# Homoclinic Orbits for Second-Order Hamiltonian Systems with Some Twist Condition

Authors: Qi Wang; Qingye Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systemsq¨−L(t)q+∇qW(t,q)=0, where L(t) is unnecessarily positive definite for all t∈ℝ, and ∇qW(t,q) is of at most linear growth and satisfies some twist condition between the origin and the infinity.


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## Body

## 1.  Introduction

Consider the following second-order non-autonomous Hamiltonian system(1.1)q̈-L(t)q+∇qW(t,q)=0,
where L(t)∈C(ℝ,ℝN×N) is a symmetric matrix-valued function, W:ℝ×ℝN→ℝ and ∇qW(t,q) denotes the gradient of W(t,q) with respect to q. As usual, we say that a nonzero solution q(t) of (1.1) is homoclinic (to 0) if q(t)→0 and q̇(t)→0 as |t|→∞.As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics and nuclear physics. While it is well known that homoclinic solutions play an important role in analyzing the chaos of Hamiltonian systems, if a system has the transversely intersected homoclinic solutions, then it must be chaotic. If it has the smoothly connected homoclinic solutions, then it cannot stand the perturbation, its perturbed system probably produces chaotic phenomena. For the chaos theory, the readers can refer to [1–3] and the references therein for more details. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from 0.In the past years, the existence and multiplicity of homoclinic orbits for (1.1) have been extensively investigated in many papers via the variational methods. Most of them (see [4–13]) treated the case where L(t) and W(t,u) are either independent of t or periodic in t. In this kind of problem, the function L(t) plays an important role. If L(t) is neither a constant nor periodic, the problem is quite different from the ones just described, because of the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka [13], many results (see, e.g., [9, 14–22]) were obtained for the case where L(t) is neither a constant nor periodic. Among them, except for [13, 16, 18, 20–22], all known results were obtained under the following assumption that L(t) is positive definite for all t∈ℝ, that is,
(1.2)(L(t)u,u)>0,∀t∈R,u∈RN∖{0}.In the present paper, we will study the existence and multiplicity of homoclinic orbits for (1.1) under the condition that L(t) is coercive but unnecessarily positive definite for all t∈ℝ. More precisely, L satisfies the following conditions:(L1)
There exists anα<2 such that
(1.3)l(t)|t|α-2⟶∞as|t|⟶∞,
where l(t) is the smallest eigenvalue of L(t), that is,
(1.4)l(t)≡inf⁡|ξ|=1(L(t)ξ,ξ).Before presenting the conditions on the nonlinearity of (1.1), we note that in the recent paper [23], under a twisting of the nonlinearity between the origin and the infinity, the authors studied the existence and multiplicity of nontrivial solutions for nonlinear elliptic equations and also for nonlinear elliptic systems. Subsequently, this kind of twist conditions and the idea of the methods in [23] were also applied to first-order Hamiltonian systems in [24].Inspired by these works, we will present some similar twist condition on the nonlinearity of (1.1) to those in [23, 24], which will be specified in what follows.Here, we introduce some notations. Denote byℬ the set of all uniformly bounded symmetric N×N matric functions. That is to say, B∈ℬ if and only if BT(t)=B(t) for all t∈ℝ and B(t) is uniformly bounded in t as the operator on ℝN. For any B∈ℬ, in the next section, we will define an index pair (i(B),ν(B)), satisfying 0≤i(B),ν(B)<∞.With this index, we can present the conditions onW(t,q) and the nonlinearity ∇qW(t,q) as follows. For notational simplicity, we set B0(t)=∇q2W(t,0), and in what follows the letter c will be repeatedly used to denote various positive constants whose exact value is irrelevant. Besides, for two N×N symmetric matrices M1 and M2, M1≤M2 means that M2-M1 is semipositive definite.(W1)
W∈C2(ℝ×ℝN,ℝ), and there exists a constant c>0 such that(1.5)|∇q2W(t,q)|≤c,∀(t,q)∈R×RN.(W0)
∇qW(t,0)≡0 and B0∈ℬ,(W∞)
there exists someR0>0 and continuous symmetric matrix functions B1,B2∈ℬ with i(B1)=i(B2) and ν(B2)=0 such that(1.6)B1(t)≤∇q2W(t,z)≤B2(t),∀t∈R,|z|>R0.
