multiplicity of lagrangian orbits on symmetric star-shaped hypersurfaces

Nonlinear Analysis 69 (2008) 1425–1436www.elsevier.com/locate/na

Multiplicity of Lagrangian orbits on symmetricstar-shaped hypersurfaces

Fei Guoa, Chungen Liub,∗

a Department of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of Chinab School of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China

Received 18 April 2007; accepted 29 June 2007

Abstract

In this paper, the multiplicity of Lagrangian orbits on C2 smooth compact symmetric star-shaped hypersurfaces Σ with respectto the origin in R2n is studied. We prove an infinitely many existence result via Z2-index theory. This is a multiplicity result aboutthe Arnold chord in some sense.c© 2008 Published by Elsevier Ltd

Keywords: Symmetric star-shaped hypersurfaces; Multiplicity; Lagrangian orbits; Z2-index theory

1. Introduction and main result

A compact hypersurface Σ in R2n is called a star-shaped hypersurface, if it bounds an open set Γ (Σ ), and thereexists a point x0(Σ ) ∈ Γ (Σ ) such that the tangent plane of Σ at any point x ∈ Σ does not pass through x0(Σ ). Inthis paper, we fix the point x0(Σ ) = o (the origin), and say that Σ is a star-shaped hypersurface with respect to theorigin. In addition, in this paper, we suppose that Σ is symmetric with its center at the origin. We call this kind ofhypersurfaces the symmetric star-shaped hypersurfaces with respect to the origin.

For z ∈ Σ , let NΣ (z) be the unit outward normal vector of Σ at z. We consider the problem of finding τ > 0 andan absolutely continuous curve z : [0, τ ] → Σ such that{

z(t) = J NΣ (z(t)), ∀t ∈ [0, τ ],

z(0), z(τ ) ∈ L ,(1.1)

where J =

(0 −InIn 0

)is the 2n × 2n standard symplectic matrix with In being the n × n identity matrix, L is a

fixed Lagrangian subspace in symplectic vector space (R2n, ω0) defined by the maximal-dimensional space satisfyingω0|L = 0, and ω0 is the standard symplectic form defined by ω0 =

∑ni=1 dxi ∧ dyi . Note that L ∩ Σ 6= ∅. We denote

by (τ, z) the L-Lagrangian orbit on Σ which solves the problem (1.1).

∗ Corresponding author.E-mail address: [email protected] (C. Liu).

0362-546X/$ - see front matter c© 2008 Published by Elsevier Ltddoi:10.1016/j.na.2007.06.042

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1426 F. Guo, C. Liu / Nonlinear Analysis 69 (2008) 1425–1436

If there exists a function H ∈ C1(R2n, R) satisfying H−1(1) = Σ with ∇ H(z) 6= 0 for all z ∈ Σ , then we cantransform the problem (1.1) into the following nonlinear Hamiltonian system with fixed energyz(t) = J∇ H(z(t)),

H(z(t)) = 1, t ∈ [0, τ ],

z(0), z(τ ) ∈ L .

(1.2)

Similarly for the problem of periodic orbits, the existence and multiplicity of the solutions of the problem (1.1) areindependent of the choice of the function H such that Σ = H−1(1). Moreover, we can transform the problem offinding the solution(s) of (1.2) to that of finding the critical point(s) of a functional via variational principle.

In [13], the second author of this paper transformed the problem of Lagrangian intersections into a Hamiltoniansystem similar to the problem (1.1). From the viewpoint of the contact geometry, any compact star-shapedhypersurface Σ with respect to the origin in R2n is a closed contact manifold with a contact form induced from thesymplectic form ω0, and Σ ∩ L is a Legendrian submanifold of Σ for any Lagrangian subspace L . So the existenceof the problem (1.1) is a special case of the Arnold’s Chord Conjecture, see p. 15–16 in paper [2], the conjecture saidthat on a closed contact manifold (precisely, S2n−1 with a standard contact structure ξ0), for any closed Legendriansubmanifold, there always exists a Reeb chord intersecting the Legendrian submanifold at least twice for any choice ofcontact form. It is well known that one can transform this problem in the case of (S2n−1, ξ0) to the existence problemof (1.1) with Σ being a star-shaped hypersurface and L being a Lagrangian submanifold satisfying Σ ∩ L 6= ∅. Anystar-shaped hypersurface (with the contact form λ =

∑ni=1 xi ∧dyi induced from ω0) in R2n is a contact manifold. The

intersection of star-shaped hypersurface with respect to the origin with a Lagrangian subspace in R2n is a Legendriansubmanifold. This problem is also related to the study of the second-order system{

x(t) + ∇V (t, x) = 0,

x(0) = x(τ ) = 0,

where V ∈ C2(R × Rn, R). One can transform it into a first-order Hamiltonian system with L0 boundary value

condition by a classical method, where L0 = range(

0In

)is the standard Lagrangian subspace in (R2n, ω0).

