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Volume 2008, Article ID 293987, 17 pagesdoi:10.1155/2008/293987 Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System Tacksun Jung 1 and Q-Heung Choi 2 Corres

Volume 2008, Article ID 293987, 17 pages

doi:10.1155/2008/293987

Research Article

Existence of Four Solutions of Some Nonlinear

Hamiltonian System

Tacksun Jung 1 and Q-Heung Choi 2

Correspondence should be addressed to Q-Heung Choi, qheung@inha.ac.kr

Received 25 August 2007; Accepted 3 December 2007

Recommended by Kanishka Perera

We show the existence of four 2π-periodic solutions of the nonlinear Hamiltonian system with some

conditions We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category.

Copyright q 2008 T Jung and Q.-H Choi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction and statements of main results

Let Ht, z be a C2function defined on R1× R 2n which is 2π-periodic with respect to the first variable t.In this paper, we investigate the number of 2π-periodic nontrivial solutions of the

following nonlinear Hamiltonian system

˙z  J

where z : R → R 2n , ˙z  dz/dt,

J 



0 −I n

I n 0



I n is the identity matrix on R n , H : R1× R 2n → R, and H z is the gradient of H Let z  p, q,

p  z1, , z n , q  z n1 , , z 2n  ∈ R n Then 1.1 can be rewritten as

˙p  −H q t, p, q,

We assume that H ∈ C2R1× R 2n , R1 satisfies the following conditions

H1 There exist constants α < β such that

αI ≤ d2z Ht, z ≤ βI ∀t, z ∈ R1× R 2n 1.4

H2 Let j1, j2  j1 1 and j3  j2 1 be integers and α, β be any numbers without loss of generality, we may assume α, β / ∈ Z such that j1− 1 < α < j1 < j2 < β < j2 1  j3

Suppose that there exist γ > 0 and τ > 0 such that j2< γ < β and

Ht, z ≥ 1

2γz2L2 − τ ∀t, z ∈ R1× R 2n 1.5

H3 Ht, 0  0, H z t, 0  0, and j ∈ j1, j2 ∩ Z such that

jI < d2z Ht, 0 < j  1I ∀t ∈ R1. 1.6

H4 H is 2π-periodic with respect to t.

We are looking for the weak solutions of 1.1 Let E  W 1/2,2 0, 2π, R 2n  The 2π-periodic weak solution z  p, q ∈ E of 1.3 satisfies

2π

0



˙p  H q t, zt· ψ −˙q − H p t, zt· φdt  0 ∀ζ  φ, ψ ∈ E 1.7 and coincides with the critical points of the induced functional

Iz 

2π

0

p ˙q dt −

2π

0

Ht, ztdt  Az −

2π

0

Ht, ztdt, 1.8

where Az  1/2 2π

0 ˙z · Jz dt.

Our main results are the following

Theorem 1.1 Assume that H satisfies conditions (H1)–(H4) Then there exists a number δ > 0 such

that for any α and β with j1− 1 < α < j1< j2< β < j2 δ < j2 1  j3, α > 0, system 1.1 has at least

four nontrivial 2π-periodic solutions.

Theorem 1.2 Assume that H satisfies conditions (H1)–(H4) Then there exists a number δ > 0 such

that for any α and β, and j1− 1 < α < j1< j2< β < j2 δ < j2 1  j3, β < 0,system 1.1 has at least

four nontrivial 2π-periodic solutions.

Chang proved in1 that, under conditions H1–H4, system 1.1 has at least two

non-trivial 2π-periodic solutions He proved this result by using the finite dimensional variational

reduction method He first investigate the critical points of the functional on the finite dimen-sional subspace and theP.S. condition of the reduced functional and find one critical point

of the mountain pass type He also found another critical point by the shape of graph of the reduced functional

For the proofs of Theorems1.1and1.2, we first separate the whole space E into the four mutually disjoint four subspaces X0, X1, X2, X3which are introduced inSection 3and then we investigate two pairs of sphere-torus variational linking inequalities of the reduced functional

I and ˇI into the two pairs of torus-sphere variational

links of

C and ˇ I and ˇI are strongly indefinite functinals, we use the notion of

theP.S.∗c condition and the limit relative category instead of the notion ofP.S. c condition and the relative category, which are the useful tools for the proofs of the main theorems We

respectively By the critical point theory induced from the limit relative category theory we

obtain two nontrivial 2π-periodic solutions in each subspace X1 and X2, so we obtain at least

four nontrivial 2π-periodic solutions of 1.1

InSection 2, we introduce some notations and some notions ofP.S.∗ccondition and the limit relative category and recall the critical point theory on the manifold with boundary We also prove some propositions InSection 3, we proveTheorem 1.1and inSection 4, we prove Theorem 1.2