Our first result reads as follows.Theorem 1.1.
Assume(L1), (W1), (W0), and (W∞) hold. If
(1.7)i(B1)∉[i(B0),i(B0)+ν(B0)],
then (1.1) has at least one nontrivial homoclinic orbit. Moreover, if ν(B0)=0 and |i(B1)-i(B0)|≥N, the problem possesses at least two nontrivial homoclinic orbits.Condition(W∞) is a two-side pinching condition near the infinity, learning from the idea of [23, 24], we can relax (W∞) to condition (W∞±) as follows.(W∞±)
There exist someR0>0 and a continuous symmetric matrix function B∞∈ℬ with ν(B∞)=0 such that(1.8)±∇q2W(t,q)≥±B∞(t),∀t∈R,|q|>R0.The uniform boundary of∇q2W(t,q) displayed in condition (W1) can also be relaxed as(W1*)
W∈C2(ℝ×ℝN,ℝ), and there exists a constant c>0 such that
(1.9)|∇qW(t,q)|≤c|q|,∀(t,q)∈R×RN.(W1**)
For anyM>0, ∇q2W(t,q) is bounded on ℝ×[-M,M]n.On the other hand, we need some sharply twisted conditions than the above theorem, and we have the following theorems.Theorem 1.2.
Assume(L1), (W1*), (W1**), (W0), (W∞+)(or(W∞-)), and ν(B0)=0 hold. If i(B∞)≥i(B0)+2 (or i(B∞)≤i(B0)-2), then (1.1) has at least one nontrivial homoclinic orbit.Theorem 1.3.
Suppose that(L1), (W1*), (W1**), (W0), (W∞+)(or(W∞-)), and ν(B0)=0 are satisfied. If, in addition, W is even in q and i(B∞)≥i(B0)+2 (or i(B∞)≤i(B0)-2), then (1.1) has at least |i(B∞)-i(B0)|-1 pairs of nontrivial homoclinic orbits.Remark 1.4.
Note that the assumptionν(B∞)=0 in (W∞±) is not essential for our main results. For the case of (W∞+) with ν(B∞)≠0, let B̃∞=B∞-εIn×n with ε>0 small enough, where In×n is the identity map on ℝN, then i(B̃∞)=i(B∞) and ν(B̃∞)=0, and hence (W∞+) holds for B̃∞. Therefore, Theorems 1.2 and 1.3 still hold in this case. While for the case of (W∞-) with ν(B∞)≠0, if we replace i(B∞) by i(B∞)+ν(B∞) in Theorems 1.2 and 1.3, then similar results hold. Indeed, let B̃∞=B∞+εIn×n with ε>0 small enough such that i(B̃∞)=i(B∞)+ν(B∞) and ν(B̃∞)=0, then this case is also reduced to the case of (W∞-) for B̃∞ with ν(B̃∞)=0.Remark 1.5.
ChooseW¯(t,q)=W(t,q)-W(t,0) instead of W in (1.1), then conditions (W1*), (W1**), (W0), and (W∞+)(or(W∞-)) still hold for W¯, so we can always assume W(t,0)≡0.


## 2.  Preliminaries

Denote byÃ the self-adjoint extension of the operator -d2/dt2+L(t) with domain D(Ã)⊂L2≡L2(ℝ,ℝN). Let {E(λ):-∞<λ<∞} and |Ã| be the spectral resolution and the absolute value of Ã, respectively, and let |Ã|1/2 be the square root of |Ã| with domain D(|Ã|). Set U=I-E(0)-E(-0), where I is the identity map on L2. Then, U commutes with Ã, |Ã|, and |Ã|1/2, and Ã=U|Ã| is the polar decomposition of Ã. Let E=D(|Ã|1/2), and define on E the inner product and norm by(2.1)(u,v)0=(|Ã|1/2u,|Ã|1/2v)2+(u,v)2,‖u‖0=(u,u)01/2,
where (·,·)L2 denotes the usual inner product on L2(ℝ,ℝN). Then, E is a Hilbert space. It is easy to see that E is continuously embedded in W1(ℝ,ℝN). In fact, we further have the following lemmas.Lemma 2.1 (see [16], Lemma 2.2).