Similar to the problem of the periodic solutions of Hamiltonian systems, problem (1.2) has a Maslov-type indextheory, which was studied by the second author of this paper in [14], moreover, the existence and the multiplicityof Lagrangian boundary solutions of asymptotically linear Hamiltonian systems were studied in [15] by using thisMaslov-type index theory. The problem (1.2) is also related to the Bolza problem (see for example [7]).

In paper [10], by using a variational method, the following existence result on Lagrangian orbit was proved.

Proposition 1.1. For every Lagrangian subspace L and C1 smooth compact star-shaped hypersurface Σ with respectto the origin, there exists at least one L-Lagrangian orbit on Σ .

In 2001, Mohnke [17] using some symplectic topological methods proved a similar but in some sense more generalresult: “For every closed Legendrian submanifold in (S2n−1, ξ0) with standard contact structure ξ0 and any contactform for this contact structure, there is a Reeb chord”, i.e. ]

{L ′∩{ϕt (L ′), t > 0}} ≥ 1, where L ′ is a closed Legendrian

submanifold and ϕt is the Reeb flow. It is a positive answer to the Arnold’s chord conjecture. We note that Ma andXu [16] using different methods proved a result similar to Proposition 1.1.

In this paper, if Σ is a symmetric star-shaped hypersurface with respect to the origin, we will prove the followingmultiplicity result :

Theorem 1.2. If Σ is an arbitrary C2 smooth compact symmetric star-shaped hypersurface with respect to the origin,then for every Lagrangian subspace L in R2n , Σ possesses infinitely many L-Lagrangian orbits.

Suppose ϕtH is the Hamiltonian flow of the Hamiltonian system z(t) = J∇ H(z(t)). We know that its restriction

to the hypersurface Σ is a contact flow. We call the restriction of ϕtH to Σ the contact Hamiltonian flow of Σ . The

subset ϕΣ (L ′) :=⋃

t>0 ϕtH (L ′) of Σ is independent of the choice of H (in fact, it is an n-dimensional immersion

submanifold of Σ for L ′= Σ ∩ L). From Theorem 1.2, we have the following consequence about the intersection

numbers:

F. Guo, C. Liu / Nonlinear Analysis 69 (2008) 1425–1436 1427

Corollary 1.3. For every Lagrangian subspace L in R2n , if Σ is a C2 smooth compact symmetric star-shapedhypersurface with respect to the origin, then

]{L ′

∩ ϕΣ (L ′)} = ∞,

where the Legendrian submanifold L ′ is defined by L ′= Σ ∩ L.

Remark 1.4. We only need to prove Theorem 1.2 for the special case L = L0. It is well known that any Lagrangiansubspace L can be transformed by an orthogonal symplectic transformation to L0. That is, there is an orthogonalsymplectic matrix P such that P L = L0. Any orthogonal transformation preserves the symmetric star-shapedhypersurfaces with respect to the origin.

The result of Corollary 1.3 in some sense is related with the Lagrangian (or Legendrian) intersections. On thistopic, one can refer [4,6,8,9,11–13,19,18], etc.

Another consequence of Theorem 1.2 is the following multiple solutions result on some autonomous second-ordersystems.

Corollary 1.5. Suppose that V ∈ C2(Rn, R) satisfies the following conditions

(i) (∇V (x), x) > 0, ∀x 6= 0.(ii) V (x) > 0, ∀x 6= 0, V (0) = 0.

(iii) V (x) → +∞, |x | → +∞.(iv) V (−x) = V (x), ∀x ∈ Rn .

Then the following problem{x(t) + ∇V (x) = 0,

x(0) = x(τ ) = 0, f or some τ > 0(1.3)

possesses infinitely many solutions (τ, x) such that |x(0)| = 1.

Proof. By taking y(t) = x(t), one can transform the problem (1.3) into the following problem{z(t) = J∇ H(z(t)), z(t) = (x(t), y(t))z(0) ∈ L0, z(τ ) ∈ L0

(1.4)

with the Hamiltonian function

H(x, y) =12|y|

2+ V (x).

We set Σ = {(x, y) ∈ R2n| H(x, y) =

12 }. Then the conditions (ii) and (iii) imply that Σ is compact and the origin is

contained in the bounded open set around Σ . The condition (i) implies that Σ is star-shaped with respect to the originand the condition (iv) implies that Σ is symmetric with the center at the origin. �

Inspired by the result of Theorem 1.2, we have the following

Conjecture. For every Lagrangian subspace L in (R2n, ω0), every star-shaped hypersurface Σ with respect to theorigin possesses infinitely many L-Lagrangian orbits. i.e.

