R E S E A R C H
Open Access
Singular potential Hamiltonian system
Tacksun Jung
1and Q-Heung Choi
2**Correspondence:
2Department 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 multiple solutions for the Hamiltonian system with singular potential nonlinearity and periodic condition. We get a theorem which shows the existence of the nontrivial weak periodic solution for the Hamiltonian system with singular potential nonlinearity. We obtain this result by using the variational method, critical point theory for indefinite functional.
MSC: 35Q70
Keywords: Hamiltonian system; singular potential nonlinearity; variational method; critical point theory for indefinite functional; (P.S.)ccondition
1 Introduction
LetDbe an open subset inRnwith a compact complementC=Rn\D,n≥. Letp∈Rn,
q∈Rn, (p(t),q(t))∈C([, π],D) andG(t, (p(t),q(t)))∈C([, π]×D,R). In this paper
we investigate the number of solutions (p(t),q(t))∈C([, π],D) with singular potential nonlinearity and periodic condition
˙
p(t) = –Gq
t,p(t),q(t), ˙
q(t) =Gp
t,p(t),q(t),
p() =p(π), q() =q(π).
(.)
Let us setz= (p,q) and
J=
–In
In
.
Then (.) can be rewritten as
–Jz˙=Gz
t,z(t),
z() =z(π),
(.)
wherez˙=dzdt. We assume thatG(t,z(t))∈C([, π]×D,R) satisfies the following condi-tions:
(G) There existsR> such that
supGt,z(t)+gradzGt,z(t)Rn|(t,z)∈[, π]×
Rn\BR < +∞.
(G) There is a neighborhoodZofCinRnsuch that
G(t,z)≥ A
d(z,C) for(t,z)∈[, π]×Z,
whered(z,C)is the distance function toCandA> is a constant.
Several authors ([–],etc.) studied the Hamiltonian system with nonsingular potential nonlinearity. Jung and Choi [, ] considered (.) with nonsingular potential nonlinearity or jumping nonlinearity crossing one eigenvalue, or two eigenvalues, or several eigenval-ues. Chang [] proved that (.) has at least two nontrivial π-periodic weak solutions under some asymptotic nonlinearity. Jung and Choi [] proved that (.) has at leastm
weak solutions, which are geometrically distinct and nonconstant under some jumping nonlinearity.
In this paper we are trying to find the weak solutionsz∈C([, π],D) of system (.) such that
π
˙
z·w–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 . Our main result is as follows.
Theorem . Assume that G satisfies conditions(G)-(G).Then system(.)has at least oneπ-periodic solution.
For the proof of Theorem ., we introduce the perturbed operatorA, such thatA– is a compact operator and the associated functionalI(z) corresponding to the operatorA, and approach the variational method, the critical point theory. In Section , we investigate the Fréchet differentiability of the associated functionalI(z) and recall the critical point theorem for indefinite functional. In Section , we show that the associated functionalI(z) satisfies the geometrical assumptions of the critical point theorem for indefinite functional and prove Theorem ..
2 Variational method
LetL([, π],Rn) denote the set of n-tuples of the square integrable π-periodic func-tions and choosez∈L([, π],Rn). Then it has a Fourier expansionz(t) =k=+∞
k=–∞akeikt, withak=π
π
z(t)e–iktdt∈Cn,a–k=a¯kand
k∈Z|ak|<∞. Let
A:z(t)→–J˙z(t)
with the domain
D(A) =z(t)∈H[, π],Rn|z() =z(π)
=
z(t)∈L[, π],Rn
k∈Z
+|k||ak|< +∞
where is a positive small number. Then Ais a self-adjoint operator. Let{Mλ} be the spectral resolution ofA, and letα be a positive number such thatα∈/ σ(A) and [–α,α] contains only one element ofσA. Let
P=
α
–α
dMλ, P+=
+∞
α
dMλ, P–=
–α
–∞ dMλ.
Let
L=PL
[, π],Rn, L+=P+L
[, π],Rn, L–=P–L
[, π],Rn.
For eachu∈L([, π],Rn), 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],Rn 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],Rn). The scalar product inLnaturally extends as the duality pairing between
E andE=W–,([, π],Rn). We note that the operator (+|A|)–is a compact linear
operator fromL([, π],Rn) toEsuch that
+|A|–w,zE= π
w(t),z(t)dt.
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|–
Thus we have
zE=uL, z+E+=u+L+, z–E–=u–L–
and thatE,E+,E–are isomorphic toL,L+,L–, respectively.
For the sake of simplicity, from now on we shall denote the subset of C([, π],D),
satisfying the π-periodic condition, byC(S,D).
Let us introduce an open set of the Hilbert spaceEas follows:
X=z∈E|z(t)∈D⊂Rn,t∈S .
Let
X=E∩X, X+=E+∩X, X–=E–∩X.
ThenX=X⊕X+⊕X–and forz∈X,zhas the decompositionz=z+z++z–∈X, where
z∈X, z+∈X+, z–∈X–.