Suppose thatL satisfies (L1). Then, E is compactly embedded in Lp(ℝ,ℝN) with the usual norm ∥·∥Lp for any 1≤p∈(2/(3-α),∞].From [16], under the assumption (L1) on L and by Lemma 2.1, we know that Ã possesses a compact resolvent. Therefore, the spectrum σ(Ã) consists of only eigenvalues numbered in η1≤η2≤⋯→∞ (counted with multiplicity), and the corresponding system of eigenfunctions {en:n∈ℕ}(Ãen=ηnen) forms an orthogonal basis in L2.Let(2.2)n-=#{i∣λi<0},n0=#{i∣λi=0},n¯=n-+n0,E-=span⁡{e1,…,en-},E0=span⁡{en-+1,…,en¯},E+=span⁡{en¯+1,…}¯,
where the closure is taken with respect to the norm ∥·∥0. Then, one has the orthogonal decomposition E=E-⊕E0⊕E+ with respect to the inner product (·,·)0. Now, we introduce on E the following inner product and norm:(2.3)(u,v)E=(|Ã|1/2u,|Ã|1/2v)L2+(u0,v0)L2,‖u‖E=(u,u)E1/2,
where u,v∈E=E-⊕E0⊕E+ with u=u-+u0+u+ and v=v-+v0+v+ correspondingly. Clearly, norms ∥·∥E and ∥·∥0 are equivalent (cf. [16]). From now on, we take E with inner product (·,·)E and norm ∥·∥E as our working space.Remark 2.2.
Note that the decompositionE=E-⊕E0⊕E+ with respect to the inner product (·,·)0 is also orthogonal with respect to both inner products (·,·)E and (·,·)L2. In what follows, we always denote by E=E-⊕E0⊕E+ the orthogonal decomposition with respect to the inner products (·,·)E unless specified otherwise.In view of Lemma2.1 and the equivalence of the norms ∥·∥E and ∥·∥0, there exists a constant c∞>0 such that(2.4)∥q∥L∞≤c∞∥q∥E,∀q∈E.Define the quadratic forma on E by(2.5)a(u,v)=∫R((u̇,v̇)+(L(t)u,v))dt,∀u,v∈E.
Then by definition, we have(2.6)a(u,u)=((P+-P-)u,u)E=∥u+∥E2-∥u-∥E2
for all u=u-+u0+u+∈E=E-⊕E0⊕E+, where P±:E→E± are the respective orthogonal projections. Define the self-adjoint operators A:E→E and K:L2→E by(2.7)Au=u+-u-,∀u∈E,(Ku,v)E=(u,v)L2,∀u∈L2,v∈E,
it is easy to check that K is a compact operator and a(u,v)=(Au,v)E for all u,v in E.For anyB∈ℬ, it is easy to see B determines a bounded self-adjoint operator on L2, by z(t)↦B(t)z(t), for any z∈L2, we still denote this operator by B, then KB:E⊂L2→E is a self-adjoint compact operator on E and satisfies(2.8)(KBu,v)E=(Bu,v)L2,∀u,v∈E.
We decompose the space E as E-(B)⊕E0(B)⊕E+(B), so that A-KB is negatively definite on E-(B), null on E0(B), and positively definite on E+(B). From the definition of A and the compactness of KB, we know that E-(B) and E0(B) are both finite dimensional. Denote(2.9)i(B)=dim⁡(E-(B)),ν(B)=dim⁡(E0(B)).Define the functionalsΨ and Φ on E by(2.10)Ψ(u)=∫RW(t,u)dt,Φ(u)=12a(u,u)-Ψ(u)=12∫R((u̇,u̇)+(L(t)u,u))dt-Ψ(u).
