# Homoclinic Solutions for a Second-Order Nonperiodic Asymptotically Linear Hamiltonian Systems

Authors: Qiang Zheng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish a new existence result on homoclinic solutions for a second-order nonperiodic Hamiltonian systems. This homoclinic solution is obtained as a limit of solutions of a certain sequence of nil-boundary value problems which are obtained by the minimax methods. Some recent results in the literature are generalized and extended.


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## Body

## 1. Introduction

Consider the following second-order Hamiltonian system:(HS)u¨(t)+∇V(t,u(t))=0,t∈ℝ,
where u=(u1,u2,…,uN)∈ℝN, V(t,u)=-K(t,u)+W(t,u), K,W:ℝ×ℝN→ℝ are  C1 maps.  We will say that a solution u:ℝ→ℝN of (HS) is homoclinic (to 0), if u(t)→0, as |t|→∞. In addition, if u≢0, then u is called a nontrivial homoclinic solution.Inspired by the excellent monographs [1, 2], by now, the existence and multiplicity of homoclinic solutions for Hamiltonian systems have been extensively investigated in many papers via variational methods; see [3–7] for the first order systems and [8–19] for the second systems, and most of them treat the following system:
(1)u¨(t)-L(t)u(t)+∇W(t,u(t))=0,t∈ℝ,
where L(t) is a symmetric matrix-valued function and W∈C1(ℝ,ℝN).For the periodic case, the periodicity is used to control the lack of compactness due to the fact that (1) is set on all ℝ. In 1990, Rabinowitz [12] first proved that (1) has a 2kT-periodic solution uk, which is bounded uniformly for k, and obtained a homoclinic solution for (1) as a limit of 2kT-periodic solution.For the nonperiodic case, the problem is quite different from the one described in nature. Rabinowitz and Tanaka [13] introduced a type of coercive condition on the matrix L:(L1)
l(t):=inf|x|=1L(t)x·x→+∞, as |t|→∞.They first obtained the existence of homoclinic solution for the nonperiodic system (1) under the well-known (AR) growth condition by using Ekeland’s variational principle.In 1995, Ding [8] strengthened condition (L1) by(L2)
there exists a constantα>0 such that
(2)l(t)|t|-α→+∞as|t|→∞.Under the condition (L2) and some subquadratic conditions on W(t,u), Ding proved the existence and multiplicity of homoclinic solutions for the system (1). From then on, the condition (L1) or (L2) is extensively used in nonperiodic second-order Hamiltonian systems. However, the assumption (L1) or (L2) is a rather restrictive and not very natural condition as it excludes, for example, the case of constant matrices IN.In 2005, Izydorek and Janczewska [9] first presented the “pinching” condition (see the following (V2)) and relaxed the conditions (L1) and (L2). They studied the general periodic Hamiltonian system
(3)u¨(t)+∇V(t,u(t))=f(t),t∈ℝ,
where V(t,u)=-K(t,u)+W(t,u) and obtained the following result.Theorem A (see [9]).
Let the following conditions hold:(V1)
V(t,u)=-K(t,u)+W(t,u), where V is continuous and T periodic with respect to t, T>0;(V2)
there existb1,b2>0 such that
(4)b1|u|2≤K(t,u)≤b2|u|2,∀(t,u)∈ℝ×ℝN;(V3)
K(t,u)≤(u,∇K(t,u))≤2K(t,u) for all (t,u)∈ℝ×ℝN;(V4)
∇W(t,u)=o(|u|) as u→0 uniformly in t;(V5)
there is a constantμ>2 such that
(5)0<μW(t,u)≤(∇W(t,u),u),∀(t,u)∈ℝ×ℝN;(V6)
b-1:=min{1,2b1}>2M and ∥f∥L2<(b-1-2M)/2C*, where M=sup{W(t,u):t∈[0,T],|u|=1} and C* is a positive constant depending on T. Then the system (3) possesses a nontrivial homoclinic solution u∈W1,2(ℝ,ℝN) such that u˙(t)→0 as t→±∞.From then on, following the idea of [9], some researchers are devoted to relaxing the conditions (L1) and (L2) and studying the existence of homoclinic solutions of system (HS) or (3) under the periodicity assumption of the potential, such as [10, 11, 16, 19].Very recently, Daouas [3] removed the periodicity condition and studied the existence of homoclinic solutions for the nonperiodic system (3), when W is superquadratic at the infinity. Motivated by [3], in this work, we will study the existence of homoclinic solutions of the nonperiodic system (HS), when W satisfies the asymptotically quadratic condition at the infinity. It is worth noticing that there are few works concerning this case for system (HS) or (3) up to now.Our result is presented as follows.Theorem 1.