2 Recall of the critical point theory induced from the limit relative category

Let E  W 1/2,2 0, 2π, R 2n  The scalar product in L2 naturally extends as the duality pairing

between E and E  W −1/2,2 0, 2π, R 2n  It is known that if z ∈ C∞R, R 2n  is 2π-periodic, then

it has a Fourier expansion zt  k∞

k−∞ a k e ikn with a k ∈ C 2n and a −k  a k : E is the closure of

such functions with respect to the norm

z 

k∈Z

1  |k||a k|2

1/2

Let us set the functional

Az  1

2

2π

0 ˙z · Jz dt 

2π

0 p ˙q dt, z  p, q ∈ E, p, q ∈ R n , 2.2

so that

Iz  Az −

2π

0

Ht, ztdt. 2.3

Let e1, , e 2n denote the usual bases in R 2nand set

E0 span e1, , e 2n



,

E span sin jte k − cos jte kn , cos jte k  sin jte kn | j ∈ N, 1 ≤ k ≤ n,

E− span sin jte k  cos jte kn , cos jte k − sin jte kn | j ∈ N, 1 ≤ k ≤ n.

2.4

Then E  E0⊕E⊕E−and E0, E, E−are the subspaces of E on which A is null, positive definite

and negative definite, and these spaces are orthogonal with respect to the bilinear form

Bz, ζ ≡

2π

0

p · ˙ψ  φ · ˙q dt 2.5

associated with A Here, z  p, q and ζ  φ, ψ If z ∈ Eand ζ ∈ E−, then the bilinear form is

zero and Az  ζ  Az  Aζ We also note that E0, E, and E−are mutually orthogonal in

L20, 2π, R 2n  Let Pbe the projection from E onto Eand P−the one from E onto E− Then

the norm in E is given by

z2z02 Az

− Az−

z02Pz2P−z2 2.6

which is equivalent to the usual one The space E with this norm is a Hilbert space.

We need the following facts which are proved in2

Proposition 2.1 For each s ∈ 1, ∞, E is compactly embedded in L s 0, 2π, R 2n  In particular,

there is an α s > 0 such that

for all z ∈ E.

Proposition 2.2 Assume that Ht, z ∈ C2R1× R 2n , R Then Iz is C1, that is, Iz is continuous and Fr´echet differentiable in E with Fr´echet derivative

DIzω 

2π

0



˙z − JH z t, z· Jω 

2π

0



˙p  H q t, z· ψ −˙q − H p t, z· φdt, 2.8

where z  p, q and ω  φ, ψ ∈ E Moreover, the functional z → 2π

0 Ht, zdt is C1 Proof For z, w ∈ E,

Iz  w − Iz − DIzw



122π

0

 ˙z ˙w · Jzw −

2π

0

Ht, zw− 1

2

2π

0

˙z · Jz

2π

0

Ht, z −

2π

0



˙z−J

H z t, z· Jw





122π

0



˙z · Jw  ˙ w · Jz  ˙ w · Jw

−

2π

0



Ht, z  w − Ht, z

−

2π

0



˙z − J

H z t, z· Jw.

2.9

We have



2π

0

Ht, z  w − Ht, z

 ≤2π

0

H z t, z · w  o|w|dt

  O|w|. 2.10

Thus, we have

Iz  w − Iz − DIzw   O|w|2. 2.11

Next, we prove that Iz is continuous For z, w ∈ E,

Iz  w − Iz 1

2

2π

0

 ˙z  ˙w · Jz  w −

2π

0

Ht, z  w − 1

2

2π

0

˙z · Jz 

2π

0

Ht, z





122π

0



˙z · Jw  ˙ w · Jz  ˙ w · Jw

−

2π

0



Ht, z  w − Ht, z

 O|w|.

2.12

Similarly, it is easily checked that I is C1

Now, we consider the critical point theory on the manifold with boundary induced from

the limit relative category Let E be a Hilbert space and X be the closure of an open subset of E such that X can be endowed with the structure of C2manifold with boundary Let f : W → R

be a C 1,1 functional, where W is an open set containing X The P.S.∗ccondition and the limit relative categorysee 3 are useful tools for the proof of the main theorem

LetE nn be a sequence of a closed finite dimensional subspace of E with the following assumptions: E n  E−

n ⊕ E

n where E n ⊂ E, E−n ⊂ E− for all n En and E n−are subspaces of E, dim E n < ∞, E n ⊂ E n1,

n∈N E n are dense in E Let X n  X ∩ E n , for any n, be the closure of

an open subset of E n and has the structure of a C2manifold with boundary in E n We assume

that for any n there exists a retraction r n : X → X n For a given B ⊂ E, we will write B n  B ∩E n

Let Y be a closed subspace of X.

