∞). It follows that gx(r) has a unique critical point, which is a maximum.Remark 8.
Fixingx∈E∖{0}, then x/∥x∥∈S. Obviously, φ(rx)=φ(r∥x∥·x/∥x∥) and sup⁡r∈R*φ(rx)=sup⁡r∈R*φ(rx/∥x∥). If we extend the definition of gx to E∖{0}, then Lemma 7 implies that there exists a unique rx>0 such that gx(rx)=sup⁡r∈R*φ(rx). Also, gx′(r)>0 for 0<r<rx and gx′(r)<0 for r>rx.Now, we study the properties ofφ restricted on M.Lemma 9.
Consider the following:
c
0
=
inf
⁡
x
∈
M
φ
(
x
)
>
0.Proof.
For anyx∈M, both Lemma 7 and Remark 8 imply that φ(x)=sup⁡r∈R*φ(rx)>r12/16. Hence c0=inf⁡x∈Mφ(x)≥r12/16>0.Lemma 10.
φ satisfies the (PS) condition on M.Proof.
Assume that{xn}⊂M is a (PS) sequence of φ. Then there exists d≥0 such that φ(xn)≤d and φ′(xn)→0 as n→∞. We show that {xn} is bounded. Suppose, to the opposite, that ∥xn∥→∞ as n→∞. Set yn=xn/∥xn∥. Then {yn}⊂S. Passing to a subsequence, yn⇀y0. If y0=0, then yn(t)⇒0 uniformly for t∈[0,T]. Hence, for any r>0, we have
(25)d≥φ(xn)=φ(∥xn∥yn)≥φ(ryn)=12r2-∫0T‍V(ryn(t))dt⟶12r2.
This is a contradiction if r>2d. So y0≠0. Arguing similarly to [23], there exist δ1>0, δ2>0 such that
(26)meas⁡({t∈[0,T]∣|y0(t)|≥δ1})≥δ2.
Denote A={t∈[0,T]∣|y0(t)|≥δ1}. But (V3) implies that
(27)0≤φ(xn)∥xn∥2≤12-∫A‍V(∥xn∥yn)∥xn∥2|yn|2|yn|2dt⟶-∞,asn⟶∞.
This is a contradiction. Hence {∥xn∥} is bounded. Since ϕ′ is compact, passing to a subsequence, {ϕ′(xn)} converges strongly. Since xn=φ′(xn)+ϕ′(xn), then {xn} contains a convergent subsequence.Define the maps as follows:(28)m^:E∖{0}⟶M,m^(x)=rxx,m=m^|S:S⟶M,m=m|S,Ψ^:E∖{0}⟶R,Ψ^(x)=φ(m^(x)),Ψ:S⟶R,Ψ=Ψ^|S,
where rx is defined on Remark 8.Lemma 11.
Assume that (V1)–(V4) hold. Then the following statements hold:(B1)
there exists a normalization functionφ such that
(29)u⟼ψ(u):=∫0∥u∥‍φ(t)dt∈C1(E∖{0},R);J:=ψ′ is bounded on bounded sets and J(w)w=1 for all w∈S;(B2)
for eachw∈E∖{0} there exists sw such that if αw(s):=Φ(sw), then αw′(s)>0 for 0<s<sw and αw′(s)<0 for s>sw;(B3)
there existsδ>0 such that sw≥δ for all w∈S and for each compact subset W⊂S there exists a constant CW such that sw≤CS  for all w∈W.Proof.
It follows from the discussion in [13] and Remark 8 that (B1), (B2) hold, and rx≥r1/2 for all x∈S. Next, we show that the last part of (B3) holds. Arguing by contradiction, we suppose that there exists a compact subset W∈S and a sequence {rxnxn}⊂M, where {xn}⊂W and {rxn}⊂R*∖{0}, such that rxn→∞ as n→∞. Since W is compact and xn∈W, passing to a subsequence, xn→x0∈S. Since the embedding of H into C(ST,RN) is compact and xn∈W, then xn(t)⇒x0(t) uniformly for t∈[0,T]. Denote by Ax0={t∈[0,T]∣|x0(t)|≥1/T∫0T‍|x0(t)|dt}. Since rxn|xn(t)|→∞, if t∈Ax0, then Fatou’s lemma yields
(30)ϕ(rxnxn)rxn2≥∫Ax0‍V(rxnxn(t))rxn2|xn(t)|2|xn(t)|2dt⟶∞,ϕ(rxnxn)rxn2≥∫Ax0‍V(rxnxn(t))rxn2|xn(t)|2|xni|2asn⟶∞.
