# Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

Authors: Qi Wang; Qingye Zhang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.


---

## Body

## 1. Introduction and Main Results

Consider the following first-order nonautonomous Hamiltonian systems(HS)z˙=JHz(t,z),
where z:ℝ→ℝ2N, J=(0-ININ0), H∈C1(ℝ×ℝ2N,ℝ) and ∇zH(t,z) denotes the gradient of H(t,z) with respect to z. As usual we say that a nonzero solution z(t) of (HS) is homoclinic (to 0) if z(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. However 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, and its perturbed system probably produces chaotic phenomena. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from 0.In the last years, the existence and multiplicity of homoclinic orbits for the first-order system(HS) were studied extensively by means of critical point theory, and many results were obtained under the assumption that H(t,z) depends periodically on t (see, e.g., [1–12]). Without assumptions of periodicity, the problem is quite different in nature and there is not much work done so far. To the best of our knowledge, the authors in [13] firstly obtained the existence of homoclinic orbits for a class of first-order systems without any periodicity on the Hamiltonian function. After this, there were a few papers dealing with the existence and multiplicity of homoclinic orbits for the first-order system (HS) in this situation (see, e.g., [14–17]).In the present paper, with the Maslov index theory of homoclinic orbits introduced by Chen and Hu in [18], we will study the existence and multiplicity of homoclinic orbits for (HS) without any periodicity on the Hamiltonian function. To the best of the author's knowledge, the Maslov index theory of homoclinic orbits is the first time to be used to study the existence of homoclinic solutions. We are mainly interested in the Hamiltonian functions of the form
(1)H(t,z)=-L(t)z·z+R(t,z),
where L is a 2N×2N symmetric matrix valued function. We assume that (L1)L∈C(ℝ,ℝN2), and there are α,c>0, t0≥0 and a constant matrix P, satisfying
(2)PL(t)-c|t|αI2N≥0,∀|t|≥t0,
where I2N is the identity map on ℝ2N, and for a 2N×2N matrix M, we say that M≥0 if and only if
(3)infξ∈ℝ2N,|ξ|=1Mξ·ξ≥0.
In (L1), if P=(0ININ0), then (L1) is similar to the condition (R0) in [14]. But the restrictions on R(t,z) will be different from [14], and we will give some examples in Remark 5. If P=±I2N or P=(IN+m00-IN-m) in condition (L1), for example, it is quite different from the existing results as authors known. In short, condition (L1) means that the eigenvalues of L(t) will tend to ±∞ with the speed no less than |t|α. But (L1) does not contain all of these cases. For examples, let N=1 and L(t)=|t|α(cos2tsin2tsin2t-cos2t), we have the eigenvalues of L(t) are ±|t|α, but there is no constant matrix P satisfying PL(t)-c|t|αI2N≥0,  for all  |t|≥t0.Denote byF~ the self-adjoint operator -𝒥(d/dt)+L(t) on L2≡L2(ℝ,ℝ2N), with domain D(F~)=H1(ℝ,ℝ2N) if L(t) is bounded and D(F~)⊂H1(ℝ,ℝ2N) if L(t) is unbounded. Let |F~| be the absolute value of F~, and let |F~|1/2 be the square root of |F~|. D(F~) is a Hilbert space equipped with the norm
(4)∥z∥F~=∥(I+|F~|)z∥L2,∀z∈D(F~).
Let E=D(|F~|1/2), and define on E the inner product and norm by
(5)(u,v)E=(|F~|1/2u,|F~|1/2v)2+(u,v)2,∥u∥E=(u,u)E1/2,
where (·,·)L2 denotes the usual inner product on L2(ℝ,ℝ2N). Then, E is a Hilbert space. It is easy to see that E is continuously embedded in H1/2(ℝ,ℝ2N), and we further have the following lemma.Lemma 1.
Suppose thatL satisfies (L1). Then E is compactly embedded in Lp(ℝ,ℝ2N) with the usual norm ∥·∥Lp for any 1≤p∈(2/(1+α),∞).This lemma is similar to Lemmas  2.1–2.3 in [13], and we will prove it in Section 3. Define the quadratic form 𝒬 on E by
(6)𝒬(u,v)=∫ℝ‍((-Ju˙,v)+(L(t)u,v))dt,∀u,v∈E.
It is easy to check that 𝒬(u,v) is a bounded quadratic form on E, and, hence, there exists a unique bounded self-adjoint operator F:E→E such that
(7)(Fu,v)E=𝒬(u,v),∀u,v∈E.
Besides, define a linear operator K:L2→E by
(8)(Ku,v)E=(u,v)L2,∀u∈L2,v∈E.
In view of Lemma 1, we know that F is a Fredholm operator and K is a compact operator.Denote byℬ the set of all uniformly bounded symmetric 2N×2N 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 ℝ2N. For any B∈ℬ, it is easy to see that 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 and then KB:E⊂L2→E is a self-adjoint compact operator on E and satisfies
(9)(KBu,v)E=(Bu,v)L2,∀u,v∈E.Before presenting the conditions onR(t,z), we need the concept of Maslov index for homoclinic orbits introduced by Chen and Hu in [18] which is equivalent to the relative Morse index. We will give a brief introduction of it by Definition 7, where for any B∈ℬ, we denote the associated index pair by (μF(KB), υF(KB)).Now we can present the conditions onR(t,z) as follows. For notational simplicity, we set B0(t)=∇z2R(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 2N×2N symmetric matrices M1 and M2, M1≤M2 means that M2-M1 is semipositive definite.(R1)
R∈C2(ℝ×ℝ2N,ℝ), and there exists a constant c>0 such that
(10)|∇z2R(t,z)|≤c,∀(t,z)∈ℝ×ℝ2N.(R0)
∇zR(t,0)≡0 and B0∈ℬ.(R∞)
There exists someR0>0 and continuous symmetric matrix functions B1,B2∈ℬ with μF(KB1)=μF(KB2) and υF(KB2)=0 such that
(11)B1(t)≤∇z2R(t,z)≤B2(t),∀t∈ℝ,|z|>R0.Then, we have our first result.Theorem 2.
Assume that(L1), (R1), (R0), and (R∞) hold. If
(12)μF(KB1)∉[μF(KB0),μF(KB0)+υF(KB0)],
then (HS) has at least one nontrivial homoclinic orbit. Moreover, if υF(KB0)=0 and |μF(KB1)-μF(KB0)|≥N, the problem possesses at least two nontrivial homoclinic orbits.Condition(R∞) is a two-side pinching condition near the infinity, and we can relax (R∞) to condition (R∞±) as follows. (R∞±)
There exist someR0>0 and a continuous symmetric matrix function B∞∈ℬ with υF(KB∞)=0 such that
(13)±∇z2R(t,z)≥±B∞(t),∀t∈ℝ,|z|>R0.Then, we have the following results.Theorem 3.
Assumes(L1), (R1), (R0), (R∞+) (or (R∞-)), and υF(KB0)=0 hold. If μF(KB∞)≥μF(KB0)+2 (or μF(KB∞)≤μF(KB0)-2), then (HS) has at least one nontrivial homoclinic orbit.Theorem 4.
Suppose that(L1), (R1), (R0), (R∞+) (or (R∞-)), and υF(KB0)=0 are satisfied. If in addition R is even in z and μF(KB∞)≥μF(KB0)+2 (or μF(KB∞)≤μF(KB0)-2), then (HS) has at least |μF(KB∞)-μF(KB0)|-1 pairs of nontrivial homoclinic orbits.Remark 5.
Lemma1 shows that σ(A), the spectrum of A, consists of eigenvalues numbered by (counted in their multiplicities):
(14)⋯≤λ-2≤λ-1≤0<λ1≤λ2≤⋯
with λ±k→±∞ as k→∞. Let B0(t)≡B0 and B∞(t)≡B∞, with the constants B0, B∞ satisfying λl<B0<λl+1, and λl+i<B∞<λl+i+1 for some l∈ℤ and i≥1 (or i≤-1). Define
(15)R(t,z)=δ(|z|)12B0|z|2+(1-δ(|z|))12B∞|z|2,
where δ is a smooth cutoff function satisfying δ(|z|)={1,|z|<1,0,|z|>2. By Proposition 12 that, it is easy to verify R satisfies all the conditions in Theorem 2. Furthermore, let the constant B∞ satisfy λl+i<B∞<λl+i+1 for some l∈ℤ and i≥2 (or i≤-2). Define
(16)R(t,z)=δ(|z|)12B0|z|2+(1-δ(|z|))12B∞|z|2.
Then R satisfies all the conditions in Theorems 3 and 4. However, it is easy to see that some conditions of the main results in [13–15, 17] do not hold for these examples.Remark 6.
Note that the assumptionυF(KB∞)=0 in (R∞±) is not essential for our main results. For the case of (R∞+) with υF(KB∞)≠0, let B~∞=B∞-εI2N with ε>0 small enough, where I2N is the identity map on ℝ2N; then, μF(KB~∞)=μF(KB∞) and υF(KB~∞)=0, and, hence, (R∞+) holds for B~∞. Therefore, Theorems 3 and 4 still hold in this case. While for the case of (R∞-) with υF(KB∞)≠0, if we replace μF(KB∞) by μF(KB∞)+υF(KB∞) in Theorems 3 and 4, then similar results hold. Indeed, let B~∞=B∞+εI2N with ε>0 small enough such that μF(KB~∞)=μF(KB∞)+υF(KB∞) and υF(KB~∞)=0, then this case is also reduced to the case of (W∞-) for B~∞ with υF(KB~∞)=0.