]{L ′

∩ ϕΣ (L ′)} = ∞.

As far as the authors’ knowledge is concerned, the result

]{L ′

∩ ϕΣ (L ′)} ≥ 2

has not been proved for any star-shaped hypersurface Σ with respect to the origin.


2. Reduction to a Hamiltonian system

From now on, we suppose that Σ is an arbitrary C2 smooth symmetric star-shaped hypersurface in R2n withrespect to the origin, and denote by C the region around Σ . Define the gauge function jΣ : R2n

→ R byjΣ (z) = min{λ > 0|

zλ

∈ C}, ∀z 6= 0 and jΣ (0) = 0, then jΣ ∈ C(R2n, R+), where R+:= [0, +∞), some

properties of jΣ can be found in p. 69 in [7]. Define H2 = j2Σ then H2 ∈ C1(R2n, R+) ∩ C2(R2n

\ {0}, R+) ishomogeneous of degree two.

A function φ ∈ C2(R+, R+) is said to be admissible, if it satisfies the following conditions:

(i) φ(0) = 0, φ′(+∞) := limt→+∞φ(t)

t > 0;(ii) φ′′(t) < 0, ∀t ≥ 0;

(iii) supt≥0 |φ′′(t)t | < +∞.

Such functions do exist, for example, φ(t) = t + ln(t + 1), ∀t ≥ 0. If function φ is admissible, from the above threeconditions we know that

φ′ is strictly decreasing, so 0 < φ′(+∞) < φ′(t) ≤ φ′(0), ∀t ≥ 0, (2.1)

∀t > 0,

∫ t

0τφ′′(τ )dτ < 0, so φ′(t)t − φ(t) < 0, ∀t > 0. (2.2)

Choose an admissible function φ, which will be precisely determined later, define

H(z) = φ(H2(z)), ∀z ∈ R2n .

Because Σ is symmetric with its center at the origin, we have that H2(z) = H2(−z), ∀z ∈ R2n . Thus we get an evenfunction H ∈ C1(R2n, R+) ∩ C2(R2n

\ {0}, R+), that is,

H(z) = H(−z), ∀z ∈ R2n .

If we normalize the outward normal vector NΣ (z) by (NΣ (z), z) = 2φ′(1) for every z ∈ Σ , then

∇ H(z) = NΣ (z), ∀z ∈ Σ .

So we reduce the problem (1.1) for L = L0 to the following fixed energy problem of Hamiltonian system withL0-boundary value conditionz(t) = J∇ H(z) = Jφ′(H2(z))∇ H2(z),

H(z(t)) = φ(1) > 0, t ∈ [0, T ],

z(0), z(T ) ∈ L0.

(2.3)

By direct computation, H ′′(z) = φ′′(H2(z))∇ H2(z)∇ H t2(z) + φ′(H2(z))H ′′

2 (z). In view of the fact that H2 ishomogeneous of degree two and φ is admissible, H ′′ is bounded in R2n

\ {0}. So we can choose a constant M > 0such that

|H ′′(z)| ≤ M, ∀z ∈ R2n\ {0}. (2.4)

Define

HK (z) = H(z) +K

2|z|2,

then we can choose K > M with K T 6∈ πZ such that HK (z) ∈ C1(R2n, R+) ∩ C2(R2n\ {0}, R+) is strictly convex

in the sense that

(∇ HK (z1) − ∇ HK (z2), z1 − z2) ≥ (K − M)|z1 − z2|2 > 0, ∀z1 6= z2 ∈ R2n .

The Fenchel dual of HK is defined by

H∗

K (z∗) = supz∈R2n

{(z∗, z) − HK (z)}, ∀z∗∈ R2n,


then H∗

K ∈ C1(R2n, R+) ∩ C2(R2n\ {0}, R+) is also strictly convex, moreover, H∗

K has the following properties (seep. 82–85 in [7]):

∇ H∗

K (z∗) = z if and only if z∗= ∇ HK (z), (2.5)

H∗

K (z∗) = (z, z∗) − HK (z) if and only if z∗= ∇ HK (z). (2.6)

From now on, fix a real number T > 0 and define two spaces as follows

W := {z ∈ W 1,2([0, T ], R2n)|z(0), z(T ) ∈ L0}, L2:= L2([0, T ], R2n).

These two spaces are Hilbert spaces with inner products defined respectively by 〈z1, z2〉W =1T

∫ T0 {(z1, z2) +

(z1, z2)}dt and 〈z1, z2〉L2 =1T

∫ T0 (z1, z2)dt . From now on, denote the inner product and norm in R2n by (·, ·) and | · |

respectively, and denote the inner product and norm in Ł2 by 〈·, ·〉 and ‖ · ‖ respectively.Define a dual functional FK : W → R by

FK (z) =

∫ T

0

[H∗

K (−J z + K z) −12(−J z + K z, z)

]dt, ∀z ∈ W.

