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Volume 2008, Article ID 293987,17pages doi:10.1155/2008/293987

Research Article

Existence of Four Solutions of Some Nonlinear

Hamiltonian System

Tacksun Jung1and Q-Heung Choi2

1Department of Mathematics, Kunsan National University, Kunsan 573-701, South Korea 2Department of Mathematics Education, Inha University, Incheon 402-751, South Korea

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.

Copyrightq2008 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

LetHt, zbe aC2function defined onR1×R2nwhich is 2π-periodic with respect to the first variablet.In this paper, we investigate the number of 2π-periodic nontrivial solutions of the following nonlinear Hamiltonian system

˙

zJHzt, zt, 1.1

wherez:RR2n, ˙zdz/dt,

J

0 −In In 0

, 1.2

Inis the identity matrix onRn,H : RR2nR, andHzis the gradient ofH. Letz p, q, p z1, . . . , zn, q zn1, . . . , z2nRn.Then1.1can be rewritten as

˙

pHqt, p, q, ˙

qHpt, p, q. 1.3

(2)

H1There exist constantsα < βsuch that

αId2zHt, zβIt, zRR2n. 1.4

H2Letj1, j2 j11 andj3 j21 be integers andα,βbe any numberswithout loss of

generality, we may assume α, β /Zsuch that j1−1 < α < j1 < j2 < β < j21 j3.

Suppose that there existγ >0 andτ >0 such thatj2< γ < βand

Ht, z≥ 1 2γz

2

L2 −τt, zRR2n. 1.5

H3Ht,0 0,Hzt,0 0, andjj1, j2∩Zsuch that

jI < d2zHt,0<j1ItR1. 1.6

H4H is 2π-periodic with respect tot.

We are looking for the weak solutions of 1.1. Let E W1/2,20,, R2n. The 2π-periodic weak solutionz p, qEof1.3satisfies

2π

0

˙

pHqt, zt

·ψq˙−Hpt, zt

·φdt0 ∀ζ φ, ψE 1.7

and coincides with the critical points of the induced functional

Iz

2π

0

pq dt˙ −

2π

0

Ht, ztdtAz

2π

0

Ht, ztdt, 1.8

whereAz 1/2 02πz˙·Jz dt.

Our main results are the following.

Theorem 1.1. Assume thatH satisfies conditions (H1)–(H4). Then there exists a numberδ >0such that for anyαandβwithj1−1< α < j1< j2< β < j2δ < j21j3,α >0, system1.1has at least

four nontrivial2π-periodic solutions.

Theorem 1.2. Assume thatH satisfies conditions (H1)–(H4). Then there exists a numberδ >0such that for anyαandβ, andj1−1< α < j1< j2< β < j2δ < j21j3,β <0,system1.1has at least

four nontrivial2π-periodic solutions.

(3)

For the proofs of Theorems1.1and1.2, we first separate the whole spaceEinto the four mutually disjoint four subspacesX0,X1,X2,X3which are introduced inSection 3and then we

investigate two pairs of sphere-torus variational linking inequalities of the reduced functional

Iand ˇIofIon the submanifold with boundaryCand ˇC, respectively, and translate these two pairs of sphere-torus variational links ofIand ˇIinto the two pairs of torus-sphere variational links of−Iand−I, whereˇ Iand ˇIare the restricted functionals ofIto the manifold with bound-aryCand ˇC, respectively. SinceIand ˇIare 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 also investigate the limit relative category of torus intorus, boundary of torusonC and ˇC, respectively. By the critical point theory induced from the limit relative category theory we obtain two nontrivial 2π-periodic solutions in each subspaceX1 andX2, so we obtain at least

four nontrivial 2π-periodic solutions of1.1.

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

LetE W1/2,20,, R2n.The scalar product inL2 naturally extends as the duality pairing

betweenEandEW−1/2,20,2π, R2n. It is known that ifzCR, R2nis 2π-periodic, then it has a Fourier expansionzt kk−∞akeikn withakC2n andak ak:Eis the closure of such functions with respect to the norm

z

kZ

1|k||ak|2

1/2

. 2.1

Let us set the functional

Az 1 2

2π

0

˙

z·Jz dt

2π

0 p

˙

q dt, z p, qE, p, qRn, 2.2

so that

Iz Az

2π

0

Ht, ztdt. 2.3

Lete1, . . . , e2ndenote the usual bases inR2nand set

E0spane1, . . . , e2n

,

Espansinjtek−cosjtekn,cosjtek sinjtekn|jN, 1≤kn,

E−spansinjtek cosjtekn,cosjtek−sinjtekn|jN, 1≤kn.