## 2. Preliminaries

In this section, we recall the definition of relative Morse index and saddle point reduction and give the relationship between them. For this propose, the notion of spectral flow will be used.


### 2.1. Relative Morse Index

Letℋ be a separable Hilbert space; for any self-adjoint operator A on ℋ, there is a unique A-invariant orthogonal splitting
(17)ℋ=ℋ+(A)⊕ℋ-(A)⊕ℋ0(A),
where ℋ0(A) is the null space of A, A is positive definite on ℋ+(A) and negative definite on ℋ-(A), and PA denotes the orthogonal projection from ℋ to ℋ-(A). For any bounded self-adjoint Fredholm operator ℱ and a compact self-adjoint operator 𝒯 on ℋ, Pℱ-Pℱ-𝒯 is compact (see [19, Lemma  2.7]), where Pℱ:ℋ→ℋ-(ℱ) and Pℱ-𝒯:ℋ→ℋ-(ℱ-𝒯) are the respective projections. Then, by Fredholm operator theory, Pℱ|ℋ-(ℱ-𝒯):ℋ-(ℱ-𝒯)→ℋ-(ℱ) is a Fredholm operator. Here and in the sequel, we denote by ind(·) the Fredholm index of a Fredholm operator.Definition 7.
For any bounded self-adjoint Fredholm operatorℱ and a compact self-adjoint operator 𝒯 on ℋ, the relative Morse index pair (μℱ(𝒯),υℱ(𝒯)) is defined by
(18)μℱ(𝒯)=ind(Pℱ|ℋ-(ℱ-𝒯)),υℱ(𝒯)=dimℋ0(ℱ-𝒯).


