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,[email protected]
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:R→R2n, ˙zdz/dt,
J
0 −In In 0
, 1.2
Inis the identity matrix onRn,H : R1×R2n → R, andHzis the gradient ofH. Letz p, q, p z1, . . . , zn, q zn1, . . . , z2n∈Rn.Then1.1can be rewritten as
˙
p−Hqt, p, q, ˙
qHpt, p, q. 1.3
H1There exist constantsα < βsuch that
αI≤d2zHt, z≤βI ∀t, z∈R1×R2n. 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 −τ ∀t, z∈R1×R2n. 1.5
H3Ht,0 0,Hzt,0 0, andj∈j1, j2∩Zsuch that
jI < d2zHt,0<j1I ∀t∈R1. 1.6
H4H is 2π-periodic with respect tot.
We are looking for the weak solutions of 1.1. Let E W1/2,20,2π, R2n. The 2π-periodic weak solutionz p, q∈Eof1.3satisfies
2π
0
˙
pHqt, zt
·ψ−q˙−Hpt, zt
·φdt0 ∀ζ φ, ψ∈E 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.
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 W1/2,20,2π, R2n.The scalar product inL2 naturally extends as the duality pairing
betweenEandEW−1/2,20,2π, R2n. It is known that ifz∈C∞R, R2nis 2π-periodic, then it has a Fourier expansionzt kk−∞∞akeikn withak ∈ C2n anda−k ak:Eis the closure of such functions with respect to the norm
z
k∈Z
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, q∈E, p, q∈Rn, 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|j ∈N, 1≤k ≤n,
E−spansinjtek cosjtekn,cosjtek−sinjtekn|j ∈N, 1≤k≤n.
ThenEE0⊕E⊕E−andE0, E, E−are 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.Ifz∈Eandζ∈E−, then the bilinear form is zero andAzζ Az Aζ. 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
z2z02Az−Az−z02Pz2P−z2 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 Ls0,2π, R2n. In particular, there is anαs >0such that
zLs ≤αsz 2.7
for allz∈E.
Proposition 2.2. Assume thatHt, z∈C2R1×R2n, R. ThenIzisC1, that is,Izis continuous and Fr´echet differentiable inEwith Fr´echet derivative
DIzω
2π
0
˙
z−JHzt, z·Jω
2π
0
˙
pHqt, z·ψ−q˙−Hpt, z·φdt, 2.8
wherez p, qandω φ, ψ∈E.Moreover, the functionalz→ 02πHt, zdtisC1.
Proof. Forz, w ∈E,
Izw−Iz−DIzw
1
2
2π
0
z˙w˙·Jzw−
2π
0
Ht, zw− 1 2
2π
0
˙ z·Jz
2π
0
Ht, z−
2π
0
˙
z−JHzt, z·Jw
1
2
2π
0
˙
z·Jww˙ ·Jzw˙ ·Jw−
2π
0
Ht, zw−Ht, z−
2π
0
˙
z−JHzt, z·Jw.
2.9
We have
2π
0
Ht, zw−Ht, z≤
2π
0
Thus, we have
Izw−Iz−DIzwO|w|2
. 2.11
Next, we prove thatIzis continuous. Forz, w∈E,
Izw−Iz1 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, zw−Ht, 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 :W →R 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 En ⊂ E,E−n ⊂ E− for allnEn andEn−are subspaces ofE, dimEn <∞,En ⊂En1,n∈NEnare dense inE. LetXn X∩En, 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:X→Xn. For a givenB⊂E, we will writeBn B∩En. LetY be a closed subspace ofX.
Definition 2.3. LetBbe a closed subset ofXwithY ⊂B. 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
B⊂X|cat∗X,YB≥i,
ci inf B∈Bi
sup x∈B
fx. 2.14
We have the following multiplicity theoremfor the proof, see4.
Theorem 2.4. Leti∈Nand assume that
1ci<∞,
2supx∈Yfx< ci,
Then there exists a lower critical pointxsuch thatfx ci. If
cici1· · · cik−1c, 2.15
then
catX
x∈X|fx c, gradX−fx 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
Cx∈H |PX2x≥1
, 2.17
and letf :W → Rbe 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
,
Sx∈X2⊕X3| xρ
,
Bx∈X2⊕X3| x ≤ρ
.