(4)

ThenEE0EEandE0, E, Eare the subspaces ofEon whichAis 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 withA. Here,z p, qandζ φ, ψ.IfzEandζE−, then the bilinear form is zero andAzζ Az . We also note thatE0, E, andE−are mutually orthogonal in L20,2π, R2n. LetPbe the projection fromEontoEandP−the one fromEontoE−. Then the norm inEis given by

z2z02AzAzz02Pz2Pz2 2.6

which is equivalent to the usual one. The spaceEwith 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 Ls0,, R2n. In particular, there is anαs >0such that

zLsαsz 2.7

for allzE.

Proposition 2.2. Assume thatHt, zC2RR2n, R. ThenIzisC1, that is,Izis continuous and Fr´echet differentiable inEwith Fr´echet derivative

DIzω

2π

0

˙

zJHzt, z·

2π

0

˙

pHqt, z·ψq˙−Hpt, z·φdt, 2.8

wherez p, qandω φ, ψE.Moreover, the functionalz02πHt, zdtisC1.

Proof. Forz, wE,

IzwIzDIzw

1

2

2π

0

z˙w˙·Jzw

2π

0

Ht, zw− 1 2

2π

0

˙ z·Jz

2π

0

Ht, z

2π

0

˙

zJHzt, z·Jw

1

2

2π

0

˙

z·Jww˙ ·Jzw˙ ·Jw

2π

0

Ht, zwHt, z

2π

0

˙

zJHzt, z·Jw.

2.9

We have

2π

0

Ht, zwHt, z

2π

0

(5)

Thus, we have

IzwIzDIzwO|w|2

. 2.11

Next, we prove thatIzis continuous. Forz, wE,

IzwIz1 2

2π

0

z˙ w˙·Jzw

2π

0

Ht, zw−1 2

2π

0

˙ z·Jz

2π

0

Ht, z

1

2

2π

0

˙

z·Jww˙ ·Jzw˙ ·Jw

2π

0

Ht, zwHt, z

O|w|.

2.12

Similarly, it is easily checked thatIisC1.

Now, we consider the critical point theory on the manifold with boundary induced from the limit relative category. LetEbe a Hilbert space andXbe the closure of an open subset ofE such thatXcan be endowed with the structure ofC2manifold with boundary. Letf :WR be aC1,1functional, whereW is an open set containingX. TheP.S.ccondition and the limit relative categorysee3are useful tools for the proof of the main theorem.

LetEnnbe a sequence of a closed finite dimensional subspace ofEwith the following assumptions:En En−⊕Enwhere EnE,EnE− for allnEn andEn−are subspaces ofE, dimEn <∞,EnEn1,nNEnare dense inE. LetXn XEn, for anyn, be the closure of

an open subset ofEnand has the structure of aC2manifold with boundary inEn. We assume

that for anynthere exists a retractionrn:XXn. For a givenBE, we will writeBn BEn. LetY be a closed subspace ofX.

Definition 2.3. LetBbe a closed subset ofXwithYB. Let catX,YBbe the relative category ofBinX, Y. We define the limit relative category ofBinX, Y, with respect toXnn, by

cat∗X,YB lim sup

n→∞catXn,YnBn. 2.13

We set

Bi

BX|cat∗X,YBi,

ci inf B∈Bi

sup xB

fx. 2.14

We have the following multiplicity theoremfor the proof, see4.

Theorem 2.4. LetiNand assume that

1ci<,

2supxYfx< ci,

(6)

Then there exists a lower critical pointxsuch thatfx ci. If

cici1· · · cik−1c, 2.15

then

catX

xX|fx c, gradXfx 0≥k. 2.16 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. LetHbe a Hilbert space and letH X1⊕X2⊕X3, whereX1,X2,X3are three closed

subspaces ofH withX1,X2of finite dimension. For a given subspaceXofH, letPX be the orthogonal

projection fromHontoX. Set

CxH |PX2x≥1

, 2.17

and letf :WRbe aC1,1function defined on a neighborhoodW ofC. Let1 < ρ < R,R

1 >0. One

defines

Δ x1x2|x1∈X1, x2∈X2, x1≤R1, 1≤x2≤R

,

Σ x1x2|x1∈X1, x2∈X2, x1≤R1, x21

x1x2|x1∈X1, x2 ∈X2, x1≤R1, x2R

x1x2|x1∈X1, x2 ∈X2, x1R1, 1≤x2≤R

,

SxX2⊕X3|

,

BxX2⊕X3| xρ

.

2.18

Assume that

supfΣ<inffS 2.19

and that theP.S.c condition holds forf onC, with respect to the sequrnce Cnn, for allca, b, where

ainffS, b supfΔ. 2.20

Moreover, one assumes b <andf|X1⊕X3 has no critical pointszinX1⊕X3 withafzb.