### 2.2. Saddle Point Reduction

In this subsection, we describe the saddle point reduction in [20–22]. Recall that ℋ is a real Hilbert space, and A is a self-adjoint operator with domain D(A)⊂ℋ. Let Φ∈C1(ℋ,ℝ), with Φ′(θ)=0. Assume that  (1)
there exist real numbersα<β such that α,β∉σ(A), and that σ(A)∩[α,β] consists of at most finitely many eigenvalues of finite multiplicities;(2)
Φ′ is Gateaux differentiable in ℋ, which satisfies
(19)∥dΦ′(u)-α+β2I∥≤β-α2,∀u∈ℋ,
and without loss of generality, we may assume that α=-β,β>0;(3)
Φ∈C2(V,ℝ),  V=D(|A|1/2), with the norm
(20)∥z∥V=(∥|A|1/2z∥ℋ2+ε2∥z∥ℋ2)1/2,
where ε>0 small and -ε∉σ(A).Consider the solutions of the following equation:(21)Az=Φ′(z),z∈D(A).
Let
(22)P0=∫-ββ‍dEλ,P+=∫β+∞‍dEλ,P-=∫-∞-β‍dEλ,
where {Eλ} is the spectral resolution of A, and let
(23)ℋ*=P*ℋ,*=0,±.
Decompose the space V as follows:
(24)V=V0⊕V-⊕V+,
where V*=|Aε|-1/2ℋ*,*=0,±, and Aε=A+εI.For eachu∈ℋ, we have the decomposition
(25)u=u++u0+u-,
where u*∈ℋ* and *=0,±; let z=z++z0+z-, with
(26)z*=|Aε|-1/2u*,*=0,±.
Define a functional f on ℋ as follows:
(27)f(u)=12(∥u+∥2+Q+∥u0∥2-Q-∥u0∥2-∥u∥2)-Φε(z),
where Q+=∫0∞‍dEλ,  Q-=∫-∞0‍dEλ, and Φε(z)=(ε/2)∥z∥ℋ+Φ(z).The Euler equation of this functional is the system(28)u±=±|Aε|-1/2P±Φε′(z),(29)Q±u0=±|Aε|-1/2Q±P0Φε′(z).
Thus, z=z++z0+z- is a solution of (21) if and only if u=u++u0+u- is a critical point of f. The implicit function can be applied, yielding a solution z±(z0) for fixed z0∈V0, such that z±∈C1(V0,V±). Since dimV0 is finite, all topologies on V0 are equivalent, and we choose ∥·∥ℋ as it norm. We have
(30)u±(z0)=|Aε|1/2z±(z0)∈C1(ℋ0,ℋ),
which solves the system (28).Let(31)a(z0)=f(u+(z0)+u-(z0)+u0(z0)),
where u0(z0)=|Aε|1/2z0, and let z0=x we have
(32)a(x)=12(A(z(x),z(x)))-Φ(z(x)),
where  z(x)=ξ(x)+x, ξ(x)=z+(x)+z-(x)∈D(A). Then, we have the following theorem due to Amann [20], Chang [21], and Long [22].Theorem 8.
Under assumptions (1), (2), and (3), there is a one-one correspondence(33)x↦z=z(x)=z+(x)+z-(x)+x
between the critical points of the C2-function a∈C2(ℋ0,ℝ) with the solutions of the operator equation
(34)Az=Φ′(z),z∈D(A).
Moreover, the functional a satisfies
(35)a(x)=12(A(z(x),z(x)))-Φ(z(x)),a′(x)=A(z(x))-Φ′(z(x))=Ax-P0Φ′(z(x)),a′′(x)=[A-Φ′′(z(x))]z′(x)=AP0-P0Φ′′(z(x))z′(x).Sinceℋ0 is a finite dimensional space, for every critical point x of a in ℋ0, the Morse index and nullity are finite, and we denote them by (ma-(x),ma0(x)).Now, let the Hilbert spaceℋ be L2(ℝ,ℝ2N), and the operator A be F~=-J(d/dt)+L, Φ(u)=∫-∞∞‍R(t,u). Then we have V=E. For R∈C2(ℝ×ℝ2N,ℝ) and |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N, let -α=β≥2(CR+1) and β∉σ(F~); we have A and Φ satisfying the previous conditions. Thus, from Theorem 8, we can solve our problems on the finite dimensional space. Similar to Lemma  2.2 and Remark  2.3 in [23], we have the following estimates.Lemma 9.
Assume thatR∈C2(ℝ×ℝ2N,ℝ), |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N and ∇zR(t,0)≡0; then one has
(36)∥u±(x)∥L2≤2β(CR+1)β-2CR-3ε∥x∥L2,∀x∈ℋ0.
Moreover, one has
(37)∥(u±)′(x)∥L2⟶0,β⟶∞.Proof.
Note that(38)u±(x)=±|Aε|-1/2P±Φε′(z++z-+x).
From ∇zR(t,0)=0, |∇z2R|≤CR, we have Φ′(0)=0 and ∥Φ′(z)∥L2≤CR∥z∥L2. Since ∥|Aε|-1/2P±∥≤1/(β-ε), we have
(39)∥u±(x)∥L2≤1β-ε∥Φ′(z++z-+x)+ε(z++z-+x)∥L2≤CR+εβ-ε∥z++z-+x∥L2≤CR+εβ-ε(∥u+(x)∥L2β+∥u-(x)∥L2β+∥x∥L2).
Therefore,
(40)∥u+(x)∥L2+∥u-(x)∥L2≤2β(CR+ε)β-2CR-3ε∥x∥L2.
Next, since
(41)(u±)′(x)=±|Aε|-1/2P±Φε′×(z++z-+x)((z+)′(x)+(z-)′(x)+I),
where I is the identity map on ℋ0, we have
(42)∥(u+)′(x)∥L2+∥(u-)′(x)∥L2≤2β(CR+ε)β-2CR-3ε,∀x∈ℋ0.Remark 10.
Forz(x), we also have that there is a constant C>0 dependent of CR, but independent of β, such that
(43)∥z±(x)∥V≤Cβ∥x∥L2,∥z′±(x)∥V≤Cβ,∀x∈ℋ0.
IfR satisfies the condition (R1), then for any homoclinic orbit z of (HS), ∇z2R(·,z)∈ℬ, and, hence, we have the associated index pair (μF(KB), υF(KB)). For notation simplicity, in what follows, we set
(44)μF(z)=μF(K∇z2(R(t,z))),υF(z)=υF(K∇z2(R(t,z))).Theorem 11.
LetR∈C2(ℝ×ℝ2N,ℝ) satisfying |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N and ∇zR(t,0)≡0. For each critical point x of a in ℋ0, z(x) is a homoclinic orbit of (HS) and one has
(45)ma-(x)=dim(E-(ℋ0))+μF(K∇z2R(t,z(x)))=dim(E-(ℋ0))+μF(z(x)),ma0(x)=υF(K∇z2R(t,z(x))=υF(z(x)),
where dim(E-(ℋ0)) is the dimension of the space ∫-β0-‍dEλ(ℋ0).This theorem shows the relations between the relative Morse index and the Morse index of the saddle point reduction, and it will play an important role in the proof of our main results. The proof of this theorem will be postponed to the next subsection where the notion of spectral flow will be used.


### 2.3. The Relationship betweenμF(T), Spectral Flow, and the Morse Index of Saddle Point Reduction

It is well known that the concept of spectral flow was first introduced by Atiyah et al. in [24] and then extensively studied in [19, 25–28]. Here, we give a brief introduction of the spectral flow as introduced in [18]. Let ℋ be a separable Hilbert space as defined before, and {ℱθ∣θ∈[0,1]} be a continuous path of self-adjoint Fredholm operators on the Hilbert space ℋ. The spectral flow of ℱθ represents the net change in the number of negative eigenvalues of ℱθ as θ runs from 0 to 1, where the counting follows from the rule that each negative eigenvalue crossing to the positive axis contributes +1 and each positive eigenvalue crossing to the negative axis contributes -1, and for each crossing, the multiplicity of eigenvalue is taken into account. In the calculation of spectral flow, a crossing operator introduced in [28] will be used. Take a C1 path {ℱθθ∣∈[0,1]} and let 𝒫θ be the projection from ℋ to ℋ0(ℱθ). When eigenvalue crossing occurs at ℱθ, the operator
(46)𝒫θ∂∂θℱθ𝒫θ:ℋ0(ℱθ)⟶ℋ0(ℱθ)
is called a crossing operator, denoted by Cr[ℱθ]. As mentioned in [28], an eigenvalue crossing at ℱθ is said to be regular if the null space of Cr[ℱθ] is trivial. In this case, we define
(47)signCr[ℱθ]=dimℋ+(Cr[ℱθ])-dimℋ-(Cr[ℱθ]).
A crossing occurring at ℱθ is called simple crossing if dimℋ0(ℱθ)=1.As indicated in [19], the spectral flow Sf(ℱθ) will remain the same after a small disturbance of ℱθ, that is, Sf(ℱθ)=Sf(ℱθ+εid) for ε>0 and small enough, where id is the identity map on ℋ. Furthermore, we can choose suitable ε such that all the eigenvalue crossings occurring in ℱθ, 0≤θ≤1 are regular [28]. Thus, without loss of generality, we may assume that all the crossings are regular. Let 𝒟 be the set containing all the points in [0,1] at which the crossing occurs. The set 𝒟 contains only finitely many points. The spectral flow of ℱθ is
(48)Sf(ℱθ,0≤θ≤1)=∑θ∈𝒟*‍signCr[ℱθ]-dimℋ-(Cr[ℱ0])=+dimℋ+(Cr[ℱ1]),
where 𝒟*=𝒟∩(0,1). In what follows, the spectral flow of ℱθ will be simply denoted by Sf(ℱθ) when the starting and end points of the flow are clear from the contents. And Pℱθ will be simply denoted by Pθ.Proposition 12 (see [18, Proposition  3]).
Suppose that, for eachθ∈[0,1], ℱθ-ℱ0 is a compact operator on ℋ then
(49)ind(P0|ℋ-(ℱ1))=-Sf(ℱθ).Thus, from Definition7,
(50)μℱ0(𝒯)=-Sf(ℱθ,0≤θ≤1),
where ℱθ=ℱ-θ𝒯,  𝒯 is a compact operator. Moreover, if σ(𝒯)⊂[0,∞) and 0∉σP(𝒯), from the definition of spectral flow, we have
(51)μℱ0(𝒯)=-Sf(ℱθ,0≤θ≤1)=∑θ∈[0,1)‍υℱ(θ𝒯)=∑θ∈[0,1)‍dimℋ0(ℱ-θ𝒯).
The proof of Theorem 11 is the direct consequence of the aformentioned Proposition 12 and Theorem  3.2 in [19], so we omit it here.Remark 13.
The case ofR∞- can be transformed into the case of R∞+. More concretely, the R∞- case follows from the R∞+ case by applying to the function R~(t,z)=-R(-t,z). If z(t) is a homoclinic solution of F~z(t)=∇zR(t,z(t)), let z~(t)=z(-t), it is easy to check that z~(t) is a homoclinic solution of F~z~(t)=∇zR~(t,z~(t)), and this is a one-one correspondence between the two systems. By the definition of spectral flow and it is catenation property [19], we have μF(-B∞(-t))-μF(-B0(-t))=μF(B0(t))-μF(B∞(t)). Thus, we only consider the case of R∞+ from now on.