From the fact that H(−z) = H(z) and the definition of Fenchel dual, we know that H∗

K (−z) = H∗

K (z), so FK is aneven functional on W .

Proposition 2.1. From the above K and T with K T 6∈ πZ, z is a critical point of FK if and only if z is C1 and z is asolution of{

z(t) = J∇ H(z(t)) = Jφ′(H2(z))∇ H2(z), ∀t ∈ [0, T ],

z(0), z(T ) ∈ L0.(2.7)

Proof. By direct computation,

〈F ′

K (z), h〉 =

∫ T

0[(∇ H∗

K (−J z + K z) − z, −J h + K h)]dt, ∀h ∈ W. (2.8)

If z is the solution of (2.7), then −J z + K z = ∇HK (z), using (2.5), we have ∇ H∗

K (−J z + K z) = z, so〈F ′

K (z), h〉 = 0, ∀h ∈ W .Suppose that z satisfies 〈F ′

K (z), h〉 = 0, ∀h ∈ W . From the condition K T 6∈ πZ we know that the operator−J d

dt + K id : W → L2 is invertible, so there exists h ∈ W such that ∇ H∗

K (−J z + K z) − z = −J h + K h. For this

h ∈ W , we have∫ T

0 |∇ H∗

K (−J z + K z) − z|2dt = 0, then we get ∇ H∗

K (−J z + K z) = z, a.e. on [0, T ], using (2.5),we get −J z + K z = ∇ HK (z), a.e. on [0, T ], i.e.

z = J∇ H(z), a.e. on [0, T ]. (2.9)

By the Sobolev imbedding theorem, we have z ∈ C([0, T ], R2n). Since z satisfies (2.9), the derivation of z can beextended continuously to all t on [0, T ]. This proves that z ∈ C1([0, T ], R2n), i.e., z(t) is the solution of (2.7). �

So we reduce the solution(s) of the system (2.7) to the critical point(s) of the functional FK .

Proposition 2.2. z ≡ 0 is the unique trivial critical point of FK and FK (0) = 0; if z is a nontrivial critical point,then FK (z) < 0.

Proof. In view of the fact that ∇ H(0) = 0 and ∇ H(z) 6= 0, ∀z 6= 0, from Proposition 2.1 we know that 0 is the uniquetrivial critical point of FK . From HK (0) = 0, ∇ HK (0) = 0 and (2.6), we know that H∗

K (0) = 0. So FK (0) = 0.If z is a nontrivial critical point, from Proposition 2.1, we know that z = J∇ H(z), so −J z + K z = ∇ HK (z), using

(2.6) we have H∗

K (−J z + K z) = (−J z + K z, z) − HK (z). Now we can estimate the critical value FK (z) by using(2.2):


FK (z) =

∫ T

0

[H∗

K (−J z + K z) −12(−J z + K z, z)

]dt

=

∫ T

0

[12(−J z + K z, z) − HK (z)

]dt

=

∫ T

0

[12(∇ H(z), z) − H(z)

]dt

=

∫ T

0

[12φ′(H2(z))(∇ H2(z), z) − H(z)

]dt

=

∫ T

0[φ′(H2(z))H2(z) − φ(H2(z))]dt < 0.

We complete the proof. �

The operator −J ddt : W → L2 is self-adjoint in the inner product of L2, and its spectra are {

kπT |k ∈ Z}, which are

the eigenvalues of the operator −J ddt , the corresponding eigen-subspace is

Ek := span

ek j (t) =

− sin(

kπ

Tt

)e j

cos(

kπ

Tt

)e j

∣∣∣∣∣∣∣∣ j = 1, . . . , n

,

where {e j | j = 1, . . . , n} is the standard basis of Rn . Obviously, dim Ek = n and{ek j |k ∈ Z, j = 1, . . . , n

}is the

standard orthogonal basis of W in the inner product of L2. So every z ∈ W can be written as

z(t) =

∑k∈Z

n∑j=1

zk j ek j (t)

with∑

k∈Z∑n

j=1

(1 +

k2π2

T

)z2

k j < +∞. Through direct computation, we have

1T

∫ T

0| − J z + K z|2dt =

∑k∈Z

n∑j=1

(K +

kπ

T

)2

z2k j , (2.10)

1T

∫ T

0(−J z + K z, z)dt =

∑k∈Z

n∑j=1

(K +

kπ

T

)z2

k j . (2.11)

3. Proof of Theorem 1.2 via Z2-index theory

Now, we recall the definition and some properties of Z2-index (or genus, see paper [1] or book [5]). Suppose thatE is a Banach space and define the following set

E := {A ⊂ E |A is closed in E and symmetric with center at the origin}.