Then there exist two lower critical pointsz1,z2forf onCsuch thatafzib,i1.2.

3. Proof ofTheorem 1.1

We assume that 0< α < β. Lete1, . . . , e2ndenote the usual bases inR2nand set

X0≡ span

sinjtek −cosjtekn,cosjtek sinjtekn,sinjtek cosjtekn,

cosjtek −sinjtekn, e1, e2, . . . , e2n|jj1−1, j ∈N, 1≤kn

,

X1≡ span

sinjtek −cosjtekn,cosjtek sinjtekn|jj1, 1≤kn

,

X2≡ span

sinjtek−cosjtekn,cosjtek sinjtekn |j j2, 1≤kn

,

X3≡ span

sinjtek −cosjtekn,cosjtek sinjtekn|jj21j3, jN, 1≤kn

.

(7)

Then E is the topological direct sum of subspacesX0, X1, X2, andX3, where X1 andX2 are

finite dimensional subspaces. We also set

S1ρ

zX1|

,

Sr1X0X1zX0X1| zr1,

Br1X0X1zX0X1| zr1,

ΣR1S1ρ, X2X3zz1z2z3X1X2X3|z1S1ρ, z1z2z3R1,

ΔR1S1ρ, X2X3zz1z2z3X1X2X3|z1S1ρ, z1z2z3R1,

S2ρ {zX2| },

Sr2X0X1X2zX0X1X2| zr2,

Br2X0X1X2zX0X1X2| zr2,

ΣR2S2ρ, X3zz2z3X2X3|z2S2ρ, z2z3R2,

ΔR2S2ρ, X3zz2z3X2X3|z2S2ρ, z2z3R2.

3.2

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

Lemma 3.1first sphere-torus variational linking. Assume thatH satisfies the conditions (H1), (H3), (H4), and the condition

H2 suppose that there existγ >0andτ >0such thatj1< γ < βand

Ht, z≥ 1 2γz

2

τt, zRR2n. 3.3

Then there existδ1 > 0,ρ >0,r1 > 0, andR1 > 0such thatr1 < R1, and for anyαandβwith

j1−1< α < j1< β < j2δ1< j21j3andα >0,

sup zSr1X0⊕X1

Iz<0< inf z∈ΣR1S1ρ,X2⊕X3

Iz,

inf z∈ΔR1S1ρ,X2⊕X3

Iz>−∞, sup zBr1X0⊕X1

Iz<. 3.4

Proof. Letzz0z1∈X0⊕X1. ByH2, we have

Iz 1 2

2π

0

˙

z·Jz dt

2π

0

Ht, ztdt

≤ 1

2z0z1

2 γ

2z0z1

2

L2τ ≤ 1

2j1−γz0z1

2

L2τ

(8)

for some τ > 0. Since j1−γ < 0, there exists r1 > 0 such that if z0z1 ∈ Sr1X0 ⊕X1,

then Iz < 0. Thus, supzS

r1X0⊕X1Iz < 0. Moreover, if zBr

1X0X1, then Iz

1/2j1−γz0z12L2 τ < τ <∞, so we have supzB

r1X0⊕X1Iz<∞. Next, we will show

that there existδ1 >0,ρ > 0 andR1 >0 such that ifj1−1 < α < j1 < β < j2δ1 < j21 j3,

then infz∈ΣR1S1ρ,X2⊕X3Iz>0. Letz z1z2z3 ∈X1⊕X2⊕X3withz1 ∈S1ρ,z2 ∈X2,

z3 ∈X3, whereρis a small number. Letj1−1< α < j1 < β < j2δ < j21 j3for someδ >0

andα >0. ThenX1⊕X2⊕X3⊂EandPz1z2z3 0. ByH1, there existsd >0 such that

Iz 1 2

2π

0

˙

z·Jz dt

2π

0 H

t, ztdt

≥ 1

2P z

1z2z32−β

2P z

1z2z32L2−d

≥ 1

2j1−βP z

12L2

1

2j2−βP z

22L2

1

2j3−βP z

32L2 −d

1

2j1−βρ

2 1

2δP z

22L2

1

2j3−βP z

32L2−d.

3.6

Since j1−β < 0,j2−β >δ, and j3−β > 0, there exist a small number δ1 > 0 andR1 > 0

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

z ∈ΣR1S1ρ, X2X3, thenIz >0. Thus, we have infzΣ

R1S1ρ,X2⊕X3Iz >0. Moreover,

if j1 −1 < α < j1 < β < j2δ1 < j2 1 j3 and z ∈ ΔR1S1ρ, X2X3, then we have Iz >1/2j1−βρ2−1/2δ1Pz22L2 −d > −∞. Thus, infΔ

R1S1ρ,X2⊕X3Iz> −∞. Thus,

we prove the lemma.

