At least three solutions for the Hamiltonian system and reduction method

R E S E A R C H

Open Access

At least three solutions for the Hamiltonian

system and reduction method

Tacksun Jung

1

_{and Q-Heung Choi}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics}

Education, Inha University, Incheon, 402-751, Korea

Full list of author information is available at the end of the article

Abstract

We investigate the multiplicity of solutions for the Hamiltonian system with some asymptotically linear conditions. We get a theorem which shows the existence of at least three 2

π

-periodic solutions for the asymptotically linear Hamiltonian system. We obtain this result by the variational reduction method which reduces the infinite dimensional problem to the finite dimensional one. We also use the critical point theory and the variational method.

MSC: 35A15; 37K05

Keywords: Hamiltonian system; asymptotical linearity; variational reduction method; critical point theory; variational method; (P.S.)ccondition

1 Introduction and statement of the main result

LetG(t,z(t)) be aC _{function defined onR}_×Rn_{which is π-periodic with respect to} the first variablet. In this paper we investigate the number of π-periodic solutions of the following Hamiltonian system:

˙

p(t) = –Gq

t,p(t),q(t),

˙

q(t) =Gp

t,p(t),q(t),

(.)

wherep,q∈Rn_{,z= (p,q). LetJbe the standard symplectic structure onR}n_,i.e.,

J=

 –In In 

,

whereInis then×nidentity matrix. Then (.) can be rewritten as –Jz˙=Gz

t,z(t), (.)

wherez˙=dz_dt andGzis the gradient ofG. We assume thatG∈C(R×Rn,R) satisfies the following asymptotically linear conditions:

(G) G(t,z(t)) =o(|z|_{)as|z| →,G(t,θ) = ,G}

z(t,θ) =θ, whereθ= (, . . . , ).

(G) There exist constantsα,β(without loss of generality, we may assumeα,β∈/Z) such that

αI≤d_zG(t,z)≤βI ∀(t,z)∈R×Rn.

(G) Letjbe an integer within[α,β]such that

j–  <α<dzG(t, ) =_|zlim_|→

Gz(t,z)·z

|z| <j.

(G) lim_|z|→∞Gz_|(_zt_|,z)·z exists and there existsj=j+ which satisfies

j<dzG(t,∞) =_|lim z|→∞

Gz(t,z)·z

|z| <β<j. (G) Gisπ-periodic with respect tot.

We are looking for the weak solutions of (.) with conditions (G)-(G). The π-periodic weak solutionz= (p,q)∈Eof (.) satisfies

π



˙

z–JGz

t,z(t)·Jw dt=  for allw∈E,

i.e., π



˙

p+Gq

t,z(t)·ψ–q˙–Gp

t,z(t)·φdt=  for allζ= (φ,ψ)∈E,

whereEis introduced in Section . By Lemma . in Section , the weak solutions of (.) coincide with the critical points of the functional

f(z) =  

π



(–J˙z)·z dt– π



Gt,z(t)dt= π



pq dt˙ – π



Gt,z(t)dt. (.)

Several authors [–] considered the multiplicity of solutions for the Hamiltonian system. Chang proved in [] that ifG∈C_(R_×Rn_,R_{) satisfies conditions (G), (G) and the}

following additional conditions:

(G) Letj,j+ ,. . ., andjbe all integers within[α,β](without loss of generality, we may assumeα,β∈/Z) such thatj–  <α<j<j<β<j+  =j. Suppose that there exist

γ > andτ > such thatj<γ <βand

G(t,z)≥  γ z



L–τ ∀(t,z)∈R×Rn.

(G) Gz(t,θ) =θandj∈[j,j)∩Zsuch that

jI<d_zG(t,θ) < (j+ )I ∀t∈R,

then (.) has at least two nontrivial π-periodic weak solutions. Jung and Choi proved in [] that ifGsatisfies the following conditions:

(G) G:Rn_→RisC_{withG(θ) = .} (G) There existsh∈Nsuch that

h<lim inf

|z|→∞

G(z)·z

(G) There existsm∈Nsuch that

h+ m<lim inf

|z|→

G(z)·z

|z| <h+ m+  or

h– m–  <lim sup

|z|→ G(z)·z

|z| <h– m.