2.18
Assume that
supfΣ<inffS 2.19
and that theP.S.c condition holds forf onC, with respect to the sequrnce Cnn, for allc ∈ a, b, where
ainffS, b supfΔ. 2.20
Moreover, one assumes b < ∞andf|X1⊕X3 has no critical pointszinX1⊕X3 witha ≤ fz ≤ b.
Then there exist two lower critical pointsz1,z2forf onCsuch thata≤fzi≤b,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|j≤j1−1, j ∈N, 1≤k≤n
,
X1≡ span
sinjtek −cosjtekn,cosjtek sinjtekn|jj1, 1≤k≤n
,
X2≡ span
sinjtek−cosjtekn,cosjtek sinjtekn |j j2, 1≤k≤n
,
X3≡ span
sinjtek −cosjtekn,cosjtek sinjtekn|j≥j21j3, j∈N, 1≤k≤n
.
Then E is the topological direct sum of subspacesX0, X1, X2, andX3, where X1 andX2 are
finite dimensional subspaces. We also set
S1ρ
z∈X1| zρ
,
Sr1X0⊕X1z∈X0⊕X1| zr1,
Br1X0⊕X1z∈X0⊕X1| z ≤r1,
ΣR1S1ρ, X2⊕X3zz1z2z3∈X1⊕X2⊕X3|z1∈S1ρ, z1z2z3R1,
ΔR1S1ρ, X2⊕X3zz1z2z3∈X1⊕X2⊕X3|z1∈S1ρ, z1z2z3≤R1,
S2ρ {z∈X2| zρ},
Sr2X0⊕X1⊕X2z∈X0⊕X1⊕X2| zr2,
Br2X0⊕X1⊕X2z∈X0⊕X1⊕X2| z ≤r2,
ΣR2S2ρ, X3zz2z3∈X2⊕X3|z2∈S2ρ, z2z3R2,
ΔR2S2ρ, X3zz2z3∈X2⊕X3|z2∈S2ρ, z2z3≤R2.
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−
τ ∀t, z∈R1×R2n. 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 z∈Sr1X0⊕X1
Iz<0< inf z∈ΣR1S1ρ,X2⊕X3
Iz,
inf z∈ΔR1S1ρ,X2⊕X3
Iz>−∞, sup z∈Br1X0⊕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
for some τ > 0. Since j1−γ < 0, there exists r1 > 0 such that if z0z1 ∈ Sr1X0 ⊕X1,
then Iz < 0. Thus, supz∈S
r1X0⊕X1Iz < 0. Moreover, if z ∈ Br
1X0 ⊕ X1, then Iz ≤
1/2j1−γz0z12L2 τ < τ <∞, so we have supz∈B
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⊂EandP−z1z2z3 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, X2⊕X3, thenIz >0. Thus, we have infz∈Σ
R1S1ρ,X2⊕X3Iz >0. Moreover,
if j1 −1 < α < j1 < β < j2δ1 < j2 1 j3 and z ∈ ΔR1S1ρ, X2 ⊕X3, 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< β≤j2< 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
Cz∈E|PX1z≥1
. 3.8
Then C is the smooth manifold with boundary. Let Cn C ∩En. Let us define afunctional
Ψ:E\ {X0⊕X2⊕X3} →Eby
Ψz z− PX1z
PX1z PX0⊕X2⊕X3z
1− 1 PX1z
We have
∇Ψ zw w− 1 PX1z
PX1w−
PX1z
PX1z, w
PX1z
PX1z
. 3.10
Let us define the functionalI:C→Rby
I I◦Ψ. 3.11
ThenI∈C1loc,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−
CIz≥PX0⊕X2⊕X3∇IΨ z ∀z∈∂C. 3.12
Let us set
Sr1 Ψ−1Sr1X0⊕X1,
Br1 Ψ−1Br1X0⊕X1,
ΣR1 Ψ−1ΣR1S1ρ, X2⊕X3,
ΔR1 Ψ−1ΔR1S1ρ, X2⊕X3.