## 2.1. Relative Morse Index

Letℋ be a separable Hilbert space; for any self-adjoint operator A on ℋ, there is a unique A-invariant orthogonal splitting
(17)ℋ=ℋ+(A)⊕ℋ-(A)⊕ℋ0(A),
where ℋ0(A) is the null space of A, A is positive definite on ℋ+(A) and negative definite on ℋ-(A), and PA denotes the orthogonal projection from ℋ to ℋ-(A). For any bounded self-adjoint Fredholm operator ℱ and a compact self-adjoint operator 𝒯 on ℋ, Pℱ-Pℱ-𝒯 is compact (see [19, Lemma  2.7]), where Pℱ:ℋ→ℋ-(ℱ) and Pℱ-𝒯:ℋ→ℋ-(ℱ-𝒯) are the respective projections. Then, by Fredholm operator theory, Pℱ|ℋ-(ℱ-𝒯):ℋ-(ℱ-𝒯)→ℋ-(ℱ) is a Fredholm operator. Here and in the sequel, we denote by ind(·) the Fredholm index of a Fredholm operator.Definition 7.
For any bounded self-adjoint Fredholm operatorℱ and a compact self-adjoint operator 𝒯 on ℋ, the relative Morse index pair (μℱ(𝒯),υℱ(𝒯)) is defined by
(18)μℱ(𝒯)=ind(Pℱ|ℋ-(ℱ-𝒯)),υℱ(𝒯)=dimℋ0(ℱ-𝒯).