Combining this with Lemma 2.1, we know that Ψ and Φ are both well defined. Furthermore, we have the following:Proposition 2.3.
Let(L1) and (W1) be satisfied. Then, Ψ∈C2(E,ℝ), and hence Φ∈C2(E,ℝ). Moreover, from the similarly argument in [13], one has all critical points of Φ on E are homoclinic orbits of (1.1).Lemma 2.4.
Let(L1), (W1), and (W∞) be satisfied. Then, one has(1)
Φ satisfies (PS) condition,(2)
Hj(E,Φ;ℝ)≅δj,rℝ,j=0,1,… for -a∈ℝ large enough, where r=i(B1).Proof.
Assume{un}⊂E with Φ′(un)→0 as n→∞. That is,
(2.11)∥Aun-K∇qW(t,un)∥E⟶0.
First, we prove {un} is bounded in E. For each ε∈(0,1), define Cn∈ℬ by
(2.12)Cn(t)={∫01∇q2W(t,sun)ds,|un(t)|≥R0ε,B1(t),|un(t)|<R0ε.
It is easy to verify that {Cn} satisfies
(2.13)B1(t)-ε(B1(t)+c⋅I)≤Cn(t)≤B2(t)+ε(c⋅I-B2(t)),∀t∈R,
where c is the constant in condition (W1) and I is the identity map on ℝN. Since B1≤B2, i(B1)=i(B2), and ν(B1)=ν(B2)=0, we can choose ε small enough, such that for each n∈ℕ+, satisfying i(Cn)=i(B1) and ν(Cn)=0. Thus A-KCn is reversible on E and there is a constant δ>0, such that
(2.14)∥(A-KCn)u∥E≥δ∥u∥E,∀u∈E,n∈N+.
On the other hand, for b∈(0,1), there is a constant c>0 depending on b, such that for each n∈ℕ+,
(2.15)|∇qW(t,un(t))-Cnun(t)|≤c|un(t)|b,∀t∈R.
Choose b>(α-1)/(3-α) in (2.15), we have
(2.16)‖(Aun-K∇qW(t,un))-(A-KCn)un‖E2=‖K(∇uW(t,un)-Cnun)‖E2≤‖∇uW(t,un)-Cnun‖L22≤c∫R|∇uW(t,un)-Cnun||un|b|un|1+bdt≤c‖un‖L1+b1+b.
As we claimed in the part of introduction, in (2.15) and (2.16), the letter c denotes different positive constants whose exact value is irrelevant. Thus, from (2.11), (2.14), (2.16), and Lemma 2.1, we have {un} in bounded in E. Thus, there exists a subsequence {unk} such that {K∇qW(t,unk)} is convergent in E. By the definition of Φ, we have
(2.17)unk+-unk--K∇qW(t,unk)=Φ′(unk)⟶0inE,
which yields the convergence of {unk+} and {unk-} in E. Additionally, passing to a subsequence, if necessary, {unk0} is convergent in E since dim(E0)<∞. Thus, {unk} is convergent and the (PS) condition is verified. By Lemma 5.1 in Chapter II of [25], we have
(2.18)Hq(X,(a)α;R)≅δq,rR,q=0,1,…,
for -α∈ℝ large enough.In order to prove Theorems1.2 and 1.3, we need the following lemma which is similar to Lemma  3.4 in [23] and Lemma  3.3 in [24].Lemma 2.5.
Assume(W1*), (W1**), and (W∞+) hold, then there exists a sequence of functions:
(2.19)Wm∈C2(R×RN,R),m∈N,
satisfying the following properties:(1)
there exists an increasing sequence of real numbersRm→∞(m→∞) such that
(2.20)Wm(t,q)≡W(t,q),∀t∈R,|q|≤Rm;(2)
for eachm∈ℕ, there is a cm>0 such that
(2.21)|∇q2Wm(t,q)|≤cm,∀t∈R,q∈RN,(2.22)∇q2Wm(t,q)≥B-ε:=B∞-εIn×n,∀t∈R,|q|≥R0
for ε>0 small enough;(3)
there exist somec>0 such that
(2.23)|∇qWm(t,q)|≤c|q|,∀(t,q)∈R×RN,m∈N;(4)
for eachm∈ℕ, there exists some Cm>0 and a constant γ with γIn×n>B∞, ν(γIn×n)=0 such that
(2.24)|∇qWm(t,q)-γq|<Cm,∀(t,q)∈R×RN,
where In×n is the identity map on ℝN.Proof.