(G) There exists an integersuch that≤G_|(z)·z

z| ≤+ ,

then (.) has at leastmweak solutions, which are geometrically distinct and nonconstant. Our main result is the following:

Theorem . Assume that G satisfies conditions(G)-(G).Then system(.)has at least threeπ-periodic solutions.

Theorem . will be proved by the finite dimensional reduction method, the critical point theory and the variational method for the perturbed operatorA. The finite dimensional reduction method combined with the critical point theory and the variational method reduces the critical point results of the functionalI(z) on the infinite dimensional space to those of the corresponding functional˜I(v) on the finite dimensional subspace.

The outline of this paper is organized as follows. In Section , we introduce the Hilbert normed spaceE, show that the corresponding functionalI(z) of (.) is inC(E,R),Fréchet

differentiable and prove the reduction lemma for the perturbed operatorA. In Section , we show that the reduced functional –˜I(v) satisfies (P.S.)ccondition andv=  is the strict local point of minimum of˜I(v) and prove Theorem . by the shape of graph of the reduced functional.

2 The perturbed operatorA

LetL([, π],Rn) denote the set of n-tuples of the square integrable π-periodic func-tions and choosez∈L_{([, π],R}n_{). Then it has a Fourier expansionz(t) =} k=+∞

k=–∞akeikt, withak=_π

π

 z(t)e–iktdt∈Cn,a–k=a¯kand k∈Z|ak|<∞. Let A:z(t)→–J˙z(t)

with the domain

D(A) =z(t)∈H[, π],Rn|z() =z(π) =

z(t)∈L[, π],Rn k∈Z

+|k||ak|< +∞

,

where is a positive small number. Then Ais a self-adjoint operator. Let{Mλ} be the spectral resolution ofA, and let

P=

β

α

dMλ, P+=

+∞

β

dMλ, P–=

α

Let

L=PL

[, π],Rn, L+=P+L

[, π],Rn, L–=P–L

[, π],Rn.

For eachu∈L_{([, π],R}n_{), we have the composition}

u=u+u++u–,

whereu∈L,u+∈L+,u–∈L–. According toA, there exists a small number>  such

that –∈/σ(A). Let us define the spaceEas follows:

E=D|A|₌

z∈L[, π],Rn k∈Z

+|k||ak|<∞

with the scalar product

(z,w)E=(z,w)L+

|A|_z,|_A|_w

L

and the norm

z = (z,z)  

E =

k∈Z

+|k||ak| 

 .

The spaceEendowed with this norm is a real Hilbert space continuously embedded in

L_{([, π],R}n_{). The scalar product inL}_{naturally extends as the duality pairing between}

E andE=W–,_{([, π],R}n_{). We note that the operator (+}|_A|₎–_{is a compact linear} operator fromL_{([, π],R}n_{) toEsuch that}

+|A|–w,z_E= (w,z)_L.

Let

A=I+A.

Let

E=|A|–

 _L

, E+=|A|–

 _L

+, E–=|A|–

 _L

–.

ThenE=E⊕E+⊕E–and forz∈E,zhas the decompositionz=z+z++z–∈E, where

z=|A|–

 _u

, z+=|A|–

 _u

+, z–=|A|–

 _u

–.

Thus we have

and thatE,E+,E–are isomorphic toL,L+,L–, respectively. Let us define the functional

f(u) onL_{as follows:}

f(u) =  

u+ + M+u – M–u – u– 

–ψ(z), (.)

whereM+=

∞

 dMλ,M–=



–∞dMλandψ(z) =ψ(z) +_ z(t) _L,ψ(z) =

π

 G(t,z(t))dt.

Let

F(z) =Gz

t,z(t).