## 2.2. Saddle Point Reduction

In this subsection, we describe the saddle point reduction in [20–22]. Recall that ℋ is a real Hilbert space, and A is a self-adjoint operator with domain D(A)⊂ℋ. Let Φ∈C1(ℋ,ℝ), with Φ′(θ)=0. Assume that  (1)
there exist real numbersα<β such that α,β∉σ(A), and that σ(A)∩[α,β] consists of at most finitely many eigenvalues of finite multiplicities;(2)
Φ′ is Gateaux differentiable in ℋ, which satisfies
(19)∥dΦ′(u)-α+β2I∥≤β-α2,∀u∈ℋ,
and without loss of generality, we may assume that α=-β,β>0;(3)
Φ∈C2(V,ℝ),  V=D(|A|1/2), with the norm
(20)∥z∥V=(∥|A|1/2z∥ℋ2+ε2∥z∥ℋ2)1/2,
where ε>0 small and -ε∉σ(A).Consider the solutions of the following equation:(21)Az=Φ′(z),z∈D(A).
Let
(22)P0=∫-ββ‍dEλ,P+=∫β+∞‍dEλ,P-=∫-∞-β‍dEλ,
where {Eλ} is the spectral resolution of A, and let
(23)ℋ*=P*ℋ,*=0,±.
Decompose the space V as follows:
(24)V=V0⊕V-⊕V+,
where V*=|Aε|-1/2ℋ*,*=0,±, and Aε=A+εI.For eachu∈ℋ, we have the decomposition
(25)u=u++u0+u-,
where u*∈ℋ* and *=0,±; let z=z++z0+z-, with
(26)z*=|Aε|-1/2u*,*=0,±.
Define a functional f on ℋ as follows:
(27)f(u)=12(∥u+∥2+Q+∥u0∥2-Q-∥u0∥2-∥u∥2)-Φε(z),
where Q+=∫0∞‍dEλ,  Q-=∫-∞0‍dEλ, and Φε(z)=(ε/2)∥z∥ℋ+Φ(z).The Euler equation of this functional is the system(28)u±=±|Aε|-1/2P±Φε′(z),(29)Q±u0=±|Aε|-1/2Q±P0Φε′(z).
Thus, z=z++z0+z- is a solution of (21) if and only if u=u++u0+u- is a critical point of f. The implicit function can be applied, yielding a solution z±(z0) for fixed z0∈V0, such that z±∈C1(V0,V±). Since dimV0 is finite, all topologies on V0 are equivalent, and we choose ∥·∥ℋ as it norm. We have
(30)u±(z0)=|Aε|1/2z±(z0)∈C1(ℋ0,ℋ),
which solves the system (28).Let(31)a(z0)=f(u+(z0)+u-(z0)+u0(z0)),
where u0(z0)=|Aε|1/2z0, and let z0=x we have
(32)a(x)=12(A(z(x),z(x)))-Φ(z(x)),
where  z(x)=ξ(x)+x, ξ(x)=z+(x)+z-(x)∈D(A). Then, we have the following theorem due to Amann [20], Chang [21], and Long [22].Theorem 8.
Under assumptions (1), (2), and (3), there is a one-one correspondence(33)x↦z=z(x)=z+(x)+z-(x)+x
between the critical points of the C2-function a∈C2(ℋ0,ℝ) with the solutions of the operator equation
(34)Az=Φ′(z),z∈D(A).
Moreover, the functional a satisfies
(35)a(x)=12(A(z(x),z(x)))-Φ(z(x)),a′(x)=A(z(x))-Φ′(z(x))=Ax-P0Φ′(z(x)),a′′(x)=[A-Φ′′(z(x))]z′(x)=AP0-P0Φ′′(z(x))z′(x).Sinceℋ0 is a finite dimensional space, for every critical point x of a in ℋ0, the Morse index and nullity are finite, and we denote them by (ma-(x),ma0(x)).Now, let the Hilbert spaceℋ be L2(ℝ,ℝ2N), and the operator A be F~=-J(d/dt)+L, Φ(u)=∫-∞∞‍R(t,u). Then we have V=E. For R∈C2(ℝ×ℝ2N,ℝ) and |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N, let -α=β≥2(CR+1) and β∉σ(F~); we have A and Φ satisfying the previous conditions. Thus, from Theorem 8, we can solve our problems on the finite dimensional space. Similar to Lemma  2.2 and Remark  2.3 in [23], we have the following estimates.Lemma 9.
Assume thatR∈C2(ℝ×ℝ2N,ℝ), |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N and ∇zR(t,0)≡0; then one has
(36)∥u±(x)∥L2≤2β(CR+1)β-2CR-3ε∥x∥L2,∀x∈ℋ0.
Moreover, one has
(37)∥(u±)′(x)∥L2⟶0,β⟶∞.Proof.
Note that(38)u±(x)=±|Aε|-1/2P±Φε′(z++z-+x).
From ∇zR(t,0)=0, |∇z2R|≤CR, we have Φ′(0)=0 and ∥Φ′(z)∥L2≤CR∥z∥L2. Since ∥|Aε|-1/2P±∥≤1/(β-ε), we have
(39)∥u±(x)∥L2≤1β-ε∥Φ′(z++z-+x)+ε(z++z-+x)∥L2≤CR+εβ-ε∥z++z-+x∥L2≤CR+εβ-ε(∥u+(x)∥L2β+∥u-(x)∥L2β+∥x∥L2).
Therefore,
(40)∥u+(x)∥L2+∥u-(x)∥L2≤2β(CR+ε)β-2CR-3ε∥x∥L2.
Next, since
(41)(u±)′(x)=±|Aε|-1/2P±Φε′×(z++z-+x)((z+)′(x)+(z-)′(x)+I),
where I is the identity map on ℋ0, we have
(42)∥(u+)′(x)∥L2+∥(u-)′(x)∥L2≤2β(CR+ε)β-2CR-3ε,∀x∈ℋ0.Remark 10.
Forz(x), we also have that there is a constant C>0 dependent of CR, but independent of β, such that
(43)∥z±(x)∥V≤Cβ∥x∥L2,∥z′±(x)∥V≤Cβ,∀x∈ℋ0.
IfR satisfies the condition (R1), then for any homoclinic orbit z of (HS), ∇z2R(·,z)∈ℬ, and, hence, we have the associated index pair (μF(KB), υF(KB)). For notation simplicity, in what follows, we set
(44)μF(z)=μF(K∇z2(R(t,z))),υF(z)=υF(K∇z2(R(t,z))).Theorem 11.
LetR∈C2(ℝ×ℝ2N,ℝ) satisfying |∇z2R|≤CR, for all (t,z)∈ℝ×ℝ2N and ∇zR(t,0)≡0. For each critical point x of a in ℋ0, z(x) is a homoclinic orbit of (HS) and one has
(45)ma-(x)=dim(E-(ℋ0))+μF(K∇z2R(t,z(x)))=dim(E-(ℋ0))+μF(z(x)),ma0(x)=υF(K∇z2R(t,z(x))=υF(z(x)),
where dim(E-(ℋ0)) is the dimension of the space ∫-β0-‍dEλ(ℋ0).This theorem shows the relations between the relative Morse index and the Morse index of the saddle point reduction, and it will play an important role in the proof of our main results. The proof of this theorem will be postponed to the next subsection where the notion of spectral flow will be used.


## 2.3. The Relationship betweenμF(T), Spectral Flow, and the Morse Index of Saddle Point Reduction

It is well known that the concept of spectral flow was first introduced by Atiyah et al. in [24] and then extensively studied in [19, 25–28]. Here, we give a brief introduction of the spectral flow as introduced in [18]. Let ℋ be a separable Hilbert space as defined before, and {ℱθ∣θ∈[0,1]} be a continuous path of self-adjoint Fredholm operators on the Hilbert space ℋ. The spectral flow of ℱθ represents the net change in the number of negative eigenvalues of ℱθ as θ runs from 0 to 1, where the counting follows from the rule that each negative eigenvalue crossing to the positive axis contributes +1 and each positive eigenvalue crossing to the negative axis contributes -1, and for each crossing, the multiplicity of eigenvalue is taken into account. In the calculation of spectral flow, a crossing operator introduced in [28] will be used. Take a C1 path {ℱθθ∣∈[0,1]} and let 𝒫θ be the projection from ℋ to ℋ0(ℱθ). When eigenvalue crossing occurs at ℱθ, the operator
(46)𝒫θ∂∂θℱθ𝒫θ:ℋ0(ℱθ)⟶ℋ0(ℱθ)
is called a crossing operator, denoted by Cr[ℱθ]. As mentioned in [28], an eigenvalue crossing at ℱθ is said to be regular if the null space of Cr[ℱθ] is trivial. In this case, we define
(47)signCr[ℱθ]=dimℋ+(Cr[ℱθ])-dimℋ-(Cr[ℱθ]).
A crossing occurring at ℱθ is called simple crossing if dimℋ0(ℱθ)=1.As indicated in [19], the spectral flow Sf(ℱθ) will remain the same after a small disturbance of ℱθ, that is, Sf(ℱθ)=Sf(ℱθ+εid) for ε>0 and small enough, where id is the identity map on ℋ. Furthermore, we can choose suitable ε such that all the eigenvalue crossings occurring in ℱθ, 0≤θ≤1 are regular [28]. Thus, without loss of generality, we may assume that all the crossings are regular. Let 𝒟 be the set containing all the points in [0,1] at which the crossing occurs. The set 𝒟 contains only finitely many points. The spectral flow of ℱθ is
(48)Sf(ℱθ,0≤θ≤1)=∑θ∈𝒟*‍signCr[ℱθ]-dimℋ-(Cr[ℱ0])=+dimℋ+(Cr[ℱ1]),
where 𝒟*=𝒟∩(0,1). In what follows, the spectral flow of ℱθ will be simply denoted by Sf(ℱθ) when the starting and end points of the flow are clear from the contents. And Pℱθ will be simply denoted by Pθ.Proposition 12 (see [18, Proposition  3]).
Suppose that, for eachθ∈[0,1], ℱθ-ℱ0 is a compact operator on ℋ then
(49)ind(P0|ℋ-(ℱ1))=-Sf(ℱθ).Thus, from Definition7,
(50)μℱ0(𝒯)=-Sf(ℱθ,0≤θ≤1),
where ℱθ=ℱ-θ𝒯,  𝒯 is a compact operator. Moreover, if σ(𝒯)⊂[0,∞) and 0∉σP(𝒯), from the definition of spectral flow, we have
(51)μℱ0(𝒯)=-Sf(ℱθ,0≤θ≤1)=∑θ∈[0,1)‍υℱ(θ𝒯)=∑θ∈[0,1)‍dimℋ0(ℱ-θ𝒯).
The proof of Theorem 11 is the direct consequence of the aformentioned Proposition 12 and Theorem  3.2 in [19], so we omit it here.Remark 13.
The case ofR∞- can be transformed into the case of R∞+. More concretely, the R∞- case follows from the R∞+ case by applying to the function R~(t,z)=-R(-t,z). If z(t) is a homoclinic solution of F~z(t)=∇zR(t,z(t)), let z~(t)=z(-t), it is easy to check that z~(t) is a homoclinic solution of F~z~(t)=∇zR~(t,z~(t)), and this is a one-one correspondence between the two systems. By the definition of spectral flow and it is catenation property [19], we have μF(-B∞(-t))-μF(-B0(-t))=μF(B0(t))-μF(B∞(t)). Thus, we only consider the case of R∞+ from now on.