Defineη:[0,∞)→ℝ by
(2.25)η(t)={0,0≤t<1,29(t-1)3-19(t-1)4,1≤t<2,1-1289(12+t2),2≤t<∞.
It is easy to see that η∈C2([0,∞),ℝ). Choose a sequence {Rm} of positive numbers such that R0<R1<R2<⋯<Rm<⋯→∞ as m→∞. For each m∈ℕ, let ηm(t)=η(t/Rm) and
(2.26)W̃m(t,q)=(1-ηm(|q|))W(t,q)+γ2ηm(|q|)|q|2,m∈N.
As in [23, 24], we can easily check that W̃m satisfies (2.20), (2.22) (with ε=0), (2.23) and (2.24) for each m∈ℕ.
Define(2.27)Wm(t,q)=ρ(|q|-TmTm)W̃m(t,q)+γ2[1-ρ(|q|-TmTm)]|q|2,
where ρ∈C2(ℝ,[0,1]) is a cut-off function with ρ(s)≡1 for s≤0 and ρ(s)≡0 for s≥1. If we choose Tm large enough, then Wm will satisfy (2.20)–(2.24).Remark 2.6.
Similar to Remark1.4, we can choose ε>0 small enough in (2.22) such that E-(B-ε)=E-(B∞) and E0(B-ε)=E0(B∞)={0}, that is, i(B-ε)=i(B∞) and ν(B-ε)=ν(B∞)=0.Remark 2.7.
For the case of(W∞-), the sequence {Wm} constructed in (2.26) will also satisfy (2.20), (2.21), (2.23), (2.24) (with γIn×n<B∞ and ν(γIn×n)=0) and
(2.28)∇q2Wm(t,q)≤Bε:=B∞+εIn×n,∀t∈R,|q|>R0,
with ε>0 small enough, therefore, i(Bε)=i(B∞) and ν(Bε)=ν(B∞)=0.Define(2.29)Ψm(u)=∫RWm(t,u)dt,∀u∈E,Φm(u)=12a(u,u)-Ψm(u),∀u∈E.
Form Lemma 2.1 and (2.21) in Lemma 2.5, we have Φm∈C2(E,ℝ). Similarly, we can define Ψγ(u)=1/2(γKu,u)E and Φγ(u)=1/2a(u,u)-Ψγ(u) for all u in E, and it follows that Φγ∈C2(E,ℝ).Lemma 2.8.
For eachm∈ℕ, Φm satisfies the (PS) condition and the critical-point set 𝒦m={z∈E∣Φm′(z)=0} is a compact set.Proof.
For anym∈ℕ, assume {un}⊂E with Φm′(un)→0 as n→∞:
(2.30)∥Φm′(un)-Φγ′′(0)un∥E=∥K(∇qWm(t,un)-γun)∥E≤∥∇qWm(t,un)-γun∥L2.
By (2.23) and (2.27), there hold
(2.31)∇qWm(t,q)-γq≡0fort∈R,|q|>2Tm,|∇qWm(t,q)-γq|<c|q|,∀(t,q)∈R×RN,
for some c>0. Choose b∈(0,1) with b>(α-1)/(3-α), then there exists a constant c>0 such that
(2.32)‖∇qWm(t,un)-γun‖L22=∫(∇qWm(t,un)-γun,∇qWm(t,un)-γun)dt≤c∫|∇qWm(t,un)-γun||un|b|un|1+bdt≤c‖un‖L1+b1+b.
Since α<2, ν(γIn×n)=0 and Ψγ′′(0) has bounded inverse, from Lemma 2.1 and (2.30), (2.32), we have {un} is bounded in E. From the similar argument in Lemma 2.4, {un} has a convergent subsequence and the (PS) condition is verified. From the same reason, we can also prove that 𝒦m is compact set.From Lemma II.5.1 in [25], by standard argument, we have the following.Lemma 2.9.