Definition 2.3 Let B be a closed subset of X with Y ⊂ B Let cat X,Y B be the relative category

of B in X, Y  We define the limit relative category of B in X, Y , with respect to X nn, by

cat∗X,Y B  lim sup

n→∞

We set

Bi B ⊂ X | cat∗X,Y B ≥ i,

c i inf

B∈B i

sup

x∈B

We have the following multiplicity theoremfor the proof, see 4

Theorem 2.4 Let i ∈ N and assume that

1 c i < ∞,

2 supx∈Y fx < c i ,

3 the P.S.∗c condition with respect to X nn holds.

Then there exists a lower critical point x such that fx  c i If

then

catX x ∈ X | fx  c, grad−X fx  0

Now, we state the following multiplicity result for the proof, see 4, Theorem 4.6 which will be used in the proofs of our main theorems

Theorem 2.5 Let H be a Hilbert space and let H  X1 ⊕ X2⊕ X3, where X1, X2, X3are three closed subspaces of H with X1, X2of finite dimension For a given subspace X of H, let P X be the orthogonal projection from H onto X Set

C  x ∈ H |P X

and let f : W → R be a C 1,1 function defined on a neighborhood W of C Let 1 < ρ < R, R1 > 0 One defines

Δ  x1 x2| x1∈ X1, x2∈ X2, x1 ≤ R1, 1 ≤x2 ≤ R,

Σ  x1 x2| x1∈ X1, x2∈ X2, x1 ≤ R1, x2  1

∪ x1 x2| x1∈ X1, x2 ∈ X2, x1 ≤ R1, x2  R

∪ x1 x2| x1∈ X1, x2 ∈ X2, x1  R1, 1 ≤x2 ≤ R,

S  x ∈ X2⊕ X3| x  ρ,

B  x ∈ X2⊕ X3| x ≤ ρ.

2.18

Assume that

and that the P.S. c condition holds for f on C, with respect to the sequrnce C nn , for all c ∈ a, b, where

a  inf fS, b  sup fΔ. 2.20

Moreover, one assumes b < ∞ and f| X1⊕X3 has no critical points z in X1⊕ X3 with a ≤ fz ≤ b Then there exist two lower critical points z1, z2for f on C such that a ≤ fz i  ≤ b, i  1.2.

3 Proof of Theorem 1.1

We assume that 0 < α < β Let e1, , e 2n denote the usual bases in R 2nand set

X0≡ span sin jte k − cos jte kn , cos jte k  sin jte kn , sin jte k  cos jte kn ,

cos jte k − sin jte kn , e1, e2, , e 2n | j ≤ j1− 1, j ∈ N, 1 ≤ k ≤ n,

X1≡ span sin jte k − cos jte kn , cos jte k  sin jte kn | j  j1, 1 ≤ k ≤ n

,

X2≡ span sin jte k − cos jte kn , cos jte k  sin jte kn | j  j2, 1 ≤ k ≤ n

,

X3≡ span sin jte k − cos jte kn , cos jte k  sin jte kn | j ≥ j2 1  j3, j ∈ N, 1 ≤ k ≤ n

.

3.1

Then E is the topological direct sum of subspaces X0, X1, X2, and X3, where X1 and X2 are finite dimensional subspaces We also set

S1ρ  z ∈ X1| z  ρ,

S r1

X0⊕ X1



 z ∈ X0⊕ X1| z  r1

,

B r1

X0⊕ X1



 z ∈ X0⊕ X1| z ≤ r1

,

ΣR1

S1ρ, X2⊕ X3



 z  z1 z2 z3∈ X1⊕ X2⊕ X3| z1∈ S1ρ, z1 z2 z3  R1

,

ΔR1

S1ρ, X2⊕ X3



 z  z1 z2 z3∈ X1⊕ X2⊕ X3| z1∈ S1ρ, z1 z2 z3 ≤ R1

,

S2ρ  {z ∈ X2| z  ρ},

S r2

X0⊕ X1⊕ X2



 z ∈ X0⊕ X1⊕ X2| z  r2

,

B r2

X0⊕ X1⊕ X2



 z ∈ X0⊕ X1⊕ X2| z ≤ r2

,

ΣR2

S2ρ, X3



 z  z2 z3∈ X2⊕ X3| z2∈ S2ρ, z2 z3  R2

,

ΔR2

S2ρ, X3



 z  z2 z3∈ X2⊕ X3| z2∈ S2ρ, z2 z3 ≤ R2

.