## 3. Proof of Our Main Results

Proof of Lemma1.
Recall the operatorF~=-𝒥(d/dt)+L(t), with domain D(F~)=H1(ℝ,ℝ2N) if L(t) is bounded and D(F~)⊂H1(ℝ,ℝ2N) if L(t) is unbounded. D(F~) is a Hilbert space equipped with the norm ∥z∥F~=∥(I+|F~|)z∥L2, for all z∈D(F~). Recall the Hilbert space E=D(|F~|1/2), with the inner product and norm by
(52)(u,v)E=(|F~|1/2u,|F~|1/2v)2+(u,v)2,∥u∥E=(u,u)E1/2,
where (·,·)L2 denotes the usual inner product on L2(ℝ,ℝ2N). From (L1), there is a matrix L0 such that P(L(t)-L0)≥0 for all t∈ℝ. We have L(t)=F~-F~0+L0, with F~0=-𝒥(d/dt)+L0 and D(F~0)=H1. Thus, for any z∈E,
(53)|(L(t)z,PTz)L2|≤|(F~z,PTz)L2|+|(F~0z,PTz)L2|+|(L0z,PTz)L2|≤c∥z∥E2.
Let K⊂E be a bounded set. We will show that K is precompact in Lp for 1≤p∈(2/(1+α),∞). We divide the proof into three steps.
Step  1 (the case of
p
=
2
). For R>0, from (L1) and (53) we have
(54)∫|t|>R‍|z|2dt≤c|R|α-2∫|t|>R‍〈L(t)z,PTz〉ℝ2Ndt≤c|R|α-2∥z∥E2.
For any ε>0, from (54), we can choose R0 large enough, such that
(55)∫|t|>R0‍|z|2dt<ε24,∀z∈K.
On the other hand, by the definition of ∥·∥E, we have
(56)∫|t|≤R0‍|z|2dt≤∥z∥E≤C,∀z∈K.
Thus, by the Sobolev compact embedding theorem, there exist z1,z2,…,zm∈K, such that for any z∈K, there is zi satisfying
(57)∥z-zi∥Lp((-R0,R0),ℝ2N)p<ε22.
From (55) and (57), we have ∥z-zi∥L2<ε; thus, K has a finite ε-net in L2, and so the embedding E↪L2 is compact.
Step  2 (the case of
p
>
2
). Since E is continuously embedded in H1/2, hence, by the Sobolev embedding theorem, E is continuously embedded in Lp, for all p>2. For any p>2, by the Hölder inequality we have
(58)∫ℝ‍|z|pdt≤∥z∥L2∥z∥L2(p-1)p-1≤C∥z∥L2∥z∥Ep-1,
thus, the embedding E↪Lp is compact, for all p>2.
Step  3 (the case of
1
≤
p
∈
(
2
/
(
1
+
α
)
,
2
)
). First, we have (α/(2-p))·p>1; so we can choose αp satisfying αp∈(0,α) and (αp/(2-p)·p)>1. Denote that r=αp/(2-p). For R>0 and z∈E, denote that ER1(z)={t;|t|≥Rand|t|r|z(t)|>1} and ER2(z)={t;|t|≥Rand|t|r|z(t)|≤1}. Then, from (53),
(59)∫ER1(z)‍|z|pdt=∫ER1(z)‍(|t|r|z|)p|t|-rpdt≤∫ER1(z)‍|z|2|t|αpdt≤c|R|α-αp|(L(t)z,PTz)L2|≤c|R|α-αp∥z∥E2,
and so
(60)∫|t|≥R‍|z|pdt=∫ER1(z)‍|z|pdt+∫ER2(z)‍|z|pdt≤c|R|α-αp∥z∥E2+2(rp-1)Rrp-1,∀z∈E.
Let K⊂E be a bounded set. For any ε>0, from (60), choose R0>0 large enough, such that
(61)∫|t|≥R0‍|z|pdt<εp4,∀z∈K.
On the other hand, by the Sobolev compact embedding theorem, there are z1,z2,…,zm∈K, such that for any z∈K, there exists zi satisfying
(62)∥z-zi∥Lp((-R0,R0),ℝ2N)p<εp2.
From (61) and (62), we have
(63)∥z-zi∥Lp<ε;
that is to say, K has a finite ε-net in Lp, and the embedding E↪Lp is compact. The proof of the lemma is complete.Consider the homoclinic orbits of the linear Hamiltonian systems(64)z˙(t)=JB(t)z(t),∀t∈ℝ,z(t)⟶0,|t|⟶∞,
where z(t):ℝ→ℝ2N,  J=(0-ININ0), and B(t) is a continuous symmetric matrix function. Denote by S the set of homoclinic orbits of linear systems (64); then, S is a linear subspace of L2(ℝ,ℝ2N) and we have the following lemma.Lemma 14.
The dimension of the solution spaceS will be less than or equal to N. Thus for any homoclinic orbit z(t) of (HS), if R satisfies (R1), one has
(65)0≤υF(z)≤N.Proof.
As usual, we define the symplectic groups onℝ2N by
(66)Sp(2N)={M∈ℒ(ℝ2N),∣MTJM=J},
where ℒ(ℝ2N) is the set of all 2N×2N real matrices, and MT denotes the transpose of M. Let W(t) be the fundamental solution of (64); then, W(t) is a path in Sp(2N). Let z(t) be a nontrivial homoclinic orbit of (64); that is to say, z(0)≠0 and satisfies
(67)z(t)=W(t)z(0),limt→∞W(t)z(0)=0.
Denote by S0 the subset of ℝ2N satisfying
(68)S0={z∈ℝ2N∣limt→∞W(t)z=0},
Then, we have dimS=dim(S0). We claim that Jz0∉S0 if z0∈S0 and z0≠0. We prove it indirectly. Assume that z0,Jz0∈S0 with z0≠0; that is to say,
(69)limt→∞W(t)z0=0,limt→∞W(t)Jz0=0.
Since W(t) is a path in Sp(2N), WT(t)JW(t)=J, for all t∈ℝ, thus
(70)0=limt→∞(JW(t)z0,W(t)Jz0)ℝ2N=-limt→∞(z0,WT(t)JW(t)Jz0)ℝ2N=(z0,z0)ℝ2N,
which contradicts z0≠0. Since J is an isomorphism on ℝ2N, we have dimS0≤N. And from the definition of υF(z) in the last part of Section 2.2, we have completed the proof.Before the proof of Theorem2, we need the following lemma. Since R satisfies condition (R1), performing on (HS) the saddle point reduction, choose a suitable number β, which is used in the projection for the saddle point reduction in Section 2.2. Let
(71)P=∫-ββ‍dEλ,X=PL2(ℝ,ℝ2N).