LetA:=sup{K(t,u):t∈ℝ,|u|≤1}<+∞ hold. Moreover, assume that the following conditions hold:(H1)
K(t,0)≡0, and there exists a constant a>0 such that
(6)K(t,u)≥a|u|2,∀(t,u)∈ℝ×ℝN;(H2)
there existsβ∈(1,2] such that
(7)K(t,u)≤(u,∇K(t,u))≤βK(t,u),∀(t,u)∈ℝ×ℝN;(H3)
W(t,0)≡0 and ∇W(t,u)=o(|u|) as u→0 uniformly in t, and there exist, M0>0 such that
(8)|∇W(t,u)||u|≤M0,
for any t∈ℝ and u∈ℝN;(H4)
W(t,u)-w(t)|u|2=o(|u|2) as |u|→∞ uniformly in t, where w∈L∞(ℝ,ℝ) with w∞:=inft∈ℝw(t)>2A;(H5)
W~(t,u):=(1/2)(∇W(t,u),u)-W(t,u)→+∞ as |u|→+∞, and
(9)inf{W~(t,u)|u|2:t∈ℝwithc≤|u|<d}>0,
for any c,d>0. Then the system (HS) possesses a nontrivial homoclinic solution u∈W1,2(ℝ,ℝN) such that u˙(t)→0 as t→±∞.Remark 2.
Theorem1 treats the asymptotically quadratic case on W. Consider the functions
(10)K(t,u)=(1+e-|t|)|u|2,W(t,u)=d(t)|u|2(1-1ln(e+|u|)),
where d∈L∞(ℝ,ℝ) and inft∈ℝd(t)>4+32π2.
A straightforward computation shows thatK and W satisfy the assumptions of Theorem 1, but K does not satisfy the conditions (L1) and (L2). Hence, Theorem 1 also extends the results in [8, 13].The remainder of this paper is organized as follows. In Section2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.


## 2. Preliminaries

Following the similar idea of [20], consider the following nil-boundary value problems:
(11)u¨(t)+∇V(t,u(t))=0,∀t∈[-T,T],u(-T)=u(T)=0.
For each T>0, let ET=W1,2([-T,T],ℝN), where
(12)W1,2([-T,T],ℝN)={u:[-T,T]→ℝNisanabsolutelycontinuousfunction,u(-T)=u(T)=0andu˙∈L2([-T,T],ℝN)},
equipped with the norm
(13)∥u∥=(∫-TT‍[|u˙(t)|2+|u(t)|2]dt)1/2.
Furthermore, for p>1, let LTp=Lp([-T,T],ℝN) and LT∞=L∞([-T,T],ℝN) under their habitual norms. We need the following result.Proposition 3 (see [9]).
There is a positive constantC such that for each T>0 and u∈ET the following inequality holds:
(14)∥u∥LT∞≤C∥u∥.Note that the inequality (14) holds true with constant C=2 if T≥1/2 (see [9]). Subsequently, we may assume this condition is fulfilled.Consider a functionalI:ET→ℝ defined by
(15)I(u)=∫-TT‍[12|u˙(t)|2-V(t,u(t))]dt=12∫-TT‍|u˙(t)|2dt+∫-TT‍K(t,u(t))dt-∫-TT‍W(t,u(t))dt.