For every closed set A ∈ E , we define the Z2-index of A by

γ (A) =

min{m ∈ N|∃ϕ : A → Rm\ {0} odd and continuous }, if A 6= ∅,

0, if A = ∅,

+∞, otherwise .

The Z2-index of a closed set A has the following properties (see [1], for example).

Proposition 3.1. (1) γ (A) = 0 if and only if A = ∅.(2) γ (Sn) = n + 1, where Sn is the sphere in Rn+1.


(3) If γ (A) ≥ 2, then A possesses infinitely many points.(4) Let V be a k-dimensional subspace of E, V ⊥ is an algebraically or topologically complementary subspace, if

γ (A) > k, then A ∩ V ⊥6= ∅.

(5) If A is compact, then γ (A) < +∞.

The following lemma is obtained from book [5].

Lemma 3.2. Suppose that E is a Banach space, f is an even functional and satisfies (PS) condition, that is, for asequence {z j } ⊂ E with f (z j ) bounded and f ′(z j ) → 0 in E, as j → +∞, there exists a subsequence convergingto z in E. Define

ck = infγ (A)≥k

supx∈A∈E

f (x), k = 1, 2, . . . ,

then

(1) if −∞ < ck < +∞, then ck is the critical value of f ,(2) if −∞ < c := ck = ck+1 = · · · = ck+m−1 < +∞, then γ (Kc) ≥ m, where Kc := {x ∈ E | f ′(x) = 0

and f (x) = c},(3) ck ≤ ck+1.

The following lemma is similar to that in p. 219 in [5].

Lemma 3.3. Suppose that E is a Banach space, f is an even functional and satisfies (PS) condition, f (0) = 0, then

(1) if there exists an m-dimensional subspace V1 and a sphere Sρ(0) in V1 such that supx∈V1∩Sρf (x) < 0, then

ck < 0 for 1 ≤ k ≤ m,(2) if there exists a p-dimensional subspace V2 such that infx∈V ⊥

2f (x) > −∞, then ck > −∞ for k > p,

(3) if m > p in (1) and (2) above, then f has at least m − p pairs of distinct critical pointsx p+1, x p+2, . . . , xm, −x p+1, −x p+2, . . . ,−xm , which correspond to the critical value cp+1 ≤ cp+2 ≤ · · · ≤

cm < 0 defined by

ck = infγ (A)≥k

supx∈A∈E

f (x), k = p + 1, p + 2, . . . , m.

Proof. Step 1. From the definition of cm and the fact that γ (V1 ∩ Sρ) = m, we have

cm := infγ (A)≥m

supx∈A∈E

f (x) ≤ supx∈V1∩Sρ

f (x) < 0.

So ck ≤ cm < 0 for k ≤ m.Step 2. From the result (4) in Proposition 3.1, we know that if γ (A) > p, then A ∩ V ⊥

2 6= ∅. From the definition ofcp+1 we know that

cp+1 := infγ (A)≥p+1

supx∈A

f (x) ≥ infγ (A)≥p+1

supx∈A∩V ⊥

2

f (x) ≥ infx∈V ⊥

2

f (x) > −∞.

So ck ≥ cp+1 > −∞ for k ≥ p + 1.Step 3. By Steps 1, 2 and Lemma 3.2, we know that ck defined by ck := infγ (A)≥k supx∈A f (x) is the critical value

of f for k = p + 1, p + 2, . . . , m. If cp+1 < cp+2 < · · · < cm , then we have at least m − p pairs of distinct criticalpoints

x p+1, x p+2, . . . , xm, −x p+1, −x p+2, . . . ,−xm,

which correspond to m − p distinct critical values. If there exist i 6= j ∈ {p +1, p +2, . . . , m} such that ci = c j , thenfrom Lemma 3.2 we know that γ (Kci ) ≥ 2, where Kci := {x ∈ E | f ′(x) = 0 and f (x) = ci }, using Proposition 3.1,Kci possesses infinitely many points. The proof is completed. �


If for every s ∈ (0, 14R ), the following systemz(t) =

(1

2R− s

)π

TJ∇ H2(z(t)) := bs J∇ H2(z(t)),

z(0), z(T ) ∈ L0,

(3.1)

always has a nontrivial solution zs(t), then zs(t) := ρ−1/2s zs

(t

bs

)is a solution of the fixed energy problem:z(t) = J∇ H2(z(t)),

H2(z(t)) = 1,

z(0), z (bs T ) ∈ L0,

where ρs = H2(zs(t)) > 0, that is, (bs T, zs) solves the problem (1.1) with L = L0. Due to the infinitely differentchoices of bs , we get infinitely many solutions of (1.1) with L = L0, that is, Theorem 1.2 is proved in this case withL = L0.