Lemma 3.2. Letδ1be the number introduced inLemma 3.1. Then for anyαandβwithj1−1 < α <

j1< βj2< j21j3andα >0, ifuis a critical point forI|X0⊕X2⊕X3, thenIu 0.

Proof. We notice that fromLemma 3.1, for fixedu0 ∈ X0, the functional u23 → Iu0u23is

weakly convex inX2⊕X3, while, for fixed u23 ∈ X2⊕X3, the functionalu0 → Iu0u23is

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

uu0u23is another critical point forI|X0⊕X2⊕X3, then we have

0I0≤Iu23≤Iu0u23≤Iu0≤I0 0. 3.7

So we haveIu I0 0.

LetPX1 be the orthogonal projection fromEontoX1and

CzE|PX1z≥1

. 3.8

Then C is the smooth manifold with boundary. Let Cn CEn. Let us define afunctional

Ψ:E\ {X0⊕X2⊕X3} →Eby

Ψz zPX1z

PX1z PX0⊕X2⊕X3z

1− 1 PX1z

(9)

We have

∇Ψ zw w− 1 PX1z

PX1w

PX1z

PX1z, w

PX1z

PX1z

. 3.10

Let us define the functionalI:CRby

I I◦Ψ. 3.11

ThenIC1loc,1. We note that ifzis the critical point ofIand lies in the interior ofC, then zΨ z is the critical point ofI. We also note that

grad

CIzPX0⊕X2⊕X3∇IΨ zz∂C. 3.12

Let us set

Sr1 Ψ−1Sr1X0X1,

Br1 Ψ−1Br1X0X1,

ΣR1 Ψ−1ΣR1S1ρ, X2X3,

ΔR1 Ψ−1ΔR1S1ρ, X2X3.

3.13

We note thatSr1,Br1,ΣR1,andΔR1 have the same topological structure asSr1,Br1,ΣR1, andΔR1, respectively.

Lemma 3.3.Isatisfies theP.S.ccondition with respect toCnnfor every real numbercsuch that

0< inf

z∈Ψ−1Sr1X0⊕X1

Izc≤ sup

z∈Ψ−1ΔR1S1ρ,X2⊕X3

Iz. 3.14

Proof. Letknnbe a sequence such thatkn→ ∞,znnbe a sequence inCsuch thatznCkn,

for alln,Izncand grad−CI|Eknzn → 0. Setzn Ψzn and henceznEknand

Iznc. We first consider the case in which zn/X0⊕X2⊕X3, for alln. Since fornlarge

PEnPX1 PX1◦PEn PX1, we have

PEkn∇−Izn PEknΨ

znIzn ΨznPE

kn∇−Izn−→0. 3.15

By3.9and3.10,

PEkn∇−Izn−→0 or

PX0⊕X2⊕X3PEkn∇−Izn−→0, PX1zn −→0.

3.16

(10)

w0∈X0⊕X2⊕X3. By the asymptotically linearity of∇−Iznwe have

∇−Izn

zn , wn

PX0⊕X2⊕X3PEkn

∇−Izn

zn , wn

∇−Izn

zn2 , PX1zn

−→0. 3.17

We have

∇−Izn

zn , wn

2Izn

zn2

2π

0

−2Ht, zn

zn2

Hzt, zn·wn

zn

dt, 3.18

wherezn zn1, . . . ,zn2n. Passing to the limit, we get

lim n→∞

2π

0

2Ht, zn

zn2 −

Hzt, zn·wn

zn

dt0. 3.19

SinceHandHzt, zn·znare bounded andzn → ∞inΩ,w00. On the other hand, we have

PX0⊕X2⊕X3PEkn

∇−Izn

zn , wn

2π

0

wn˙ ·Jwn

PX0⊕X2⊕X3PEkn

Hzt, zn

zn ·wn dt. 3.20

Moreover, we have

PX0⊕X2⊕X3PEkn

∇−Izn

zn , PwnPwn

PX2⊕X3P

w

n2−PX0P

w n2−

2π

0

PX0⊕X2⊕X3PEkn

Hzt, zn

zn ·PwnPwn

dt.

3.21

Since wn converges to 0 weakly and Hzt, zn·PwnPwnis bounded,PX2⊕X3Pwn

2

PX0Pwn

2 0. Since

PX1wn

2 0,

wn converges to 0 strongly, which is a contradiction. Hence, znn is bounded. Up to a subsequence, we can suppose that zn converges to z0 for

somez0∈X0⊕X2⊕X3. We claim thatznconverges toz0strongly. We have

PX0⊕X2⊕X3PEkn∇−Izn, P

znPzn

PX2⊕X3PEknP

zn2

PX0PEknP

zn2

PX0⊕X2⊕X3PEkn 2π

0 Hz

t, zn·PznPzn.