Thus φ(rxnxn)=1/2rxn2-ϕ(rxnxn)→-∞, which contradicts with the fact that φ(rxnxn)>r12/16>0. Consequently, there exists CW>0 such that rx≤CW for all x∈W and (B3) holds.Lemma 12 (see [13]).
The mappingm^ is continuous and m is a homeomorphism between S and M.Lemma 13 (see [13]).
Consider the following.(1)
Ψ^,Ψ∈C1(S,R) and
(31)〈Ψ^′(x),y〉=∥m^(x)∥∥x∥〈Ψ′(m^(x)),y〉,∀x,y∈E,x≠0,〈Ψ′(x),y〉=∥m(x)∥〈Ψ′(m(x)),y〉,∀y∈Tx(S),
where Tx(S)={y∈E∣〈φ(x),y〉=0} is the tangent space of S at x.(2)
If{xn} is a (PS) sequence for Ψ, then {m(xn)} is a (PS) sequence for φ. If {xn}⊂M is a bounded (PS) sequence for φ, then {m-1(xn)} is a (PS) sequence for Ψ.(3)
x is a critical point of Ψ if and only if m(x) is a nontrivial critical point of φ. Moreover, the corresponding values of Ψ and φ coincide and inf⁡SΨ=inf⁡Mφ.(4)
Ifφ is even, then so is Ψ.Lemma 14.
Ifφ satisfies the (PS) condition, so does Ψ.Proof.
Assume that{xn} is a (PS) sequence for Ψ. According to Lemma 13, {m(xn)}⊂M is a (PS) sequence for φ. Since φ satisfies the (PS) condition, passing to a subsequence, m(xn)→y. Thus {xn} contains a convergent subsequence, which converges to m-1(y). Hence Ψ satisfies the (PS) condition.Proof of Theorem1.
According to Lemmas9 and 13, inf⁡x∈SΨ(x)=inf⁡x∈Mφ(x)=c0. Let {yn} be a minimizing sequence for Ψ restricted to S. By Ekeland’s variational principle, we may assume that Ψ′(yn)→0 as n→∞. The (PS) condition implies that {yn} contains a converging subsequence, whose limit is denoted by y0. Thus y0 is a critical point of Ψ. According to Lemma 13 again, x0=m(y0) is a critical point of φ, which is also a nonconstant T-periodic solution of (1).
Claim.
x
0 has minimal period T.
Suppose, to the opposite, thatx0 has minimal period T/k, where k≥2 is an integer. Let y0(t)=x0(t/k). Obviously, y0∈E. It follows from Lemma 7 that there exists r¯>0 such that r¯y0∈M. It follows that
(32)inf⁡x∈Mφ(x)≤φ(r¯y0)=∫0T‍[12|r¯y˙0(t)|2-V(r¯y0(t))]dt=∫0T‍[12k2|r¯x˙0(tk)|2-V(r¯x0(tk))]dt=∫0T‍[12k2|r¯x˙0(τ)|2-V(r¯x0(τ))]dτ<∫0T‍[12|r¯x˙0(τ)|2-V(r¯x0(τ))]dτ=φ(r¯x0)≤φ(x0)=inf⁡x∈Mφ(x),
which is a contradiction. Hence x0 has minimal period T.Proof of Corollary2.
Forx∈RN with |x|=1, set a(λ)=λ-1(V′(λx),x). Then (V8) implies that
(33)a′(λ)=1λ3[(V′′(λx)λx,λx)-(V′(λx),λx)]>0;
that is, a(λ) is increasing strictly on (0,+∞). Thus (V4) is satisfied. It is easy to check that V satisfies (V1)–(V3) when (V8) is available.