For anym∈ℕ, there is an am∈ℝ with -am large enough such that
(2.33)Hq(E,(Φm)am;R)=δqrR,
where r=dim(Eγ-)=i(γIn×n).Note thatθ is an isolated critical point of Φm since ν(B0)=0. For each m∈ℕ, let 𝒦m*=𝒦m∖{θ}, then 𝒦m* is also compact since 𝒦m is compact. Then we haveLemma 2.10.
For anyϵ,μ>0 small enough, for each m∈ℕ, there exists a functional Φ̃m such that(1)
∥Φm-Φ̃m∥C2<ϵ,(2)
Φ(z)=Φ̃m, z∉N2μ(𝒦m*),(3)
Φm′′(z)=Φ̃m′′(z),z∈Nμ(𝒦m*),where Nμ(𝒦m*)={z∈X∣dist(z,𝒦m*)<μ}. Moreover, Φ̃m satisfies the (PS) condition and has only a finite number of critical points, all nontrivial critical points of Φ̃m lie in Nμ(𝒦m*) and are nondegenerate.Proof.
We follow the idea of [26], since 𝒦m* is a compact subset of E, for every μ>0, there exists a C∞ function l:E→[0,1], with all its derivatives bounded and
(2.34)l(z)=1,∀z∈Nμ(Km*),l(z)=0,∀z∈E∖N2μ(Km*).
Let M=supz∈N2μ(𝒦m*){∥z∥},Cl=∥l(z)∥C2, δ=infz∈N2μ(𝒦m*)∖Nμ(𝒦m*){∥Φm′(z)∥}>0. We use the Sard-Smale Theorem to find y∈E, with ∥y∥<min{ϵ/Cl(2+2M),δ/2(Cl(1+2M))}, and -y is a regular value for Φm′. For any z0∈N2μ(𝒦m*), the functional is defined by
(2.35)Φ̃m(z)=Φm(z)+l(z)〈y,z-z0〉.
By the fact ∥y∥<ϵ/Cl(2+2M) and the definition of l(z), it is easy to check that conclusions (1), (2), and (3) hold. Since ∥y∥<δ/2Cl(1+2M) and -y is a regular value for Φm, then all nontrivial critical points of Φ̃m are nondegenerate and lie in Nμ(𝒦m*).
In order to prove thatΦ̃m satisfies the (PS) condition for each m∈ℕ, assume there is a sequence {zn}⊂E such that Φ̃m′(zn)→0, (n→∞). From the definition of Φ̃m, we have ∥Φ̃m′(z)∥>δ/2 for all z∈N2μ(𝒦m*)∖Nμ(𝒦m*). So zn∈(E∖N2μ(𝒦m*))∪Nμ(𝒦m*), when n is large enough. From the definition of Φ̃m and the proof in Lemma 2.8, we know that Φ̃m satisfies the (PS) condition and hence has a finite number of critical points.


## 3.  Proof of the Main Results

From Lemma2.4, Theorem 1.1 is a direct consequence of Theorem 5.1 and Corollary 5.2 in chapter II of [25].Proof of Theorem1.2.
We first consider the case of(W∞+). We divide the proof into two steps and follow the ideas of [24].Step 1.
We claim thatΦm has a nontrivial critical point zm with its Morse index satisfying
(3.1)m-(zm)≤i(B0)+1.
Note that z=0 is a critical point of Φm. The Morse index of 0 for Φm is i(B0), since γIn×n≥B∞,
(3.2)i(γIn×n)≥i(B∞)>i(B0)+1.
IfΦm has only finite critical points, consider (i(B0)+1)th Morse inequality:(3.3)∑p=0q(-1)q-pMp(am,bm,Φm)≥∑p=0q(-1)q-pβp(am,bm,Φm),
where q=i(B0)+1, and bm is large enough such that 𝒦m⊂Φm-1[am,bm].