The associated functional of (.) onXis as follows:
I(z) = A
z+
L+A
M+z
–(–A)
z
–
L–(–A)
M
–z
–ψ(z), (.)
whereψ(z) =ψ(z) +zL,ψ(z) =
π
G(t,z(t))dt. Let
F(z) =Gz
t,z(t).
ByG∈C,ψ(z) =πG(t,z(t))∈C(S×D,R). Let
F(z) =I+F(z) =I+Gz
t,z(t).
System (.) can be rewritten as
A(z) =F(z). (.)
The Euler equation of the functionalI(z) is the system
u+=|A|–
P
+F(z), (.)
u–= –|A|–
P–F(z), (.)
M+u=|A|–
M+PF(z), M–u= –|A|–M–PF(z). (.)
Thusz=z+z++z–is a solution of (.) if and only ifu=u+u++u–is a critical point
ofI. System (.)-(.) is reduced to
Az–=P–F(z+z++z–) or z–= (A)–P–F(z+z++z–), (.)
AM+z=M+PF(z+z++z–), AM–z=M–PF(z+z++z–). (.)
By the conditionF(z)∈C(E,E) and (G),
F(z) –F(w)L≤(+β)u–vH ∀u,v∈E. (.)
By (G) and (G), there existsγ >min{j–α+,β–j+}such that
A– |L+⊕L–≤
γ.
By the following lemma, the weak solutions of (.) coincide with the critical points of the functionalI(z).
Lemma . Assume that G satisfies conditions(G)-(G).Then I(z)is continuous and Fréchet differentiable in X with the Fréchet derivative
DI(z)w= π
Az–F(z)
·w dt for all w∈X.
Moreover,DI∈C.That is,I∈C.
Proof First we prove thatI(z) =π[Az–G(t,z(t)) –z]dtis continuous and Fréchet differentiable inX. Forz,w∈X,
I(z+w) –I(z)
= π
A(z+w)·(z+w) – π
G(t,z+w) + (z+w)
– π
A(z)·z+ π
G(t,z) + z
= π
A(z)·w+A(w)·z+A(w)·w
– π
G(t,z+w) –G(t,z) +
z·w+w.
We have π
G(t,z+w) –G(t,z)dt
≤ π
Gz
t,z(t)·w+OwRn
dt=OwRn
. (.)
Thus we have
I(z+w) –I(z)=OwRn
Next we shall prove thatI(z) is Fréchet differentiable inX. Forz,w∈X,
I(z+w) –I(z) –DI(z)w
= π
A(z+w)·(z+w) – π
G(t,z+w) + (z+w)
– π
A(z)·z+ π
G(t,z) + z – π
A(z)·w+ π
Gz(t,z) +z·w
= π
A(w)·z+A(w)·w
– π
G(t,z+w) –G(t,z) –Gz(t,z) +
w
.
By (.), we have π
G(t,z+w) –G(t,z) –Gz(t,z)
dt=OwRn
.
Thus
I(z+w) –I(z) –DI(z)w=O wRn
.
Lemma . Assume that G satisfies conditions(G)-(G).Let{zk} ⊂X–and zkz weakly
in X with z∈∂X.Then I(zk)→–∞.
Proof To prove the conclusion, it suffices to prove that π
Gt,zk(t)
dt−→+∞.
SinceG(t,z(t)) is bounded from below, it suffices to prove that there is an interval [a,b]⊂ [, π] such that
b
a
Gt,zk(t)
dt−→+∞.
By definition,z∈∂Xmeans that there existst*∈[, π] such thatz(t*)∈∂D. By G(), there exists a constantB> such that
G(x,t,U)≥ A
d(z,C)–B.
Thus we have t*+δ
t* G
t,z(t)dt≥
t*+δ t*
A
z(t) –z(t*)
Rn –B
dt
for allδ> . By Schwarz’s inequality, we have
z(t) –zk(t)Rn≤t–t*
π
z˙(t)Rndt
Thus we have t*+δ
t* G
t,z(t)dt= +∞.
Since the embeddingH(S,Rn)→C(S,Rn) is compact, we have
maxz(t) –zk(t)
Rn|t∈S −→ ask→ ∞. Thus by Fatou’s lemma, we have
lim inf
t*+δ t* G
t,zk(t)dt≥
t*+δ t*
lim infGt,zk(t)
= t*+δ
t* G
t,z(t)dt= +∞.
Thus
lim inf
t*+δ t*
Gt,zk(t)
= +∞.
Thus if{zk} ⊂X–,
I(zk) = π
Azk(t) –ψ(zk)
dx dt
= A
(zk)+
L+A
M+(zk)
–(–A)(zk)–
L
–(–A)
M
–(zk)
–ψ(zk)
≤–(–A)
(z
k)–
L–ψ(zk)−→–∞,
so we prove the lemma.