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
−Iz≤c≤ sup
z∈Ψ−1ΔR1S1ρ,X2⊕X3
−Iz. 3.14
Proof. Letknnbe a sequence such thatkn→ ∞,znnbe a sequence inCsuch thatzn ∈Ckn,
for alln,−Izn → cand grad−C−I|Eknzn → 0. Setzn Ψzn and hencezn ∈ Eknand
−Izn→c. We first consider the case in which zn/∈X0⊕X2⊕X3, for alln. Since fornlarge
PEn◦PX1 PX1◦PEn PX1, we have
PEkn∇−Izn PEknΨ
zn∇−Izn ΨznPE
kn∇−Izn−→0. 3.15
By3.9and3.10,
PEkn∇−Izn−→0 or
PX0⊕X2⊕X3PEkn∇−Izn−→0, PX1zn −→0.
3.16
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
2−Izn
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 , Pwn−P−wn
−PX2⊕X3P
w
n2−PX0P
−w n2−
2π
0
PX0⊕X2⊕X3PEkn
Hzt, zn
zn ·Pwn−P−wn
dt.
3.21
Since wn converges to 0 weakly and Hzt, zn·Pwn−P−wnis bounded,PX2⊕X3Pwn
2
PX0P−wn
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
zn−P−zn
−PX2⊕X3PEknP
zn2−
PX0PEknP
−zn2
PX0⊕X2⊕X3PEkn 2π
0 Hz
t, zn·Pzn−P−zn.
3.22
ByH1and the boundedness ofHzt, znPzn−P−zn,
PX
2⊕X3PEknP
zn2
PX0PEknP
−zn2−→
PX0⊕X2⊕X3PEkn 2π
0
That is,PX2⊕X3PEknPzn
2
PX0PEknP−zn
2 converges. Since
PX1zn
2 → 0,
zn2converges, soznconverges tozstrongly. Therefore, we have
grad−C−Iz grad−C−Iz lim
n→∞PEkngrad −
C−Izn limn→∞PEkngrad −
C−Izn 0. 3.24
So we proved the first case.
We consider the casePX1zn0, that is,zn∈X0⊕X2⊕X3. Thenzn∈∂C, for alln. In this
case,zn Ψzn∈X0⊕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
Iz≤Izi≤ sup z∈Sr1X0⊕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 z∈Sr1X0⊕X1
−Iz inf
z∈Sr1
−Iz,
sup z∈ΔR1
−Iz sup
z∈ΔR1S1ρ,X2⊕X3
−Iz − inf
z∈ΔR1S1ρX2⊕X3
Iz<∞,
inf z∈Br1
−Iz inf
z∈Br1X0⊕X1
−Iz − sup
z∈Br1X0⊕X1
Iz>−∞.
3.26
ByLemma 3.3,−Isatisfies theP.S.c∗condition with respect toCn n for every real numberc such that
0< inf z∈Sr1
−Iz≤c≤ sup z∈ΔR1
−Iz. 3.27
Thus byTheorem 2.5, there exist two critical pointsz1,z2for the functional−Isuch that
inf z∈Sr1
−Iz≤−Izi≤ sup z∈ΔR1
−Iz, i1,2. 3.28
SettingziΨ zi,i1,2, we have
0< inf z∈Sr1
−Iz inf
z∈Sr1
−Iz≤−Iz1≤−Iz2≤ sup z∈ΔR1
−Iz sup
z∈ΔR1
−Iz.
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
zi ∈ X0⊕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 z∈Sr1X0⊕X1
−Iz≤−Izi≤ sup
z∈ΔR1S1ρ,X2⊕X3
−Iz. 3.30
Lemma 3.2implies that there is no critical pointz∈X0⊕X2⊕X3such that
0< inf z∈Sr1X0⊕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−
τ ∀t, z∈R1×R2n. 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 z∈Sr2X0⊕X1⊕X2
Iz<0< inf ΣR2S2ρ,X3
Iz,
inf z∈ΔR2S2ρ,X3
Iz>−∞, sup z∈Br2X0⊕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 ifz∈Sr2X0⊕X1⊕X2, thenIz<0.