By Lemmas2.8 and 2.9, we have(3.4)βp(am,bm,Φm)=rank⁡(Hp(E,(Φm)am))=δpr,
where r=i(γIn×n). Since i(γIn×n)>i(B0)+1, the right side of the inequality is equal to 0. If Φm has no nontrivial critical point with its Morse index less than i(B0)+1, the left side of the inequality is equal to −1, it is a contradiction.
IfΦm has infinitely many critical points. Assume for any z∈𝒦m*,(3.5)m-(z)>i(B0)+1,
then from Lemma 2.10, we can choose μ small enough such that(1)
0∉N2μ(𝒦m*), so 0 is also an isolated critical point of Φ̃m and has the same Morse index i(B0);(2)
for anyz∈Nμ(𝒦m*), m-(z) is the dimension of the negative subspace of Φ′′(z), satisfying m-(z)>i(B0)+1. (Because Φm is C2 continuous, we can assume this.)
From (3) in Lemma2.10, if z is a nontrivial critical point of Φ̃m, the Morse index mΦ̃m-(z) satisfies(3.6)mΦ̃m-(z)>i(B0)+1.
Then choose ãm satisfying N2μ(𝒦m*)∩(Φm)ãm=∅, that is, (Φm)ãm=(Φ̃m)ãm. So,
(3.7)Hq(E,(Φm)ãm;R)=Hq(E,(Φ̃m)ãm;R)=δqrR.
Then, Φ̃m will not satisfy the (i(B0)+1)th Morse inequality. It is a contradiction.Step 2.
We show that{zm} is bounded in L∞(ℝ,ℝN), so from the definition of Φm, zm is a nontrivial critical point of Φ for m large enough.
We prove it indirectly, assume∥zm∥L∞→∞, from (2.4), we have ∥zm∥E→∞. Denote vm=zm/∥zm∥E. Passing to a subsequence, we assume that for some v∈E,(3.8)vm⇀vinE,vm⟶vinL2,vm⟶va.e.inR.
Since zm satisfies z̈m(t)-L(t)zm(t)+∇qWm(t,zm)=0, we have
(3.9)‖zm‖E≤c‖zm‖L2,forsomec>0,
which implies ∥vm∥L2≥1/c for each m∈ℕ and thus ∥v∥L2≥1/c.
For eachm∈ℕ, let hm(t)=∇qWm(t,zm)/∥zm∥E. By (W0), (2.20), and (2.23), there holds |hm(t)|≤c|vm(t)| for all t∈ℝ. Assume hm(t)⇀h(t) in L2, then |h(t)|≤c|v(t)|, almost everywhere in ℝ. By standard argument, we have(3.10)v̈-L(t)v+h(t)=0,∀t∈R.
By Lemma  3.1 in [23], we have v(t)≠0 almost everywhere in ℝ, which implies zm(t)→∞almost everywhere in ℝ. For any u∈E-(B-ε)=E-(B∞),
(3.11)(Φm′′(zm)u,u)=∫R(|u̇|2-(L(t)u,u)-∇q2Wm(t,zm(t))u2)dt.
By (W1**), (2.20), and (2.22), {∇q2Wm} is uniformly bounded from below. Using Fatou’s Lemma, we have
(3.12)liminf⁡m→∞∫R∇q2Wm(t,zm)u2dt≥∫Rliminf⁡m→∞∇q2Wm(t,zm)u2dt≥∫RB-ε(t)u2dt,
which implies
(3.13)limsup⁡m→∞(Φm′′(zm)u,u)≤∫R(|u̇|2-(L(t)u,u))dt-∫RB-ε(t)u2dt<0,
which contradicts to m-(zm)≤i(B0)+1<i(B-ε). So {zm} is bounded in L∞(ℝ,ℝN), and Φ has a nontrivial critical point.
In order to prove the case of(W∞-), we choose γ with γIn×n≤B∞ and ν(γIn×n)=0 in Lemma 2.5, then, using the similar argument as in Step 1 for the case of (W∞+), we can also prove that there exists a nontrivial critical point zm of Φm satisfying(3.14)m-(zm)+ν(zm)≥i(B∞)+1.