3.22

ByH1and the boundedness ofHzt, znPznPzn,

PX

2⊕X3PEknP

zn2

PX0PEknP

zn2−→

PX0⊕X2⊕X3PEkn 2π

0

(11)

That is,PX2⊕X3PEknPzn

2

PX0PEknPzn

2 converges. Since

PX1zn

2 0,

zn2converges, soznconverges tozstrongly. Therefore, we have

grad−CIz grad−CIz lim

n→∞PEkngrad −

CIzn limn→∞PEkngrad −

CIzn 0. 3.24

So we proved the first case.

We consider the casePX1zn0, that is,znX0⊕X2⊕X3. Thenzn∂C, for alln. In this

case,zn ΨznX0⊕X2⊕X3andPX0⊕X2⊕X3∇−Izn→ 0. Thus, by the same argument

as the first case, we obtain the conclusion. So we prove the lemma.

Proposition 3.4. Assume thatHsatisfies the conditions (H1),H2, (H3), (H4). Then there exists a numberδ1 >0such that for anyαandβwithj1−1 < α < j1 < β < j2δ1 < j21 j3andα >0,

there exist at least two nontrivial critical pointszi,i1,2, inX1for the functionalIsuch that

inf z∈ΔR1S1ρ,X2⊕X3

IzIzi≤ sup zSr1X0⊕X1

Iz<0< inf z∈ΣR1S1ρ,X2⊕X3

Iz, 3.25

whereρ,r1, andR1are introduced inLemma 3.1.

Proof. First, we will find two nontrivial critical points for −I. By Lemma 3.1, −Isatisfies the torus-sphere variational linking inequality, that is, there existδ1>0,ρ >0,r1>0, andR1>0

such thatr1< R1, and for anyαandβwithj

1−1< α < j1< β < j2δ1< j21j3andα >0

sup z∈ΣR1

Iz sup

z∈ΣR1S1ρ,X2⊕X3

Iz<0< inf zSr1X0⊕X1

Iz inf

zSr1

Iz,

sup z∈ΔR1

Iz sup

z∈ΔR1S1ρ,X2⊕X3

Iz − inf

z∈ΔR1S1ρX2⊕X3

Iz<,

inf zBr1

Iz inf

zBr1X0⊕X1

Iz − sup

zBr1X0⊕X1

Iz>−∞.

3.26

ByLemma 3.3,−Isatisfies theP.S.ccondition with respect toCn n for every real numberc such that

0< inf zSr1

Izc≤ sup z∈ΔR1

Iz. 3.27

Thus byTheorem 2.5, there exist two critical pointsz1,z2for the functional−Isuch that

inf zSr1

Iz≤−Izi≤ sup z∈ΔR1

Iz, i1,2. 3.28

SettingziΨ zi,i1,2, we have

0< inf zSr1

Iz inf

zSr1

Iz≤−Iz1≤−Iz2≤ sup z∈ΔR1

Iz sup

z∈ΔR1

Iz.

(12)

We claim thatzi/∂C, that is zi /X0⊕X2⊕X3, which implies thatziare the critical points for−I inX1, soziare the critical points for I inX1. For this we assume by contradiction that

ziX0⊕X2⊕X3. From3.12,PX0⊕X2⊕X3∇−Izi 0, namely,zi,i 1,2, are the critical

points for−I|X0⊕X2⊕X3. ByLemma 3.2,−Izi 0, which is a contradiction for the fact that

0< inf zSr1X0⊕X1

Iz≤−Izi≤ sup

z∈ΔR1S1ρ,X2⊕X3

Iz. 3.30

Lemma 3.2implies that there is no critical pointzX0⊕X2⊕X3such that

0< inf zSr1X0⊕X1

Iz≤−Iz≤ sup

z∈ΔR1S1ρ,X2⊕X3

Iz. 3.31

Hence,zi/X0⊕X2⊕X3,i1,2. This provesProposition 3.4.

Lemma 3.5second sphere-torus variational linking. Assume thatHsatisfies the conditions (H1), (H3), (H4), and the condition

H2suppose that there existγ >0andτ >0such thatj2< γ < βand

Ht, z≥ 1 2γz

2

τt, zRR2n. 3.32

Then there existδ2 > 0,ρ >0,r2 > 0, andR2 > 0such thatr2 < R2, and for anyαandβwith

j1−1< α < j1< j2< β < j2δ2< j21j3andα >0,

sup zSr2X0⊕X1⊕X2

Iz<0< inf ΣR2S2ρ,X3

Iz,

inf z∈ΔR2S2ρ,X3

Iz>−∞, sup zBr2X0⊕X1⊕X2

Iz<. 3.33

Proof. Letz z0z1 z2∈X0⊕X1⊕X2. ByH2, we have

Iz 1 2

2π

0

˙

z·Jz dt

2π

0

Ht, ztdt≤ 1 2z

2 γ

2z

2

L2τ

1

2j2−γz

2

L2 τ 3.34

for someτ. Sincej2−γ <0, there existsr2>0 such that ifzSr2X0⊕X1⊕X2, thenIz<0.