3.2

We have the following two pairs of the sphere-torus variational linking inequalities

Lemma 3.1 first sphere-torus variational linking Assume that H satisfies the conditions (H1),

(H3), (H4), and the condition

H2 suppose that there exist γ > 0 and τ > 0 such that j1< γ < β and

Ht, z ≥ 1

Then there exist δ1 > 0, ρ > 0, r1 > 0, and R1 > 0 such that r1 < R1, and for any α and β with

j1− 1 < α < j1< β < j2 δ1< j2 1  j3and α > 0,

sup

z∈S r1 X0⊕X1 Iz < 0 < inf

z∈Σ R1 S1ρ,X2⊕X3 Iz,

inf

z∈Δ R1 S1ρ,X2⊕X3 Iz > −∞, sup

z∈B r1 X0⊕X1 Iz < ∞. 3.4

Proof Let z  z0 z1∈ X0⊕ X1 ByH2, we have

Iz  1

2

2π

0

˙z · Jz dt −

2π

0

Ht, ztdt

≤ 1

2z0 z12− γ

2z0 z12

L2 τ

≤ 1

2j1− γz0 z12

L2 τ

3.5

for some τ > 0 Since j1− γ < 0, there exists r1 > 0 such that if z0 z1 ∈ S r1X0 ⊕ X1,

then Iz < 0 Thus, sup z∈S

r1 X0⊕X1 Iz < 0 Moreover, if z ∈ B r1X0 ⊕ X1, then Iz ≤

1/2j1− γz0 z12

L2  τ < τ < ∞, so we have sup z∈B

r1 X0⊕X1 Iz < ∞ Next, we will show

that there exist δ1 > 0, ρ > 0 and R1 > 0 such that if j1− 1 < α < j1 < β < j2 δ1 < j2 1  j3, then infz∈Σ R1 S1ρ,X2⊕X3 Iz > 0 Let z  z1 z2 z3 ∈ X1⊕ X2⊕ X3with z1 ∈ S1ρ, z2 ∈ X2,

z3 ∈ X3, where ρ is a small number Let j1− 1 < α < j1 < β < j2 δ < j2 1  j3for some δ > 0 and α > 0 Then X1⊕ X2⊕ X3⊂ Eand P−z1 z2 z3  0 By H1, there exists d > 0 such that

Iz  1

2

2π

0 ˙z · Jz dt −

2π

0 Ht, ztdt

≥ 1

2Pz1 z2 z32−β

2Pz1 z2 z32

L2− d

≥ 1

2j1− βPz12

L21

2j2− βPz22

L2 1

2j3− βPz32

L2 − d

 1

2j1− βρ2− 1

2δPz22

L2 1

2j3− βPz32

L2− d.

3.6

Since j1− β < 0, j2− β > −δ, and j3− β > 0, there exist a small number δ1 > 0 and R1 > 0

with δ1 < δ and R1 > r1 such that if j1 − 1 < α < j1 < β < j2  δ1 < j2 1  j3 and

z ∈ Σ R1S1ρ, X2⊕ X3, then Iz > 0 Thus, we have inf z∈Σ R1 S1ρ,X2⊕X3 Iz > 0 Moreover,

if j1 − 1 < α < j1 < β < j2 δ1 < j2  1  j3 and z ∈ Δ R1S1ρ, X2 ⊕ X3, then we have

Iz > 1/2j1− βρ2− 1/2δ1Pz22

L2 − d > −∞ Thus, infΔR1 S1ρ,X2⊕X3 Iz > −∞ Thus,

we prove the lemma

Lemma 3.2 Let δ1 be the number introduced in Lemma 3.1 Then for any α and β with j1− 1 < α <

j1< β ≤ j2< j2 1  j3and α > 0, if u is a critical point for I| X0⊕X2⊕X3 , then Iu  0.

Proof We notice that fromLemma 3.1, for fixed u0 ∈ X0, the functional u23 → Iu0 u23 is

weakly convex in X2⊕ X3, while, for fixed u23 ∈ X2⊕ X3, the functional u0 → Iu0 u23 is

strictly concave in X0 Moreover, 0 is the critical point in X0⊕ X2 ⊕ X3 with I0  0 So if

u  u0 u23is another critical point for I| X0⊕X2⊕X3 , then we have

0 I0 ≤ Iu23 ≤ Iu0 u23 ≤ Iu0 ≤ I0  0. 3.7

So we have Iu  I0  0.