Now we consider our problem with the following condition(C) there exists an s0 ∈ (0, 1

4R ) such that the system (3.1) has no nontrivial solution.Because Σ is compact in R2n , there exist two balls by which Σ is pinched, that is, if C denotes the region bounded

by Σ , then there exist two balls B := {z ∈ R2n|

R2 |z|2 ≤ 1} and βB := {z ∈ R2n

|R2 |z|2 ≤ β} such that B ⊂ C ⊂ βB

for some β > 1, so

R

2β|z|2 ≤ H2(z) ≤

R

2|z|2, ∀z ∈ R2n . (3.2)

Now, for the s0 determined by condition (C) and the number r > 0 determined later, we choose an admissible functionφ by specifying φ′(0) and φ′(+∞). Precisely, we choose φ′(0) and φ′(+∞) such that

φ′(0) =

(β

R+ r

)π

T, φ′(+∞) =

(1

2R− s0

)π

T,

such admissible functions φ do exist. For example, we set

φ(t) =π

T

(1

2R− s0

)t +

π

T

[(β

R+ r

)−

(1

2R− s0

)]ln(t + 1), t ≥ 0.

In this section, we set

a =

(β

R+

r

2

)π

T∈ (0, φ′(0)), b = φ′(+∞) =

(1

2R− s0

)π

T. (3.3)

Lemma 3.4. Suppose that the function φ satisfies φ′(+∞) = ( 12R − s0)

πT , then the functional FK satisfies the strong

(PS) condition, i.e., for a sequence {z j } ⊂ W satisfying F ′

K (z j ) → 0, j → +∞ in W , there exists a subsequence of{z j } converging to z ∈ W and F ′

K (z) = 0.

Proof. We follow the ideas in [3]. By direct computation, we have

〈F ′

K (z), h〉 =

∫ T

0[(∇ H∗

K (−J z + K z) − z, −J h + K h)]dt, ∀h ∈ W. (3.4)

Suppose that {z j } ⊂ W satisfying F ′

K (z j ) → 0, j → +∞ in W , we will prove that there exists a subsequence of{z j } converging to some z ∈ W with F ′

K (z) = 0.From (3.4) we know that ΠK z j − ΠK ∇ H∗

K (−ΠK z j ) := η j → 0 in L2, where ΠK : W → L2 is defined byΠK z := J z − K z, which has a bounded inverse. So we have

z j − ∇ H∗

K (−ΠK z j ) := ε j → 0, j → +∞, in W. (3.5)

In view of (2.5), we have −J z j + K z j = ∇ HK (z j − ε j ) = φ′(H2(z j − ε j ))∇ H2(z j − ε j ) + K (z j − ε j ), that is,

φ′(H2(z j − ε j ))∇ H2(z j − ε j ) + J z j = K ε j , ∀ j. (3.6)


Claim: ‖z j‖C0,α ≤ c holds for some constant c > 0 and α ∈ (0, 1/2).Otherwise, there is a subsequence, for example, ‖z j‖C0,α → +∞, j → +∞. Set v j =

z j‖z j ‖C0,α

, w j = v j −ε j

‖z j ‖C0,α

for large j . Using (3.6) and the homogeneity of ∇ H2, we have

φ′(H2(z j − ε j ))∇ H2(w j ) + J v j = Kε j

‖z j‖C0,α

, ∀ j large. (3.7)

From (2.1) we know that φ′(H2(z j − ε j )) ∈ (φ′(+∞), φ′(0)], ∀t ∈ [0, T ], from the definition of w j we know that|w j | is bounded, then from (3.7) we get that ‖v j‖ is bounded via the homogeneity of degree 1 of ∇ H2, and from theinequality ‖z − z(0)‖ ≤ T ‖z‖, ∀z ∈ W , we get that ‖v j‖W is also bounded. Since the imbedding W ↪→ C0,α

is compact for α ∈ (0, 1/2), there exists a subsequence of v j , still denoted by v j , strongly convergent to v inC0,α , ‖v‖C0,α = 1, write v j → v ∈ C0,α, j → +∞. (v j → v in L2 is also true, since we have the imbeddingC0,α ↪→ L2.) From the fact that ‖v j‖ is bounded and φ′(H2(z j − ε j )) ∈ [φ′(0), φ′(+∞)], we get two weaklyconvergent (sub)sequence in L2: v j ⇀ v in L2 and ξ j := φ′(H2(z j − ε j )) ⇀ ξ in L2([0, T ], R+). So from (3.7) weknow that in the weak sense in L2, there holds