ByG∈C_{and (G),ψ(z) =}π

 G(t,z(t))∈C(E,R). Let

F(z) =I+F(z) =I+Gz

t,z(t).

The system (.) is equal to

A(z) =F(z). (.)

The Euler equation of the functionalf(u) is the system

u+=|A|–



_{P+F(z), (.)}

u–= –|A|–

 _P

–F(z), (.)

M+u=|A|–



_{M+PF(z), M–u= –}|_A|–_{M–PF(z). (.)} Thusz=z+z++z–is a solution of (.) if and only ifu=u+u++u–is a critical point

off. System (.)-(.) is reduced to

Az+=P+F(z+z++z–) or z+= (A)–P+F(z+z++z–), (.)

Az–=P–F(z+z++z–) or z–= (A)–P–F(z+z++z–), (.)

AM+z=M+PF(z+z++z–), AM–z=M–PF(z+z++z–). (.)

By (G),

F(u) –F(v)_L≤(+β) u–v L ∀u,v∈L. (.)

By (G), there exists aγ >β+such that

A– |L+⊕L–≤  γ.

We note that

f(u) =fu(z+z–+z+)

Let us set

I(z+z++z–) =f

u(z+z–+z+)

.

Now we will prove a reduction lemma which reduces the problem on the infinite dimen-sional spaceEto that of the finite dimensional subspace.

Letz∈Ebe fixed and consider the functionh:E–×E+→Rdefined by

h(z–,z+) =I(z+z–+z+).

The functionhhas continuous partialFréchetderivativesDhandDhwith respect to its

first and second variables given by

Dih(z–,z+)(yi) =DI(z+z–+z+)(yi) (.)

fory∈E–andy∈E+,i= , . Letv=z.

Lemma . Assume that G satisfies the conditions(G)-(G).

(i) For givenv∈E,there exists a uniquez–+z+∈C(E,E–⊕E+)satisfying the equation

A(z–+z+) = (P–+P+)F(v+z–+z+). (.)

(ii) There existsm< such that ifz–andy–are inE–andz+∈E+,then

Dh(z–,z+) –Dh(y–,z+)

(z––y–)≤m z––y– . (.)

(iii) There existsm> such that ifz+andy+are inE+andz–∈E–,then

Dh(z–,z+) –Dh(z–,y+)

(z+–y+)≥m z+–y+ . (.)

(iv) For givenv∈E,if we put the unique solutionz–(v) +z+(v)of(.)as

z–(v) +z+(v) =θ(v),thenθ(v)is continuous onEand satisfies a uniform Lipschitz condition inEwith respect toLnorm(also norm · E)and

|A|



z_–(v)∈C(E_,E_–⊕E₊),|A|z₊(v)∈C(E_,E_–⊕E₊).Moreover,

DIv+θ(v)(w) =  for allw∈E–⊕E+.

(v) IfI˜:E→Ris defined by

˜

I(v) =Iv+θ(v)=Iv+z–(v) +z+(v)

,

thenI˜has a continuousFréchetderivativeD˜Iwith respect tov,and

(vi) v∈Eis a critical point of˜Iif and only ifv+θ(v) =v+z–(v) +z+(v)is a critical point ofI.

Proof (i) Letδ=α+_β +. IfFδ

(ψ) =F(ψ) –δ, then equation (.) is equivalent to the equation

z–+z+= (A–δ)–(P–+P+)Fδ(v+z–+z+). (.)

The operator (A–δ)–(P–+P+) is a self-adjoint, compact and linear map from (P–+P+)L

into itself and its norm is (min{|j–δ|,|j–  –δ|})–. We note that

Fδ(ψ) –Fδ(ψ)_L≤

max|α–δ|,|β–δ|+ ψ–ψ L =

β–α

 +

ψ–ψ L.