Then I∈C1(ET,ℝ), and it is easy to show that for all u,v∈ET, we have
(16)I′(u)v=∫-TT‍[(u˙(t),v˙(t))-(W(t,u(t)),∇V(t,u(t)),v(t))]dt=∫-TT‍[(u˙(t),v˙(t))+(∇K(t,u(t)),v(t))-(∇W(t,u(t)),v(t))]dt.
It is well known that critical points of I are classical solutions of the problem (11). We will obtain a critical point of I by using an improved version of the Mountain Pass Theorem. For completeness, we give this theorem.Recall that a sequence{uj} is a (C)-sequence for the functional φ if φ(uj) is bounded and (1+∥uj∥)φ′(uj)→0. A functional φ satisfies the (C)-condition if and only if any (C)-sequence for φ contains a convergent subsequence.Theorem 4 (see [21]).
LetE be a real Banach space, and let φ∈C1(E,ℝ) satisfy the (C)-condition and φ(0)=0. If φ satisfies the following conditions:(A1)
there exist constantsρ,α>0 such that φ|∂Bρ(0)≥α;(A2)
there existse∈E∖B-ρ(0) such that φ(e)≤0, then φ possesses a critical value c≥α given by
(17)c=infg∈Γmaxs∈[0,1]φ(f(s)),whereBρ(0) is an open ball in E of radius  ρ at about 0, and
(18)Γ={f∈C([0,1],E):f(0)=0,f(1)=e}.Proof.
As shown in Bartolo et al. [22], a deformation lemma can be proved with the (C)-condition replacing the usual (PS)-condition, and it turns out that the standard version Mountain Pass Theorem (see Rabinowitz [21]) holds true under the (C)-condition.Lemma 5.
Assume that(H2) holds, then
(19)K(t,u)≤K(t,u|u|)|u|β,∀t∈ℝ,|u|≥1.Proof.
From(H2) it follows that for u≠0 a map given by
(20)(0,∞)∋ν↦W(t,ν-1u)
is nondecreasing. Similar to the proof in [12], we can get the conclusion.Lemma 6 (see [9]).
Letu:ℝ→ℝN be a continuous map such that u˙ is locally square integrable. Then, for all t∈ℝ, one has
(21)|u(t)|≤2(∫t-1/2t+1/2‍(|u(s)|2+|u˙(s)|2)ds)1/2.


## 3. Proof of Theorem1

Lemma 7.
Under the assumptions of Theorem1, the problem (11) possesses a nontrivial solution.Proof.
It suffices to prove that the functionalI satisfies all the assumptions of Theorem 4.
Step1. We show that the functional I satisfies the (C)-condition. Let
(22)I(uj)beboundedand(1+∥uj∥)I′(uj)→0.
Observe that, forj large, it follows from (H1) and (H2) that there exists a constant C0 such that
(23)C0≥I(uj)-12I′(uj)uj=∫-TT‍[12(∇W(t,uj),uj)-W(t,uj)]dt+∫-TT‍[K(t,uj)-12(∇K(t,uj),uj)]dt≥∫-TT‍W~(t,u)dt.
Arguing indirectly, assume as a contradiction that ∥uj∥→∞. Setting vj=uj/∥uj∥, then ∥vj∥=1, and by Proposition 3, one has
(24)∥vj∥LT∞≤2∥vj∥=2.
Note that
(25)I′(uj)uj=∥u˙j∥LT22+∫-TT‍(∇K(t,uj),uj)dt-∫-TT‍(∇W(t,uj),uj)dt≥∥u˙j∥LT22+∫-TT‍K(t,uj)dt-∫-TT‍(∇W(t,uj),uj)dt≥C1∥uj∥2-∫-TT‍|∇W(t,uj)||uj|dt,
where C1=min{1,a}>0. This implies that
(26)∫-TT‍|∇W(t,uj)||uj|∥uj∥2dt=∫-TT‍|∇W(t,uj)||vj|2|uj|dt→C1.