ξ(t)∇ H2(v(t)) = −J v(t), v 6= 0, ξ(t) ∈ [φ′(0), φ′(+∞)]. (3.8)

Since we have known that ∇ H2(z) = 0 ⇔ z = 0, from (3.8) we know that |v(t)| ≥ θ > 0 for all t ∈ [0, T ], then|v j (t)| ≥ θ/2 > 0 for all t ∈ [0, T ] and j ≥ N (N large enough), so we have that |z j (t)| = ‖z j‖C0,α |v j (t)| → +∞

uniformly in t ∈ [0, T ], as j → +∞. From the fact that ε j → 0 inW and the fact thatW is imbedded in C0, we knowthat ε j → 0 uniformly on [0, T ] as j → ∞, so we have |ε j (t)| ≤ 1/2 for j large enough and |z j (t) − ε j (t)| → +∞

on [0, T ]. Furthermore, using the homogeneity of degree 1 of ∇ H2 again, we have |∇ H2(z j − ε j )| ≤ C ′|z j − ε j |,

then |∇ H2(w j )| ≤ C ′

∣∣∣v j −ε j

‖z j ‖C0,α

∣∣∣ ≤ C ′′(1 + |v|) for j large enough. From the Lebesgue’s domain convergence

theorem, we know that φ′(H2(z j − ε j ))∇ H2(w j ) → φ′(+∞)∇ H2(v) in L2. Let j → +∞ in (3.7) then we get

−J v = φ′(+∞)∇ H2(v) =

(1

2R− s

)π

TJ∇ H2(v), ‖v‖C0,α = 1, v ∈ W, (3.9)

that is, we get a nontrivial solution of (3.1), which is a contradiction under the condition (C).From the above claim: ‖z j‖C0,α ≤ c we know that there exist subsequence of z j , still denoted by z j , such that

z j → z in C0([0, T ], R2n). From (3.6), we get z j = J [φ′(H2(z j − ε j ))∇ H2(z j − ε j ) − K ε j ] → Jφ′(H2(z)) in L2.So z j → z in W and z = Jφ′(H2(z))∇ H2(z), that is F ′

K (z) = 0. �

Proposition 3.5. There exist two subspaces V1 and V2 of W with the dimensions m and p respectively, such that

(i) supx∈V1∩SρFK (x) < 0, where Sρ := Sρ(0) is a sphere in V1 with the radius ρ,

(ii) infx∈V ⊥

2FK (x) > −∞,

(iii) m − p = n(1 + [rβ2R ]) and m = n(1 + [

K Tπ

]), where for every real number χ , [χ ] is defined by [χ ] = max{k ∈

Z | k < χ}.

Proof. The proof is divided into three steps according to Lemma 3.3.Step 1. From the fact that limt→0+

φ(t)t = φ′(0) > a we know that there exists small δ = δ(a) such that

φ(t) ≥ at, for |t | ≤ δ. (3.10)

Using (3.10) and (3.2), we get

HK (z) ≥ aH2(z) +K

2|z|2 ≥

12

(K +

a R

β

)|z|2, ∀|z| ≤ δ.

From the definition of the Fenchel dual we have

H∗

K (z) ≤1

2(

K +a Rβ

) |z|2, ∀|z| ≤ δ. (3.11)


Define

BK (z) =12

∫ T

0

[(K +

a R

β

)−1

| − J z + K z|2 − (−J z + K z, z)

]dt, ∀z ∈ W. (3.12)

Using (2.10) and (2.11), we have

BK (z) =T

2

∑k∈Z

n∑j=1

(K +

a R

β

)−1 (K +

kπ

T

) (kπ

T−

a R

β

)z2

k j . (3.13)

Define the subspace V1 in W by

V1 = span{

ek j

∣∣∣∣− K T

π< k <

a RT

βπ, k ∈ Z, j = 1, . . . , n

}.

From (3.11)–(3.13) and the fact that all the norms in finite-dimensional space are equivalent, we know that there existssmall ρ = ρ(δ) such that

FK |V1∩Sρ ≤ BK |V1∩Sρ < 0.

Step 2. From the fact that limt→+∞φ(t)

t = b < 2b we know that there exists a large number N such that φ(t)t < 2b

for t > N , and φ(t) is bounded for t ∈ [0, N ], so there exists a constant C such that

φ(t) ≤ 2bt + C, for all t ≥ 0. (3.14)

Using (3.14) and (3.2), we get

HK (z) ≤ 2bH2(z) +K

2|z|2 + C ≤

12(K + 2bR)|z|2 + C, ∀z ∈ R2n,

then from the definition of the Fenchel dual we have

H∗

K (z) ≥1

2(K + 2bR)|z|2 − C, ∀z ∈ R2n . (3.15)

Define

AK (z) =12

∫ T

0

[(K + 2bR)−1

| − J z + K z|2 − (−J z + K z, z)]

dt, ∀z ∈ W. (3.16)

Using (2.10) and (2.11), we have

AK (z) =T

2

∑k∈Z

n∑j=1

(K + 2bR)−1(

K +kπ

T

) (kπ

T− 2bR

)z2

k j .