We claim that the right-hand side of (.) is a Lipschitz mapping of (P–+P+)Linto itself

with a Lipschitz constantr< . In fact, letvbe a fixed element inEandw=v+z–+z+,

y=v+w–+w+be any elements inE. Then we have

(A–δ)–(P–+P+)Fδ(v+z–+z+) – (A–δ)–(P–+P+)Fδ(v+w–+w+)_E

=|A–δ|–



_(P–+P+)Fδ

(w) –F δ (y)_L

≤|A–δ|–



_(P–+P+)Fδ

(w) –F δ (y)_L

≤max|α–δ|,|β–δ|+|A–δ|–



_{(P–+P+)(z–+z+) – (w–+w+)}

L.

Since the operator norm of|A–δ|–



_{(P–+P+) is less than or equal to}√ 

min{|j–δ|,|j––δ|}+ and

z– L=|A|–

 _u–

L≤



min{|j–δ|,|j–  –δ|}+

u– L

= 

min{|j–δ|,|j–  –δ|}+

z– E,

z+ L=|A|–

 _u

+_L≤



min{|j–δ|,|j–  –δ|}+

u+ L

= 

min{|j–δ|,|j–  –δ|}+

z+ E,

we have

(A–δ)–(P–+P+)Fδ(w) – (A–δ)–(P–+P+)Fδ(y)_E

≤ max{|α–δ|,|β–δ|}+

min{|j–δ|,|j–  –δ|}+

(z–+z+) – (w–+w+)_E–⊕E+

since min{|j–δ|,|j–  –δ|}+>max{|α–δ|,|β–δ|}+. Therefore, by the implicit

function theorem, for givenv∈E, there exists a unique solutionz–(v) +z+(v)∈E–⊕E+

which satisfies (.). (ii) For allz–∈E–,

z– E≤(j– ) w _L. (.)

For allz+∈E+,

z+ E≥(j+ ) w _L₍₎. (.)

Ifv∈E,z–andy–are inE–,z+∈E+andz=v+z–+z+, then

Dh(z–,z+) –Dh(y–,z+)

(z––y–)

= π



A(z––y–)(z––y–) –

G_z(v+z–+z+) –Gz(v+y–+z+)

(z––y–)

dt.

Since (G

z(ξ) –Gz(ξ))(ξ–ξ) > (α+)(ξ–ξ)and (.) holds, we see that ifz–andy–

are inE–andz+∈E+, then

Dh(z–,z+) –Dh(y–,z+)

(z––y–)≤m z––y– ,

wherem=  –_jα_–< .

(iii) Similarly, using the fact that (G

z(ξ) –Gz(ξ))(ξ–ξ) < (β+)(ξ–ξ)and (.)

holds, we see that ifz+andy+are inE+andz–∈E–, then

Dh(z–,z+) –Dh(z–,y+)

(z+–y+)≥m z+–y+ ,

wherem=  –_jβ_+> .

(iv) If θ(v) denotes the unique (z– +z+)(v)∈E– ⊕E+ which solves (.), then θ ∈

C(E,E). In fact, ifv,v∈E, andp=θ(v),p=θ(v), then we have

p–p E =(A)–(P–+P+)

F(v+p) –F

v+p_E

≤C(v+p) –

v+p_E

≤Cv–v– (p–p)_E.

Thus we have

p–p E≤ C

 –Cv–v

E.

Thus θ is continuous. Since F ∈C(E,E), θ ∈C(E,E). Since dimL is finite and all

topologies onLare equivalent, we have

|A|



Letv∈E. Ifq∈E–⊕E+, then from (.) we have π  A

θ(v)·q– (P–+P+)F

v+θ(v)·qdt= .

Since_πAv·q= , we have

DIv+θ(v)(w) = π



A

v+θ(v)·q– (P–+P+)F

v+θ(v)·qdt= 

for allq∈E–⊕E+.

(v) Since the functionalI has a continuousFréchet derivativeDI, ˜Ihas a continuous

FréchetderivativeDI˜with respect tov.