Thus we have supzS

r2X0⊕X1⊕X2Iz<0. Moreover, ifzBr

2X0X1X2, thenIz< τ <∞, so we have supzB

r2X0⊕X1⊕X2Iz<∞. Next, letz z2z3∈X2⊕X3withz2 ∈S2ρ, where

ρis a small number. We also letj1−1 < α < j1 < j2 < β < j2δ < j21 j3andα > 0. Then

X2⊕X3⊂EandPz2z3 0. ByH1, there existsτ>0 such that

Iz 1 2

2π

0

˙

z·Jz dt

2π

0

Ht, ztdt

≥ 1

2P z

2z32−

β 2P

z

2z32L2−τ

1

2P z

22 1

2P z

32− β

2P z

22L2−

β 2P

z

32L2−τ

≥ 1

2

1− β j2

ρ21

2j3−βP z

32L2−τ.

(13)

Since 1−β/j2<0 andj3−β >0, there exist a small numberδ2>0 andR2>0 withδ2< δand

R2> r2such that ifj

1−1< α < j1< j2< β < j2δ2< j21j3andzz2z3∈ΣR2S2ρ, X3, thenIz>0. Thus we have infz∈ΣR2S2ρ,X3Iz>0.

Moreover, ifz∈ΔR2S2ρ, X3, thenIz≥1/21−β/j2ρ2−τ >−∞. Thus we have infΔR2S2ρ,X3Iz>−∞. Thus we prove the lemma.

Lemma 3.6. For anyΛ ∈j2, j3there exists a constantτ >0such that for anyαandβwithj1−1 <

α < j1< j2 ≤β≤Λ < j21j3andα >0, ifzis a critical point forI|X0⊕X1⊕X3 with0≤Izτ,

thenz0.

Proof. By contradiction, we can suppose that there existΛ>0, a sequenceαnn,βnnsuch that αnα,βnβwithαj1−1, j1, βj2·Λ, and a sequenceznninX0⊕X1⊕X3such that

Izn→0 andPX0⊕X1⊕X3∇Izn 0. We claim thatznnis bounded. If we do not suppose that zn →∞, let us setwn zn/zn. We have up to a subsequence, thatwn w0weakly for

somew0∈X0⊕X1⊕X3. Furthermore,

0∇Izn, PX0⊕X1zn

PPX0⊕X1zn

2

PPX0⊕X1zn

2

Hzt, zn, PX0⊕X1zn

, 3.36

so we have

PX0X1zn2

Hzt, zn, PX0⊕X1zn

. 3.37

Moreover,

0∇Izn, PX3zn

PX3zn

2

Hzt, zn, PX3zn

, 3.38

so we have

PX3zn2

Hzt, zn, PX3zn

. 3.39

Adding3.37and3.39, we have

zn2

Hzt, zn, zn. 3.40

From3.40we have

w02lim

n→∞

Hzt, zn, wn. 3.41

We also have

0PX0⊕X1⊕X3∇Izn, zn

2Izn

2π

0

−2Ht, zn Hzt, zn·zndt. 3.42

Dividing byznand going to the limit, we have

lim n→∞

2π

0

Hzt, zn·wn0. 3.43

Thus

w02

0, 3.44

which is a contradiction sincew01. Soznnis bounded and we can suppose thatzn z

forzX0⊕X1⊕X3. From3.42, we have

Hzt, zn, zn

2π

0

(14)

From3.40,

lim n→∞zn

2lim

n→∞

Hzt, zn, zn lim n→∞

2π

0 2H

t, zndt

2π

0 2H

t, zdt. 3.46

Thus,znconverges tozstrongly. We claim thatz0. Assume thatz/0. ByH1αz2L2 c1 <

2 02πHt, zdt < βz2L2 c2, for somec1andc2. If zX0⊕X1with PX0⊕X1z

2 ≥ |

j|z2L2 for

j <0 and|j|> β,

|j|PX0⊕X1z

2

L2 ≤PX0⊕X1z

2

βPX0⊕X1z

2

L2c2. 3.47

IfzX3,PX3z

2

j3PX3z

2

L2, and

j3PX3z

2

L2 ≤PX3z

2

βPX3z

2

L2 c2. 3.48

Thus, we have

|j| −βPX0⊕X1z

2

L2 j3−βPX3z

2

L2−2c2≤0, 3.49

which is absurd because of|j|> βandj3> β. Thusz0. We proved the lemma.