Let P X1 be the orthogonal projection from E onto X1and

C  z ∈ E |P X1z ≥ 1. 3.8

C n C ∩ E n Let us define afunctional

0⊕ X2⊕ X3} → E by

X1z

P X1z  P X0⊕X2⊕X3 z 



1−P X1z



We have

1

P X1z



P X1w −



P X1z

P X1z,w



P X1z

P X1z



C → R by

3.11

I ∈ C 1,1loc We note that if

is the critical point of I We also note that

grad−

Let us set



S r1 −1

S r1

X0⊕ X1



,



B r1 −1

B r1

X0⊕ X1



,



ΣR1 −1

ΣR1

S1ρ, X2⊕ X3



,



ΔR1 −1

ΔR1

S1ρ, X2⊕ X3



.

3.13

We note that S r1,  B r1, ΣR1, and ΔR1 have the same topological structure as S r1, B r1,ΣR1,

andΔR1, respectively

∗condition with respect to C nn for every real number

0 < inf

−1

S r1 X0⊕X1 

sup

−1

ΔR1 S1ρ,X2⊕X3 

3.14

Proof Let k nn be a sequence such that k n→ ∞, z nn be a sequence in C such that  z n ∈ C k n,

I z n −

C E knz n  → 0 Set z n  Ψz n  and hence z n ∈ E k n and

P E n ◦ P X1  P X1◦ P E n  P X1, we have

P E kn z n   P E knΨz n ∇−Iz n  Ψz n P E kn ∇−Iz n  −→ 0. 3.15

By3.9 and 3.10,

P E kn ∇−Iz n−→ 0 or

P X0⊕X2⊕X3 P E kn ∇−Iz n  −→ 0, P X1zn −→ 0. 3.16

In the first case, the claim follows from the limit Palais-Smale condition for−I In the second case, P X0⊕X2⊕X3 P E kn ∇−Iz n  → 0 We claim that z nnis bounded By contradiction, we sup-pose thatz n  → ∞ and set w n  z n /z n  Up to a subsequence w n  w0weakly for some

w0∈ X0⊕ X2⊕ X3 By the asymptotically linearity of ∇−Iz n we have



∇−Iz n

z n  ,w n







P X0⊕X2⊕X3 P E kn

∇−Iz n

z n  ,w n







∇−Iz n

z n2 , P X1zn



−→ 0. 3.17

We have



∇−Iz n

z n  ,w n



 2−Izn

z n2 

2π

0



−2Ht, z n

z n2 H z t, z n  · w n

z n



where z n  z n1, , z n2n Passing to the limit, we get

lim

n→∞

2π

0



2Ht, z n

z n2 − H z t, z n  · w n

z n



dt  0. 3.19

Since H and H z t, z n ·z nare bounded andz n  → ∞ in Ω, w0 0 On the other hand, we have



P X0⊕X2⊕X3 P E kn ∇−Iz n

z n  ,w n





2π

0



− ˙w n · Jw n



P X0⊕X2⊕X3 P E kn H z t, z n

z n



· w n



dt.

3.20 Moreover, we have



P X0⊕X2⊕X3 P E kn

∇−Iz n

z n  ,Pw n − P−w n



 −P X

2⊕X3Pw n2−P X0P−w n2−

2π

0

P X0⊕X2⊕X3 P E kn H z t, z n

z n ·Pw n − P−w n

dt.

3.21

Since w n converges to 0 weakly and H z t, z n  · Pw n − P−w n  is bounded, P X2⊕X3Pw n2

P X0P−w n2 → 0 Since P X1w n2 → 0, w n converges to 0 strongly, which is a contradiction Hence, z nn is bounded Up to a subsequence, we can suppose that z n converges to z0 for

some z0∈ X0⊕ X2⊕ X3 We claim that z n converges to z0strongly We have



P X0⊕X2⊕X3 P E kn ∇−Iz n , Pz n − P−z n



 −P X2⊕X3P E

kn Pz n2−P X0PE

kn P−z n2 P X0⊕X2⊕X3 P E kn

2π

0 H z t, z n ·Pz n − P−z n



.

3.22

ByH1 and the boundedness of H z t, z n Pz n − P−z n,

P X

2⊕X3PE kn Pz n2P X0PE

kn P−z n2−→ P X0⊕X2⊕X3 P E kn

2π

0

H z t, z ·Pz − P−z

. 3.23