Set for s≥0(27)h(s):=inf{W~(t,u):t∈[-T,T],u∈ℝNwith|u|≥s}.
By (H5), h(s)→∞ as s→∞.
For0≤l<m, let
(28)Ωj(l,m)={t∈[-T,T]:l≤|uj(t)|<m},Clm=inf{W~(t,u)|u|2:t∈[-T,T]withl≤|u(t)|<m}.
Then by (H5), Clm>0. One has
(29)W~(t,uj)≥Clm|uj|2,∀t∈Ωj(l,m).
It follows from (23) that
(30)C0≥∫Ωj(0,l)‍W~(t,uj)dt+∫Ωj(l,m)‍W~(t,uj)dt+∫Ωj(m,∞)‍W~(t,uj)dt≥∫Ωj(0,l)‍W~(t,uj)dt+Clm∫Ωj(l,m)‍|uj|2dt+h(m)|Ωj(m,∞)|
which implies that
(31)|Ωj(m,∞)|≤C0h(m)→0
as m→∞ uniformly in j, and for any fixed 0<l<m(32)∫Ωj(l,m)‍|vj|2dt=1∥uj∥2∫Ωj(l,m)‍|uj|2dt≤C0Clm∥uj∥2→0
as j→∞. Using (14) and (31), we have
(33)∫Ωj(m,∞)‍|vj|2dt≤∥vj∥LT∞2·|Ωj(m,∞)|≤2|Ωj(m,∞)|→0
as m→∞ uniformly in j.
Let0<ϵ<C1/3. By (H3) there is lϵ>0 such that
(34)|∇W(t,u)|<ϵ4T|u|
for all |t|≤lϵ. Consequently,
(35)∫Ωj(0,lϵ)‍|∇W(t,uj)||vj|2|uj|dt≤ϵ4T∫Ωj(0,lϵ)‍|vj|2dt≤ϵ4T∥vj∥LT∞22T<ϵ
for all j.
By (31), we can take mϵ large such that
(36)∫Ωj(mϵ,∞)‍|vj|2dt<ϵM0.
Hence, by (H3) one has
(37)∫Ωj(mϵ,∞)‍|∇W(t,uj)||vj|2|uj|dt≤M0∫Ωj(mϵ,∞)‍|vj|2dt<ϵ
for all j. By (32) there exists j0 such that
(38)∫Ωj(lϵ,mϵ)‍|∇W(t,uj)||vj|2|uj|dt≤M0∫Ωj(lϵ,mϵ)‍|vj|2dt<ϵ
for allj≥j0. By (35)–(38), one has
(39)limsupj→∞∫-TT‍|∇W(t,uj)||vj|2|uj|dt≤3ϵ<C1
which contradicts with (26). So {uj} is bounded in ET. In a similar way to Proposition B. 35 in [21], we can prove that {uj} has a convergent subsequence. Hence I satisfies the (C)-condition.
Step2. We show that the functional I satisfies the condition (A1) of Theorem 4.
Observe that, by (H3) and (H4), given 0<ϵ<a, there exists some Cϵ>0 such that
(40)|W(t,u)|≤ε|u|2+Cε|u|p
for all u∈ℝN and t∈[-T,T], where p>2. It follows from (H1), (40), and Proposition 3 that
(41)I(u)=12∫-TT‍|u˙(t)|2dt+∫-TT‍K(t,u(t))dt-∫-TT‍W(t,u(t))dt≥12∥u˙∥LT22+a∥u∥LT22-ϵ∥u∥LT22-Cϵ∫-TT‍|u(t)|pdt≥12∥u˙∥LT22+a∥u∥LT22-ϵ∥u∥LT22-2TCϵ∥u∥LT∞p≥min{12,a-ϵ}∥u∥2-2p/2+1TCϵ∥u∥p.