By Theorem 8, we have a functional a(x) with x∈X, whose critical points give rise to solutions of (HS).Lemma 15.
(1)a satisfies (PS) condition;
(2)Hq(X,a;ℝ)≅δq,rℝ,q=0,1,… for -a∈ℝ large enough, where r=dim(E-(X))+μF(KB1).Proof.
Assume that there is a sequence{xn}⊂X, satisfying a′(xn)→0(n→∞). That is,
(72)∥Fzn-K∇zR(t,zn)∥E⟶0,
where zn=z(xn) defined in Section 2.1. Since X is a finite dimensional space, and from the definition of zn, it is enough to prove that {zn} is bounded in E. For each ε∈(0,1), define Cn∈ℬ by
(73)Cn(t)={∫01∇z2R(t,szn)ds,|zn(t)|≥R0ε,B1(t),|zn(t)|<R0ε.
It is easy to verify that {Cn} satisfies
(74)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 (R1) and I is the identity map on ℝ2N. Since B1≤B2,  μF(B1)=μF(B2) and υF(KB1)=υF(KB2)=0, we can choose ε small enough, such that for each n∈ℕ+, μF(KCn)=μF(KB1) and υF(KCn)=0. Thus F-KCn is reversible on E, and there is a constant δ>0, such that
(75)∥(F-KCn)z∥E≥δ∥z∥E,∀z∈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+,
(76)|∇zR(t,zn(t))-Cnzn(t)|≤c|zn(t)|b,∀t∈ℝ.
Choose b>(1-α)/(1+α) in (76); that is, 1+b∈(2/(1+α),2), we have
(77)∥(Fzn-K∇zR(t,zn))-(F-KCn)zn∥E2=∥K(∇zR(t,zn)-Cnzn)∥E2≤∥∇zR(t,zn)-Cnzn∥L22≤c∫ℝ‍|∇zR(t,zn)-Cnzn||zn|b|zn|1+bdt≤c∥zn∥L1+b1+b.
As we claimed in the introduction, in (76) and (77), the letter c denotes different positive constants whose exact value is irrelevant. Thus, from (72), (75), (77), and Lemma 1, we have that {zn} is bounded in E, and a satisfies the (PS) conditions. And by Lemma  5.1 in Chapter II of [21], we have
(78)Hq(X,(a)α;ℝ)≅δq,rℝ,q=0,1,…,
for -α∈ℝ large enough.From Theorem11, Lemmas 14 and 15, Theorem 2 is a direct consequence of Theorem  5.1 and Corollary  5.2 in Chapter II of [21].In order to proof Theorems3 and 4, we need the following lemma which is similar to Lemma  3.4 in [29] and Lemma  3.3 in [23].Lemma 16.
Assume that(R1), (R0), and (R∞+) hold; then, there exists a sequence of functions Rk∈C2(ℝ×ℝ2N,ℝ),k∈ℕ, satisfying the following properties. (1)
There exists an increasing sequence of real numbersMk→∞(k→∞) such that
(79)Rk(t,z)≡R(t,z),∀t∈ℝ,|z|≤Mk.(2)
For eachk∈ℕ, there is a C>0 independent of k, such that
(80)|∇z2Rk(t,z)|≤C,∀t∈ℝ,z∈ℝ2N,∇z2Rk(t,z)≥B∞,∀t∈ℝ,|z|≥R0.(3)
For eachk∈ℕ, there exist some Ck>0 and a constant γ with γI2N>B∞, νF(KγI2N)=0, such that
(81)|∇zRk(t,z)-γz|<Ck,∀(t,z)∈ℝ×ℝ2N,
where I2N is the identity map on ℝ2N.Proof.
Defineη:[0,∞)→ℝ by
(82)η(s)={0,0≤s<1,29(s-1)3-19(s-1)4,1≤s<2,1-1289(12+s2),2≤s<∞.
It is easy to see that η∈C2([0,∞),ℝ). Choose a sequence {Mk} of positive numbers such that R0<M1<M2<⋯<Mk<⋯→∞ as k→∞. For each k∈ℕ, let ηk(s)=η(s/Mk) and
(83)Rk(t,z)=(1-ηk(|z|))R(t,z)+γ2ηk(|z|)|z|2,k∈ℕ.
As in [23, 29], we can check that Rk satisfies (79)–(81) for each k∈ℕ.For eachk∈ℕ, we consider the following problem
(HS)kF~z=∇zRk(t,z),z(t)⟶0,z′(t)⟶0,t⟶∞,
where Rk is given in Lemma 16. Performing on (HS)k the saddle point reduction, we choose the number β which is used in the projection for the saddle point reduction in Section 2.2. First, we choose
(84)β>max{2(C+1),2(γ+1)},β∉σ(A0).
Let
(85)Pβ=∫-ββ‍dEλ,Xβ=PβL2(ℝ,ℝ2N).
Thus, for each k and such a β fixed, by Theorem 8, we have a functional
(86)ak,β(x),x∈Xβ,
whose critical points give rise to solutions of (HS)k. Similarly we have a functional
(87)aγ,β(x),x∈Xβ,
whose critical points give rise to solutions of the following systems (HS)γ(HS)γF~z=γz,z(t)⟶0,z′(t)⟶0,t⟶∞.
For notational simplicity, we denote ak,aγ for ak,β and aγ,β. Define
(88)Φk(z)=∫-∞∞‍Rk(t,z)dt.
For the functional ak, similar to Lemma 15, we have the following lemma.Lemma 17.
(1)ak satisfies (PS) condition, and the critical point set of ak is compact;
(2)Hq(Xβ,(ak)αk;ℝ)≅δq,rβℝ,q=0,1,… for -αk∈ℝ large enough, where rβ=dim(E-(Xβ))+μF(KγI2N).Proof.
The proof is similar to Lemma15. From Theorem 11, we have
(89)∥ak′(x)-aγ′′(0)x∥L2=∥PβK(∇zΦk(zk(x))-γx)∥L2≤∥Pβ(∇zRk(zk(x))-γzk(x))∥L2≤∥(∇zRk(zk(x))-γzk(x))∥L2.
Similar to (76), for b∈(0,1), there is some c>0, such that
(90)|∇zRk(t,zk)-γzk|≤c|zk(t)|,∀t∈ℝ.
Choose b∈((1-α)/(1+α),1), similar to (77), and we have
(91)∥∇zRk(t,zk)-γzk∥L22≤c∥zk∥L1+b1+b.
From Lemma 1, Remark 10, and (91),
(92)∥∇zRk(t,zk)-γzk∥L22≤c∥zk∥E1+b≤c(β)∥x∥L21+b.
From (89) and (92), we have
(93)∥ak′(x)-aγ′′(0)x∥L22≤c(β)∥x∥L21+b.