Now, we recall the critical point theorem for indefinite functional (cf.[]). Let
Br=
u∈X| u ≤r ,
Sr=
u∈X| u=r .
Theorem .(Critical point theorem for indefinite functional) Let X be a real Hilbert space with X=X⊕Xand X=X⊥.Suppose that I∈C(X,R)satisfies(PS)and
(I) I(u) =(Lu,u) +bu,whereLu=LPu+LPuandLi:Xi→Xiis bounded and
self-adjoint,i= , , (I) bis compact,and
(I) there exists a subspaceX˜ ⊂Xand setsS⊂X,Q⊂ ˜Xand constantsα>ωsuch that
(i) S⊂XandI|S≥α,
(ii) Qis bounded andI|∂Q≤ω,
(iii) Sand∂Qlink.
3 Proof of Theorem 1.1
We shall show that the functionalI(z) satisfies the geometric assumptions of the critical point theorem for indefinite functional.
Lemma .(Palais-Smale condition) Assume that G satisfies conditions(G)and(G).
Then there exists a constant γ depending on the C norm of the function G on S×
(Rn\B
R)such that I(z)satisfies the(P.S.)γ condition in X forγ <γ.
Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence {zk} ⊂Xsatisfying
I(zk)→γ (.)
and
DI(zk) =zk–A–
Gz(t,zk) +zk
−→θ, (.)
whereA– is a compact operator andθ= (, . . . , ). SinceG(t,z(t)) is bounded from below, (.) implies that there exists a constantC> such that
π
Azk(t)dt≥C. (.)
We claim that{zk}has a convergent subsequence. It suffices to prove that the sequence {zk}is bounded inX. By contradiction, we suppose that up to a subsequence,{zk}satisfies zkRn→ ∞. Then, for largek, we have
zkRn≥R. (.)
It follows from (.) that
π
G(t,zk)dt≤πsupG(t,zk)|(t,zk)∈S×Rn\BR . (.)
By (.) and (.),
I(zk) = π
A(zk) –G
t,zk(t)
– z
k
dt
≥ C
– πsupG(t,zk)|(t,zk)∈S
×Rn\B R –
zk
L.
Letting
γ= C
– πsupG(t,zk)|(t,zk)∈S
×Rn\B R –
zk
L,
we haveI(zk)≥γ, which is a contradiction.
Let
Q=B¯r∩X–
Lemma . Assume that G satisfies conditions(G)and(G).Then there exist sets Sρ⊂
X+with radiusρ> ,Q⊂X and a constantα> such that (i) Sρ⊂X+andI|Sρ≥α,
(ii) Qis bounded andI|∂Q≤,
(iii) Sρand∂Qlink.
Proof (i) Let us choosez∈X+⊂X. Thenz(t)∈D. By (G),G(t,z) is bounded above and
there exists a constantτ>
I(z) = A
z+
L+A
M+z–(–A)
z
–
L–(–A) M
–z
– π
G(t,z)dt– z
L
≥ A
z+
L–τ–
z
L
forτ> . Then there exist constantsρ> andα> such that ifz∈Sρ∩X+, thenI(z)≥α. (ii) Let us choosee∈B∩X+. Letz∈ ¯Br∩X–⊕ {re| <r}. Thenz=w+y,w∈ ¯Br∩X–,
y=re. We note that
Ifw∈ ¯Br∩X–, then π
A(z)·z dt= –(–A)
z–
L≤.
By (G),G(t,w+re) is bounded from below. Thus, by Lemma ., there exists a constant
τ > such that ifz=w+re, then we have
I(z) = r
–(–A
)
z
–
L–
π
G(t,w+re)dt– z
L
≤ r
–(–A
)
z–
L–
τ
d(V+re,C)dx dt–
z
L.
We can choose a constantR>rsuch that ifz=w+re∈Q= (B¯r∩X–)⊕{re|e∈B∩X+, < r<R}, thenI(z) < . Thus we prove the lemma.
Proof of Theorem. By Lemma .,I(z) is continuous and Fréchet differentiable inXand, moreover,DI∈C. By Lemma ., if{zk} ⊂X–andzkzweakly inXwithz∈∂X, then
I(zk)→–∞. By Lemma .,I(z) satisfies the (P.S.)γ condition forγ <γ. By Lemma .,
there exist setsSρ⊂X+with radiusρ> ,Q⊂Xand a constantα> such thatI|Sρ≥α,Q is bounded andI|∂Q≤, andSρand∂Qlink. By the critical point theorem,I(z) possesses a critical valuec≥α. Thus (.) has at least one nontrivial weak solution. Thus we prove
Theorem ..
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Author details
1Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea.2Department of Mathematics
Acknowledgements
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-2010-0023985).
Received: 24 April 2013 Accepted: 8 October 2013 Published:19 Nov 2013
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Cite this article as:Jung and Choi:Singular potential Hamiltonian system.Journal of Inequalities and Applications