Hence there exist α>0 and ρ>0 such that I(u)≥α for  all u∈ET with ∥u∥=ρ.
Step3. We show that the functional I satisfies the condition (A2) of Theorem 4.
By(H4), there exists B>0 such that
(42)W(t,u)≥w∞|u|2-B,∀t∈[-T,T],u∈ℝN.
Let
(43)e(t)=ζ|sin(ωt)|e0,t∈[-T,T],
where ω=2π/T and e0=(1,0,…,0). Clearly, e∈ET. By (15), (42),  and Lemma 5, one has
(44)I(e)=ζ2ω22∫-TT‍|cos(ωt)|2dt+∫{t∈[-T,T];|e(t)|≤1}‍K(t,e(t))dt+∫{t∈[-T,T];|e(t)|≥1}‍K(t,e(t))dt-∫-TT‍W(t,e(t))dt≤Tζ2ω22+2TA+A∫{t∈[-T,T];|e(t)|≥1}‍|e(t)|βdt-w∞ζ2∫-TT‍|sin(ωt)|2dt+2TB=Tζ2ω22+2TA+Aζ2∫-TT‍|sin(ωt)|2dt-Tw∞ζ2+2TB=T(ω22+A-w∞)ζ2+2T(A+B).
Since w∞>2A+32π2 and T>2/Aπ, then ω2/2+A-w∞<0. So I(e)→-∞ as |ζ|→∞. So, we can choose large enough ζ∈ℝ such that ∥e∥>ρ and I(e)<0.
ClearlyI(0)=0; then, by application of Theorem 4, there exists a critical point uT∈ET of I such that I(uT)≥α for all T>2/Aπ.Lemma 8.
u
T is bounded uniformly in T>2/Aπ.Proof.
Define the set of paths(45)ΓT={f∈C([0,1],ET)∣f(0)=0,f(1)=e}.
It follows from Lemma 7 that there exists a solution uT of problem (11) at which
(46)inff∈ΓTmaxs∈[0,1]I(f(s))≡DT
is achieved. Let T->T. Since any function in ET can be regarded as belonging to ET- if one extends it by zero in [-T-,T-]∖[-T,T], then ΓT⊂ΓT-. Therefore, for any solution uT of problem (11), we obtain
(47)I(uT)=DT≤D1/2uniformlyinT>2Aπ.
Notice that I′(uT)=0, and together with (47), one has
(48)I(uT)≤D1/2,(1+uT)∥I′(uT)∥=0.
The rest of the proof is similar to that of Step 1 in Lemma 7. Hence there exists a constant M1>0, independent of T such that
(49)∥uT∥≤M1,∀T>2Aπ.
The proof is complete.Take a sequenceTn→∞, and consider the problem (11) on the interval [-Tn,Tn]. By Lemma 7, there exists a nontrivial solution un:=uTn of problem (11).Lemma 9.
Let{un}n∈ℕ be the sequence given above. Then there exists a subsequence {unj}j∈ℕ convergent to u0 in Cloc1(ℝ,ℝN).Proof.
First we prove that the sequences∥un∥LTn∞, ∥u˙n∥LTn∞, and ∥u¨n∥LTn∞ are bounded. From (14) and (49), for n large enough, one has
(50)∥un∥LTn∞≤CM1:=M2.
By (11) and (50), for all t∈[-Tn,Tn], there exists M3>0 independent of n such that
(51)∥u¨n∥LTn∞≤M3.
It follows from the Mean Value Theorem that for every n∈N and t∈ℝ, there exists τn∈[t-1,t] such that
(52)u˙n(τn)=∫t-1t‍u˙n(s)ds=un(t)-un(t-1).