As in Step 2 for the case of (W∞+), we will show that {zm} is bounded in L∞(ℝ,ℝN). We prove it indirectly, assume ∥zm∥L∞→∞, zm(t)→∞ almost everywhere in ℝ. We claim there exists m0∈ℕ such that if m>m0, then for any u∈E+(B∞)∖{0},
(3.15)(Φm′′(zm)u,u)E>0,
which implies m-(zm)+ν(zm)≤i(B∞). If (3.15) is false, then there exist mj→∞ and uj∈E+(B∞) with ∥uj∥E=1 such that (Φm′′(zmj)uj,uj)E≤0, which can be rewritten as
(3.16)(Auj,uj)E-(K∇q2Wmj(t,zmj)uj,uj)E≤0,(3.17)1-2∥uj-∥E2≤(K∇q2Wmj(t,zmj)uj,uj)E.
Passing to a subsequence if necessary, we assume that uj→u in L2 and almost everywhere in ℝ for some u∈E+(B∞)=E+(Bε) with ∥u∥E≤1. Since
(3.18)liminf⁡j→∞∫R-∇q2Wmj(t,zmj)uj2dt≥∫Rliminf⁡j→∞-∇q2Wmj(t,zmj)uj2dt≥-(Bεu,u)L2,
we see that
(3.19)limsup⁡j→∞(K∇q2Wmj(t,zmj)uj,uj)E≤(Bεu,u)L2.
From (3.17) and (3.19), we have
(3.20)1-2‖u-‖E2≤(Bεu,u)L2,
If u=0, it is a contradiction, or we have
(3.21)1-2‖u-‖E2≤(Bεu,u)L2<(Au,u)E=‖u+‖E2-‖u-‖E2≤1-2‖u-‖E2,
which is also a contradiction. Thus, {zm} is bounded in L∞(ℝ,ℝN) and Φ has a nontrivial critical point. By Proposition 2.3, (1.1) has a nontrivial homoclinic orbit. The proof is completed.The proof of Theorem1.3 is similar to the proof of Theorem 1.2. The difference is in Step 1, instead of Morse theory we make use of minimax arguments for multiplicity of critical points.LetX be a Hilbert space and assume ϕ∈C2(X,ℝ) is an even functional, satisfying the (PS) condition and ϕ(0)=0. Denote Sc={u∈X∣∥u∥=c}.Lemma 3.1 (see [27], Corollary 10.19).
AssumeY and Z are subspaces of X satisfying dimY=j>k=codimZ. If there exist R>r>0 and α>0 such that
(3.22)inf⁡ϕ(Sr∩Z)≥α,sup⁡ϕ(SR∩Y)≤0,
then ϕ has j-k pairs of nontrivial critical points {±x1,±x2,…,±xj-k}, so that μ(ui)≤k+i, for i=1,2,…,j-k.First, we consider the case of(W∞+), since W is even, we have Wm is also even, and it satisfies Lemma 2.5. Let Y=E-(B∞), and Z=E+(B0), and we have dimY=i(B∞), codimZ=i(B0), dimY>codimZ. Then, it is easy to prove that Φm satisfies Lemma 3.1 for R and 1/r large enough. So, Φm has l:=iA(B∞)-iA(B0) pairs of nontrivial critical points:(3.23){±x1,±x2,…,±xl},
and l-1 pairs of them satisfying(3.24)m-(xi)≤i(B0)+i<i(B∞),i=1,2,…,l-1.
Then, we can complete the proof. In order to prove the case of (W∞-), we need the following.Lemma 3.2 (see [25], Corollary II 4.1).
AssumeY and Z are subspaces of X satisfying dimY=j>k=codimZ. If there exist r>0, and α>0 such that
(3.25)inf⁡ϕ(Z)>-∞,sup⁡ϕ(Sr∩Y)≤-α,
then ϕ has j-k pairs of nontrivial critical points ±u1,±u2,…,±uj-k so that μ(ui)+ν(ui)≥k+i-1 for i=1,2,…,j-k.The proof is similar to the case of(W∞+), we omit it here.



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*Source: 250607-2012-06-07.xml*