(vi) Suppose that there existsv∈Esuch thatDI˜(v) = . FromDI˜(v)(h) =DI(v+θ(v))(h)

for allv,h∈E,DI(v+θ(v))(h) =  for allh∈E. SinceDI(v+θ(v))(w) for allw∈E–⊕E+, it

follows thatDI(v+θ(v)) = . Thusv+θ(v) is a solution of (.). Conversely ifuis a solution

of (.) andv=Pu, thenD˜I(v) = .

3 Proof of Theorem 1.1

Lemma . Assume that G satisfies the conditions(G)-(G).Then–˜I(v)is bounded below and satisfies(P.S.)condition.

Proof Letv∈E. By the finite dimensional reduction,

˜

I(v) =  

A

v+θ(v),v+θ(v)– π



Gt,v(t) +θv(t)dt,

whereθ(v) =θ–(v) +θ+(v),v∈E,θ–(v)∈E–,θ+(v)∈E+,G(t,v(t) +θ(v(t))) =G(t,v(t) +

θ(v(t))) +(v(t) +θ(v(t))). Letw=v+θ–(v). Then we have

˜

I(v) =  

A(w),w

– π



G_t,w(t)dt

+   A

θ(v),θ(v)–A(w),w

– π



Gt,θv(t)–Gt,w(t)dt

.

Moreover, we have

 

A

θ(v),θ(v)–A(w),w

– π



G_{t,θv(t)–Gt,w(t)dt}

= – π



G_zt,sθ+

v(t)–w(t),θ+

v(t)ds+ 

A

θ(v),θ+(v)

= π  π 

d_zGt,sθ+

v(t)+w(t)θ+

v(t),θ+

v(t)s ds dt

– 



A

θ+(v)

,θ+(v)

By (G), we have chosen a numberγ such thatj<γ <dzG(t,∞) <β. Thus we have

˜

I(v)≤  

A(w),w

– π



G_t,w(t)dt

≤ 

(j–γ) w



L+C→–∞ as v E→ ∞.

Thus –I˜(v) is bounded from below and satisfies (P.S.) condition.

Lemma . Assume that G satisfies conditions(G)-(G).Then v= is a strict local point of minimum of˜I(v)with˜I() = .

Proof

˜

I(v) =Iv+θ(v)

= 

A

v+θ(v),v+θ(v)– π



G_{t,v(t) +θv(t)dt}

= 

A(v),v

+C,

where

C= 

A

θ(v),θ(v)– π



Gt,θv(t)dt

– π



G_{t,v(t) +θv(t)–Gt,θv(t)dt}

=I˜() – π



Gt,v(t) +θv(t)–Gt,θv(t)dt,

lim

|v|→˜I(v) –I˜() =

 

A(v),v

– lim

|v|→

π



Gt,v(t) +θv(t)–Gt,θv(t)dt

= 

A(v),v

– lim

|v|→ π



G z

t,sv(t) +θv(t)v(t)dt.

Thus we have

lim

|v|→˜I(v) –I˜() =

 

j–dzG(t, )

v L> .

Thusv=  is a strict local point of minimum ofI˜(v). Sinceθ() = ,˜I() = .

Proof of Theorem . By Lemma .(v),˜I(v) is continuous andFréchetdifferentiable in

E. By Lemma .,˜I(v) is bounded above, satisfies the (P.S.) condition andI˜(v)→–∞as

v E→ ∞. By Lemma .,v=  is a strict local point of minimum of˜I(v) with a critical valueI˜() = . We note thatmaxv∈EI˜(v) >  is another critical value of˜I. By the shape of the graph of the functional˜Ion the one-dimensional subspaceE, there exists the third

critical point of˜I(v). Thus (.) has at least three solutions, one of which is a trivial solution

Competing interests

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Authors’ contributions

The authors did not provide this information.

Author details

1_{Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea.}2_{Department of Mathematics}

Education, Inha University, Incheon, 402-751, Korea.

Acknowledgements

The authors appreciate very much the referees for their kind corrections. This work (Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2011-0026920).

Received: 20 August 2012 Accepted: 9 February 2013 Published: 5 March 2013

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Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986)

doi:10.1186/1029-242X-2013-91