LetPX2 be the orthogonal projection fromEontoX2and

ˇ

CzE|PX2z≥1

. 3.50

Then ˇC is the smooth manifold with boundary. Let ˇCn Cˇ ∩En. Let us define a functional ˇ

Ψ:E\ {X0⊕X1⊕X3} →Eby

ˇ

Ψz zPX2z

PX

2z

PX0⊕X1⊕X3z

1− 1 PX2z

PX2z. 3.51

We have

Ψˇ zw w 1 PX2z

PX2w

PX2z

PX

2z

, w

PX2z

PX

2z

. 3.52

Let us define the functional ˇI: ˇCRby

ˇ

I I◦Ψˇ. 3.53

Then ˇIC1loc,1. We note that if ˇzis the critical point of ˇIand lies in the interior of ˇC, thenzΨˇ zˇ is the critical point ofI. We also note that

grad− ˇ

CIˇzˇ≥PX0⊕X1⊕X3∇I

ˇ

(15)

Let us set

ˇ

Sr2 Ψˇ−1Sr2X0X1X2,

ˇ

Br2 Ψˇ−1Br2X0⊕X1⊕X2

,

ˇ

ΣR2 Ψˇ−1ΣR2S2ρ, X3,

ˇ

ΔR2 Ψˇ−1ΔR2S2ρ, X3.

3.55

We note that ˇSr2,Bˇr2,ΣˇR2,and ˇΔR2 have the same topological structure asSr2,Br2,ΣR2, andΔR2, respectively.

We have the following lemma whose proof has the same arguments as that of Lemma 3.5except the spaceX0⊕X1⊕X3,X0⊕X1,X3instead of the spaceX0⊕X2⊕X3,X0,

X2⊕X3.

Lemma 3.7.Iˇsatisfies theP.S.cˇcondition with respect toCnˇ nfor every real numbercˇsuch that

0< inf

ˇ

zΨˇ−1

Sr2X0⊕X1⊕X2

Izˇ≤cˇ ≤ sup

˘

z∈Ψ˘−1ΔR2S2ρ,X3

Izˇ, 3.56

whereρ,r2, andR2are introduced inLemma 3.5.

Proposition 3.8. Assume thatHsatisfies the conditions (H1), (H2), (H3), and (H4). Then there exists a small numberδ2>0such that for anyαandβwithj1−1< α < j1< j2< β < j2δ < j21j3and

α >0, there exist at least two nontrivial critical pointswi,i1,2, inX2for the functionalIsuch that

inf z∈ΔR2S2ρ,X3

IzIwi≤ sup zSr2X0⊕X1⊕X2

Iz<0< inf z∈ΣR2S2ρ,X3

Iz, 3.57

whereρ,r2, andR2are introduced inLemma 3.5.

Proof. It suffices to find the critical points for−I. Byˇ Lemma 3.5,−Iˇ satisfies the torus-sphere variational linking inequality, that is, there existδ2 > 0,ρ >0, r2 > 0, andR2 >0 such that

r2< R2, and for anyαandβwithj1−1< α < j1< j2< β < j2δ2 < j21j3,

sup

ˇ

zΣˇ R2

Izˇ sup

z∈ΣR2S2ρ,X3

Iz<0< inf zSr2X0⊕X1⊕X2

Iz inf

ˇ

zSˇ r2

Izˇ,

sup

ˇ

zΔˇ R2

Izˇ sup

z∈ΔR2S2ρ,X3

Iz − inf

z∈ΔR2S2ρ,X3

Iz<,

inf

ˇ

zBˇr2

Izˇ inf

zBr2X0⊕X1⊕X2

Iz − sup

zBr2X0⊕X1⊕X2

Iz>−∞.

3.58

ByLemma 3.7,−Iˇsatisfies theP.S.c∗ˇcondition with respect toCnˇ n for every real number ˇc such that

0< inf

ˇ

zSˇ r2

Izˇ≤cˇ≤ sup

ˇ

zΔˇ R2

(16)

Then byTheorem 2.5, there exist two critical points ˇw1, ˇw2for the functional−Iˇsuch that

inf

ˇ

wSˇr2

Iwˇ≤−Iwiˇ ≤ sup

ˇ

wΔˇ R2

Iwˇ, i1,2. 3.60

SettingwiΨˇ wiˇ ,i1,2, we have

0< inf wSr2

Iw inf

ˇ

wSˇ r2

Iwˇ≤−Iw1≤ −Iw2≤ sup ˇ

wΔˇ R2

Iwˇ sup

w∈ΔR2S2ρ,X3

Iw.