Without assumption (V5),φ may not be continuous differentiable on H. However, we can use the method introduced in [1] to handle such a situation. Let K>1 and χ∈C∞(R+,R+) such that χ(s)=1 if s≤K, χ(s)=0 if s≥K+1, and χ′(s)<0 if s∈(K,K+1). Set
(34)VK(x)=χ(|x|)V(x)+[1-χ(|x|)]ρ(K)|x|4,
where ρ(K)≥max⁡K≤|x|≤K+1(V(x)/|x|4). Since V satisfies (V8), it follows from Lemma 2.3 in [18] that
(35)0<(ν+1)V(x)≤(V′(x),x),0<(ν+)Viii(x)∀x∈RN∖{0}.
Arguing similarly to Lemma 2.9 in [1], we have
(36)0<λVK(x)≤(VK′(x),x),∀x∈RN∖{0},
where λ=min⁡{ν+1,4}>2. Let a3=min⁡|x|=1Vk(x) and a4=max⁡|x|=1Vk(x). Since K>1, then a3=min⁡|x|=1V(x)>0 and a4=max⁡|x|=1V(x)>0. Obviously, a3,a4 are independent of K. Arguing similarly to Lemma 4.1 of [22], we can prove the following inequality:
(37)VK(x)≥a3|x|λ-a4,∀x∈RN.
Hence VK satisfies (V1)–(V6).
Consider the disturbed second order Hamiltonian systems(38)x¨+VK′(x)=0,
whose variational function is
(39)φK(x)=∫0T‍[12|x˙(t)|2-VK(x(t))]dt.
Since VK satisfies (V1)–(V6), applying Theorem 1, (38) possesses a periodic solution xK with minimal period T.
Claim. There exists K0>0 independent of K such that ∥xK∥L∞<K0.
Letx1(t)=b1T/(2π2)sin(2πt/T)∈E with b1=(1,0,…,0)′∈RN, where (′) denotes the transposition of a vector. Computing directly, we have
(40)∥x1∥2=∥x1∥12=1,∥x1∥L22=T24π2.
So x1∈S. It follows from (37) that
(41)φK(rx1)=12r2-∫0T‍VK(rx1)dt≤12r2-a3rλ∥x1∥Lλλ+a4T≤12r2-a3M1λrλ∥x1∥L2λ+a4T≤12r2-a3M1λrλTλ(2π)λ+a4T=h(r),
where M1 denotes the constant such that ∥x∥Lλ≥M1∥x∥L2. Since λ>2, h(r)→-∞ as r→+∞. Then there exists M2>0 independent of K such that h(r)≤M2. Thus
(42)cK=inf⁡x∈MφK(x)=inf⁡x∈Smax⁡r∈R*φK(rx)≤max⁡r∈R*φK(rx1)≤max⁡r∈R*h(r)≤M2.
Hence
(43)M2≥cK=φK(xK)=∫0T‍[12(VK′(xK),xK)-VK(xK)]dt≥λ-22∫0T‍VK(xK)dt.
Consequently, we have
(44)c∞-2∥x∥L∞2≤∥x∥2=2φK(xK)+2∫0T‍VK(xK)dt≤2λλ-2M2.
It follows that there exists K0>0 independent of K such that ∥xK∥L∞≤K0. If K>K0, by the definition of VK, VK=V and hence xK is a nonconstant T-periodic solution of (1) with A=0.Proof of Theorem3.
SinceV is even, so do φ and Ψ. Since inf⁡x∈SΨ(x)=c0>0 and Ψ satisfies the (PS) condition, then Lemma 6 yields that Ψ has infinitely many pairs of critical points. According to Lemma 13, φ has infinitely many pairs of critical points. Thus (1) has infinitely many pairs of periodic solutions.Example 15.
LetA=0 and let V be
(45)V(x)=|x|2[ln⁡⁡(1+|x|2)]2.
It is easy to check that V does not satisfy (V8) or even (V7) (cf. [23]). However, V satisfies (V1)–(V6). Applying Theorems 1 and 3, for any T>0, (1) possesses infinitely many pairs of T-periodic solutions and has at least one solution with T as its minimal period.



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