Thus we have supz∈S
r2X0⊕X1⊕X2Iz<0. Moreover, ifz∈Br
2X0⊕X1⊕X2, thenIz< τ <∞, so we have supz∈B
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⊂EandP−z2z3 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.
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≤Iz≤τ,
thenz0.
Proof. By contradiction, we can suppose that there existΛ>0, a sequenceαnn,βnnsuch that αn→α,βn→βwithα∈j1−1, j1, β∈j2·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−
P−PX0⊕X1zn
2−
Hzt, zn, PX0⊕X1zn
, 3.36
so we have
PX0⊕X1zn2
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
forz∈X0⊕X1⊕X3. From3.42, we have
Hzt, zn, zn
2π
0
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 z ∈ X0⊕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
Ifz∈X3,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
Cz∈E|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 z− PX2z
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: ˇC→Rby
ˇ
I I◦Ψˇ. 3.53
Then ˇI∈C1loc,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
Let us set
ˇ
Sr2 Ψˇ−1Sr2X0⊕X1⊕X2,
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.c∗ˇcondition with respect toCnˇ nfor every real numbercˇsuch that
0< inf
ˇ
z∈Ψˇ−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
Iz≤Iwi≤ sup z∈Sr2X0⊕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
z∈Σˇ R2
−Izˇ sup
z∈ΣR2S2ρ,X3
−Iz<0< inf z∈Sr2X0⊕X1⊕X2
−Iz inf
ˇ
z∈Sˇ r2
−Izˇ,
sup
ˇ
z∈Δˇ R2
−Izˇ sup
z∈ΔR2S2ρ,X3
−Iz − inf
z∈ΔR2S2ρ,X3
Iz<∞,
inf
ˇ
z∈Bˇr2
−Izˇ inf
z∈Br2X0⊕X1⊕X2
−Iz − sup
z∈Br2X0⊕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
z∈Sˇ r2
−Izˇ≤cˇ≤ sup
ˇ
z∈Δˇ R2
Then byTheorem 2.5, there exist two critical points ˇw1, ˇw2for the functional−Iˇsuch that
inf
ˇ
w∈Sˇr2
−Iwˇ≤−Iwiˇ ≤ sup
ˇ
w∈Δˇ R2
−Iwˇ, i1,2. 3.60
SettingwiΨˇ wiˇ ,i1,2, we have
0< inf w∈Sr2
−Iw inf
ˇ
w∈Sˇ r2
−Iwˇ≤−Iw1≤ −Iw2≤ sup ˇ
w∈Δˇ 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 thatwi ∈ X0⊕
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 w∈Sr2X0⊕X1⊕X2
−Iw≤−Iwi≤ sup
w∈ΔR2S2ρ,X3
−Iw. 3.62
It follows fromLemma 3.6that there is no critical pointw∈X0⊕X1⊕X3such that
0< inf w∈Sr2X0⊕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
Iz≤Izi≤ sup z∈Sr1X0⊕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
Iz≤Iwi≤ sup z∈Sr2X0⊕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
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≤k≤n},
X1≡span{sinjtek cosjtekn,cosjtek−sinjtekn|j≥ −j1, j∈N, 1≤k≤n},
X2≡span{sinjtek cosjtekn,cosjtek−sinjtekn|j−j2, j∈N, 1≤k≤n},
X3≡span
e1, e2, . . . , e2n,sinjtek −cosjtekn,cosjtek sinjtekn|j >0, j∈N, 1≤k≤n
∪sinjtek cosjtekn,cosjtek−sinjtekn|j≤ −j2−1−j3, j∈N, 1≤k≤n
.
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
Iz≤Izi≤ sup z∈Sr1X0⊕X1
Iz<0< inf z∈ΣR1S12ρ,X3
Iz, 4.2
and two of which are nontrivial solutionswi,i1,2,inX2with
inf z∈ΔR2S2ρ,X3
Iz≤Iwi≤ sup z∈Sr2X0⊕X1⊕X2
Iz<0< inf z∈ΣR2S2ρ,X3
Iz. 4.3
Acknowledgment
This research is supported in part by Inha University research grant.
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