3.61

We claim that ˇwi/∂C, that isˇ wi/X0⊕X1⊕X3, which implies thatwiare the critical points

for−I, sowiare the critical points forI. For this we assume by contradiction thatwiX0⊕

X1⊕X3. From3.54,PX0⊕X1⊕X3∇−Iwi 0, namely,wi,i 1,2, are the critical points for

I|X0⊕X1⊕X3. ByLemma 3.6,−Iwi 0, which is a contradiction for the fact that

0< inf wSr2X0⊕X1⊕X2

Iw≤−Iwi≤ sup

w∈ΔR2S2ρ,X3

Iw. 3.62

It follows fromLemma 3.6that there is no critical pointwX0⊕X1⊕X3such that

0< inf wSr2X0⊕X1⊕X2

Iw≤−Iw≤ sup

w∈ΔR2S2ρ,X3

Iw. 3.63

Hence,wi/X0⊕X1⊕X3,i1,2. This provesProposition 3.8.

Proof ofTheorem 1.1. Assume thatH satisfies conditions H1–H4. By Proposition 3.4, there existδ1 > 0,ρ > 0,r1 > 0, andR1 > 0 such that for anyαandβwith j1−1 < α < j1 < β <

j2δ1< j21j3,1.1has at least two nontrivial solutionszi,i1,2, inX1for the functional

Isuch that

inf z∈ΔR1S1ρ,X2⊕X3

IzIzi≤ sup zSr1X0⊕X1

Iz<0< inf z∈ΣR1S1ρ,X2⊕X3

Iz. 3.64

ByProposition 3.8, there existδ2>0,ρ >0,r2>0, andR2>0 such that for anyαandβwith

j1−1< α < j1< j2< β < j2δ2< j21j3andα >0,1.1has at least two nontrivial solutions

wi,i1,2, inX2for the functionalIsuch that

inf z∈ΔR2S2ρ,X3

IzIwi≤ sup zSr2X0⊕X1⊕X2

Iz<0< inf z∈ΣR2S2ρ,X3

Iz. 3.65

Let

δmin{δ1, δ2}. 3.66

Then for anyαandβwithj1−1 < α < j1 < j2 < β < j2δ < j21 j3andα >0,1.1has at

(17)

4. Proof ofTheorem 1.2

Assume thatHsatisfies conditionsH1–H4withα < β <0. Let us set

X0≡span{sinjtek cosjtekn,cosjtek−sinjtekn|j≥ −j11, j∈N, 1≤kn},

X1≡span{sinjtek cosjtekn,cosjtek−sinjtekn|j≥ −j1, jN, 1≤kn},

X2≡span{sinjtek cosjtekn,cosjtek−sinjtekn|jj2, jN, 1≤kn},

X3≡span

e1, e2, . . . , e2n,sinjtek −cosjtekn,cosjtek sinjtekn|j >0, j∈N, 1≤kn

∪sinjtek cosjtekn,cosjtek−sinjtekn|j≤ −j2−1−j3, jN, 1≤kn

.

4.1

Then the spaceEis the topological direct sum of the subspacesX0,X1,X2, andX3, where

X1andX2are finite dimensional subspaces.

Proof ofTheorem 1.2. By the same arguments as that of the proof of Theorem 1.1, there exist δ >0,ρ >0,r1>0,R1,r2>0, andR2 >0 such that for anyαandβwithj

1−1 < α < j1 <

j2< β < j2δ,1.1has at least four nontrivial solutions, two of which are nontrivial solutions

zi,i1,2, inX1with

inf z∈ΔR1S12ρ,X3

IzIzi≤ sup zSr1X0⊕X1

Iz<0< inf z∈ΣR1S12ρ,X3

Iz, 4.2

and two of which are nontrivial solutionswi,i1,2,inX2with

inf z∈ΔR2S2ρ,X3

IzIwi≤ sup zSr2X0⊕X1⊕X2

Iz<0< inf z∈ΣR2S2ρ,X3

Iz. 4.3

Acknowledgment

This research is supported in part by Inha University research grant.

References

1 K.-C. Chang,Infinite-Dimensional Morse Theory and Multiple Solution Problems, vol. 6 ofProgress in

Non-linear Differential Equations and their Applications, Birkh¨auser, Boston, Mass, USA, 1993.

2 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,

vol. 65 ofCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence,

RI, USA, 1986.

3 G. Fournier, D. Lupo, M. Ramos, and M. Willem, “Limit relative category and critical point theory,” in

Dynamics Reported, vol. 3, pp. 1–24, Springer, Berlin, Germany, 1994.

4 A. M. Micheletti and C. Saccon, “Multiple nontrivial solutions for a floating beam equation via critical

References

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