Now, for each k∈ℕ, we assume that {xm}⊂Xβ satisfying ∥ak′(xm)∥→0. By νF(KγI2N)=0, we have that aγ′′(0) is invertible on Xβ, since b<1 the sequence {xm} must be bounded. Thus the (PS) condition for ak holds. From the same reason, we have the compactness of the critical point set of ak. And by Lemma  5.1 in Chapter II of [21], we have
(94)Hq(Xβ,(ak)αk;ℝ)≅δq,rβℝ,q=0,1,…,
for -αk∈ℝ large enough.Lemma 18.
There existc>0, such that for any k∈ℕ, and z∈L2 satisfies the systems (HS)k, if μF(z)≤μF(KB∞)-1, one has ∥z∥L∞≤c.Proof.
We prove it indirectly. Assume there existRk and zk satisfying the conditions, and ∥zk∥L∞→∞; that is, ∥z∥F~→∞. Since |∇zRk(t,z)|<c|z|, for all t∈ℝ,z∈ℝ2N, we have ∥zk∥F~≤c∥zk∥L2. Denote that yk=zk/∥zk∥F~; then, we have yk→y in L2 for some y∈L2 with ∥y∥L2>0, and
(95)F~yk=Rk′(t,zk)∥zk∥F~.
Then, for any r>0, there exists Cr>0, satisfying
(96)|y˙k(t)|≤Cr|yk(t)|,t∈Ir,
where Ir=[-r,r]. Since ∥y∥L2>0, there is a r0>0, such that
(97)∥y∥L2(-r,r)>12∥y∥L2>0,∀r>r0.
Then, from the similar argument in [23], there is a subsequence throught which we may assume that {yk} converges in uniform norm to y, and y(t)≠0, for all t∈Ir. Therefore |zk(t)|→∞ uniformly on Ir, and there is K(r) depending on r, such that |zk(t)|≥R0, for any t∈Ir and k≥K(r).
Performing the saddle point reduction on the following systems:(HS)∞F~z=B∞z,z(t)⟶0,z′(t)⟶0,t⟶∞,
for β large enough, we have the functional a∞,β (denote by a∞ for simplicity) and the function z(x); since υF(KB∞)=0, we have the following decomposition:
(98)Xβ=Xβ++Xβ-,
where a∞′′(0) is positive definite on Xβ+ and negative definite on Xβ-. From Remark 10, and υF(KB∞)=0, there exists α>0, such that for β large enough,
(99)((F~-B∞(t))x,x)L2≤-α∥x∥L22,∀x∈Xβ-.
From the uniform boundary of ∇z2Rk(t,z) and Remark 10, we can choose β large enough, such that
(100)∥(∇z2Rk(t,zk)zk′±(xk)x,x∥L2≤α4∥x∥L2,∀x∈L2,
where xk=Pβzk, zk(xk)=zk(t) defined in Theorem 8. Choose ε>0 small enough and ε<α/4, such that μF(KB∞)=μF(K(B∞-ε·Id)). Since Xβ- is finite dimensional space, choose r large enough, such that
(101)((∇z2Rk(t,zk)-B∞)x,x)L2(Irc)≥-ε(x,x)L2,∀x∈Xβ-,
where Irc=ℝ∖Ir, and from the definition of Rk,
(102)∇z2Rk(t,zk(t))≥B∞(t),t∈Ir,k≥K(r);
that is,
(103)((∇z2Rk(t,zk)-B∞)x,x)L2(Ir)≥0,∀x∈Xβ-,k≥K(r).
From (101) and (103),
(104)(∇z2Rk(t,zk)x,x)L2≥(B∞x,x)L2-ε(x,x)L2,
for k large enough. Thus we have
(105)(ak′′(xk)x,x)L2=((F~-∇z2Rk(t,zk))x,x)L2-(∇z2(Rk(t,zk))(zk′++zk′-)x,x)L2≤((F~-B∞)x,x)L2+α2∥x∥L22+ε∥x∥L22≤-α4∥x∥L22.
That is, mak-(x)≥ma∞-(0), from Theorem 11, mak-(xk)=dim(E-(ℋ0))+μF(zx(xk)), ma∞-(0)=dim(E-(ℋ0))+μF(KB∞), and thus μF(zx)≥μF(KB∞), which contradicts the assumption.Proof of Theorem3.
As claimed in Remark13, we can only consider the case of (R∞+). Note that z=0 is a critical point of ak, and the morse index of 0 for ak is mak-(0)=dim(E-(ℋ0))+μF(KB0), since γ·I2N>B∞, we have
(106)μF(Kγ·I2N)≥μF(KB∞)≥μF(KB0).
From proposition (2) in Lemma 17, use the (mak-(0))th and (mak-(0)+1)th Morse inequalities, and ak has a nontrivial critical point xk with its morse index mak-(xk)≤mak-(0)+1; that is, μF(zk)≤μF(KB0)+1≤μF(KB∞)-1, then from Lemma 18, we have that {zk} is bounded in L∞. Thus zk is a nontrivial solution of (HS) for k large enough.The proof of Theorem4 is similar to the proof of Theorem 3. 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 19 (see [30, Corollary  10.19]).
Assume thatY and Z are subspaces of X satisfying dimY=j>k=codimZ. If there exist R>r>0 and α>0 such that
(107)infϕ(Sr∩Z)≥α,supϕ(SR∩Y)≤0,
then ϕ has j-k pairs of nontrivial critical points {±x1,±x2,…,±xj-k}, so that μ(xi)≤k+i, for i=1,2,…,j-k.First, we consider the case of(R∞+), since R is even, we have that Rk is also even and satisfies Lemma 16. Let Y=Xβ-, and Z the positive space of ak′′(0) in Xβ, and we have and dimY=E-(Xβ)+μF(KB∞), codimZ=E-(Xβ)+μF(KB0), dimY>codimZ. So, ak has l:=μF(KB∞)-μF(KB0) pairs of nontrivial critical points
(108){±x1,±x2,…,±xl},
and l-1 pairs of them satisfy
(109)m-(xi)≤μF(KB0)+i<μF(KB∞),i=1,2,…,l-1.
Then, we can complete the proof. In order to prove the case of (R∞-), we need the following lemma.Lemma 20 (see [21, Corollary  II 4.1]).
Assume thatY and Z are subspaces of X satisfying dimY=j>k=codimZ. If there exist r>0, and α>0 such that
(110)infϕ(Z)>-∞,supϕ(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(R∞+), and we omit it here.



---

*Source: 610906-2013-04-22.xml*