Define the subspace V2 in W by

V2 = span{

ek j

∣∣∣∣− K T

π< k <

2bRT

π, k ∈ Z, j = 1, . . . , n

}.

From the definition of the space V2 we know that AK (z) ≥ 0 for z ∈ V ⊥

2 . From (3.15) and (3.16) we get

FK (z) ≥ AK (z) − CT ≥ −CT, ∀z ∈ V ⊥

2 .

Step 3. Since b := φ′(+∞) = ( 12R − s0)

πT and s0 ∈ (0, 1

4R ), we know that 2bRTπ

∈ (0, 1). Since a = (βR +

r2 )π

T ,we know that a RT

βπ= 1 +

r R2β

. So we have

m − p = n]

{k ∈ Z

∣∣∣∣2bRT

π< k <

a RT

βπ

}= n

(1 +

[r R

2β

]). �


Proof of Theorem 1.2. By Remark 1.4, it is sufficient to prove Theorem 1.2 for the standard Lagrangian subspace L0in R2n . Now, we prove Theorem 1.2 for L = L0 via Z2-index indirectly under the condition (C).

In fact, we suppose that there are finitely many L0-Lagrangian orbits, without loss of generality, we suppose thatthere are exactly µ pairs of distinct L0-Lagrangian orbits.

Step 1. We will prove that we can find n(µ + 1) pairs of distinct critical points of FK .The even functional FK satisfies the (PS) condition in view of Lemma 3.4. Now choose r > 0 such that[

r R

2β

]= µ. (3.17)

Then from Proposition 3.5, Lemma 3.3 and the choice of r in (3.17), we know that the even functional FK has at leastn(1 + [

r R2β

]) = n(µ + 1) pairs of distinct critical points

zm+1, zm+2, . . . , zm+n(µ+1), −zm+1, −zm+2, . . . ,−zm+n(µ+1),

which correspond to the critical value

cm+1 ≤ cm+2 ≤ · · · ≤ cm+n(µ+1) < 0

defined by

ck = infγ (A)≥k

supx∈A

FK (x), k = m + 1, m + 2, . . . , m + n(µ + 1),

where m := n([ K Tπ

] + 1). Since all the above critical values are negative, in view of Proposition 2.2, we know thatzi , −zi , i = m + 1, m + 2, . . . , m + n(µ + 1) are nontrivial solutions of (2.7). Define the projection orbits on Σ by

zi (t) = ρ−

12

i zi

(φ′(1)

φ′(ρi )t

), i = m + 1, m + 2, . . . , m + n(µ + 1),

where ρi = H2(zi ) > 0. Then zi , i = m + 1, m + 2, . . . , m + n(µ + 1) are nontrivial solutions ofz(t) = J∇ H(z) = Jφ′(H2(z))∇ H2(z),H2(z(t)) = 1, t ∈ [0, T ],

z(0), z

(φ′(ρi )

φ′(1)T

)∈ L0,

that is, zi , −zi , i = m + 1, m + 2, . . . , m + n(µ + 1) are L0-Lagrangian orbits on Σ .Step 2. We prove that the above n(µ + 1) pairs of critical points of FK are projected on Σ to be n(µ + 1) pairs of

distinct L0-Lagrangian orbits on Σ .

Case 1. If ρi 6= ρ j , i, j ∈ {m+1, m+2, . . . , m+n(µ+1)}, from the monotone of φ′(t) we know that φ′(ρi )φ′(1)

6=φ′(ρ j )

φ′(1).

So zi and z j are two distinct L0-Lagrangian orbits which takes different times to return to L0.Case 2. If ρi = ρ j , i 6= j ∈ {m + 1, m + 2, . . . , m + n(µ + 1)}, then zi and z j are two distinct solutions of (2.7)

with the same energy. So zi and z j are two distinct L0-Lagrangian orbits on Σ which takes the same time to return toL0.

From above discussion, we know that no matter which case happens, there exist at least n(µ + 1) > µ pairs ofdistinct Lagrangian orbits on Σ , which contradicts our assumption that there are exactly µ pairs of distinct Lagrangianorbits on Σ . The proof is completed. �

Acknowledgements

Part of this work was done while CL was visiting the Chern Institute of Mathematics. He would like to thank theChern Institute of Mathematics for the hospitality. His work was supported by the NSF special funds (10531050) andinnovation group funds, and FANEDD.

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