Combining the above with (50), and (51) we get
(53)|u˙n(t)|=|∫τnt‍u¨n(s)ds+u˙n(τn)|≤∫t-1t‍|u¨n(s)|ds+|un(t)-un(t-1)|≤M3+2M2≔M4
and hence for n large enough
(54)∥u˙n∥LTn∞≤M4.
Second we show that the sequences {un}n∈ℕ and {u˙n}n∈ℕ are equicontinuous. Indeed, for any n∈ℕ and t1,t2∈ℝ, by (54), we have
(55)|un(t1)-un(t2)|=|∫t2t1‍u˙n(s)ds|≤∫t2t1‍|u˙n(s)|ds≤M3|t1-t2|.
Similarly, by (51), one gets
(56)|u˙n(t1)-u˙n(t2)|≤M2|t1-t2|.
By using the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence {unj}j∈ℕ and a function u0 such that
(57)unj→u0,asj→∞inCloc1(ℝ,ℝN).
The proof is complete.Lemma 10.
Letu0:ℝ→ℝN be the function given by (57). Then u0 is the homoclinic solution of (HS).Proof.
First we show thatu0 is a solution of (HS). Let {unj}j∈ℕ be the sequence given by Lemma 9, then
(58)u¨nj(t)+∇V(t,unj(t))=0
for every j∈ℕ and t∈[-Tnj,Tnj]. Take b,c∈ℝ with b<c. There exists j0∈ℕ such that for all j>j0; we get [b,c]⊂[-Tnj,Tnj] and
(59)u¨nj(t)=-∇V(t,unj(t)),∀t∈[b,c].
Integrating (59) from b to t∈[b,c], we have
(60)u˙nj(t)-u˙nj(b)=-∫bt‍∇V(s,unj(s))ds,∀t∈[b,c].
Since unj→u0 uniformly on [b,c] and u˙nj→u˙0 uniformly on [b,c] as j→∞, then, from (60), we obtain
(61)u˙0(t)-u˙0(b)=-∫bt‍∇V(s,u0(s))ds,∀t∈[b,c].
Because of the arbitrariness of b and c, we conclude that u0 satisfies (HS).
Second we prove thatu0(t)→0, as |t|→∞. Note that, by (49), for k∈ℕ, there exists j0∈ℕ such that, for all j>j0, one has
(62)∫-TnjTnj‍[|unj(t)|2+|u˙nj(t)|2]dt≤∥unj∥2≤M12.
Letting j→∞, one gets
(63)∫-TnjTnj‍[|u0(t)|2+|u˙0(t)|2]dt≤M12
and letting j→∞, we have
(64)∫-∞+∞‍[|u0(t)|2+|u˙0(t)|2]dt≤M12
and so
(65)∫|t|≥r‍[|u0(t)|2+|u˙0(t)|2]dt→0.
From Lemma 6 and (65), we obtain u0(t)→0 as |t|→∞.
Next we show thatu˙0(t)→0 as |t|→∞. Indeed, applying again Lemma 6 to u˙0, we obtain
(66)|u˙0(t)|≤2(∫t-1/2t+1/2‍(|u˙0(s)|2+|u¨0(s)|2)ds)1/2.
Also, from (65), we get
(67)∫t-1/2t+1/2‍|u˙0(s)|2ds→0,as|t|→∞.
Hence, it is enough to prove that
(68)∫t-1/2t+1/2‍|u¨0(s)|2ds→0,as|t|→∞.
Since u0 is a solution of (HS), one has
(69)∫t-1/2t+1/2‍|u¨0(s)|2ds=∫t-1/2t+1/2‍|∇V(s,u0(s))|2ds.
Since ∇V(t,0)=0 for all t∈ℝ and u0(t)→0, as |t|→∞, (68) follows from (69).
Finally, similar to the proof in [12], we can prove that u0 is nontrivial, and we omit it here. The proof of Theorem 1 is complete.



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