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

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.

1. Introduction and Main Results

Consider the following first-order nonautonomous Hamiltonian systems 𝑧̇=𝐽𝐻_𝑧(𝑡,𝑧), (HS) where 𝑧∶ℝ→ℝ^2𝑁, 𝐽=^0−𝐼𝑁_𝐼𝑁0, 𝐻∈𝐶¹(ℝ×ℝ^2𝑁,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 𝐻(𝑡,𝑧)=−𝐿(𝑡)𝑧⋅𝑧+𝑅(𝑡,𝑧), (1) where 𝐿 is a 2𝑁×2𝑁 symmetric matrix valued function. We assume that (𝐿₁)𝐿∈𝐶(ℝ,ℝ^𝑁2), and there are 𝛼,𝑐>0, 𝑡₀≥0 and a constant matrix 𝑃, satisfying 𝑃𝐿(𝑡)−𝑐|𝑡|^𝛼𝐼_2𝑁≥0,∀|𝑡|≥𝑡₀, (2) where 𝐼_2𝑁 is the identity map on ℝ^2𝑁, and for a 2𝑁×2𝑁 matrix 𝑀, we say that 𝑀≥0 if and only if inf_{𝜉∈ℝ2𝑁,|𝜉|=1}𝑀𝜉⋅𝜉≥0. (3) In (𝐿₁), if 𝑃=^0𝐼𝑁_𝐼𝑁0, 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=±𝐼_2𝑁 or 𝑃=^{𝐼𝑁+𝑚0}_{0−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|^{𝛼cos2𝑡sin2𝑡}_{sin2𝑡−cos2𝑡}, we have the eigenvalues of 𝐿(𝑡) are ±|𝑡|^𝛼, but there is no constant matrix 𝑃 satisfying 𝑃𝐿(𝑡)−𝑐|𝑡|^𝛼𝐼_2𝑁≥0,  for all  |𝑡|≥𝑡₀.

Denote by 𝐹 the self-adjoint operator −𝒥(𝑑/𝑑𝑡)+𝐿(𝑡) on 𝐿²≡𝐿²(ℝ,ℝ^2𝑁), with domain 𝐷(𝐹)=𝐻¹(ℝ,ℝ^2𝑁) if 𝐿(𝑡) is bounded and 𝐷(𝐹)⊂𝐻¹(ℝ,ℝ^2𝑁) 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 ‖𝑧‖_𝐹=𝐼+𝐹𝑧_𝐿2,∀𝑧∈𝐷𝐹. (4) Let 𝐸=𝐷(|𝐹|^1/2), and define on 𝐸 the inner product and norm by (𝑢,𝑣)_𝐸=|𝐹|^1/2𝑢,|𝐹|^1/2𝑣₂+(𝑢,𝑣)₂,‖𝑢‖_𝐸=(𝑢,𝑢)_𝐸^1/2, (5) where (⋅,⋅)_𝐿² denotes the usual inner product on 𝐿²(ℝ,ℝ^2𝑁). Then, 𝐸 is a Hilbert space. It is easy to see that 𝐸 is continuously embedded in 𝐻^1/2(ℝ,ℝ^2𝑁), and we further have the following lemma.

Lemma 1. Suppose that 𝐿 satisfies (𝐿₁). Then 𝐸 is compactly embedded in 𝐿^𝑝(ℝ,ℝ^2𝑁) with the usual norm ‖⋅‖_𝐿^𝑝 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 𝒬(𝑢,𝑣)=_ℝ((−𝐽𝑢̇,𝑣)+(𝐿(𝑡)𝑢,𝑣))𝑑𝑡,∀𝑢,𝑣∈𝐸. (6) 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 (𝐹𝑢,𝑣)_𝐸=𝒬(𝑢,𝑣),∀𝑢,𝑣∈𝐸. (7) Besides, define a linear operator 𝐾∶𝐿²→𝐸 by (𝐾𝑢,𝑣)_𝐸=(𝑢,𝑣)_𝐿²,∀𝑢∈𝐿²,𝑣∈𝐸. (8) In view of Lemma 1, we know that 𝐹 is a Fredholm operator and 𝐾 is a compact operator.

Denote by ℬ the set of all uniformly bounded symmetric 2𝑁×2𝑁 matric functions. That is to say, 𝐵∈ℬ if and only if 𝐵^𝑇(𝑡)=𝐵(𝑡) for all 𝑡∈ℝ, and 𝐵(𝑡) is uniformly bounded in 𝑡 as the operator on ℝ^2𝑁. 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 (𝐾𝐵𝑢,𝑣)_𝐸=(𝐵𝑢,𝑣)_𝐿²,∀𝑢,𝑣∈𝐸. (9)

Before presenting the conditions on 𝑅(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 (𝜇_𝐹(𝐾𝐵), 𝜐_𝐹(𝐾𝐵)).

Now we can present the conditions on 𝑅(𝑡,𝑧) as follows. For notational simplicity, we set 𝐵₀(𝑡)=∇²_𝑧𝑅(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×ℝ^2𝑁,ℝ), and there exists a constant 𝑐>0 such that ∇²_𝑧𝑅(𝑡,𝑧)≤𝑐,∀(𝑡,𝑧)∈ℝ×ℝ^2𝑁. (10)(𝑅₀)∇_𝑧𝑅(𝑡,0)≡0 and 𝐵₀∈ℬ.(𝑅_∞)There exists some 𝑅₀>0 and continuous symmetric matrix functions 𝐵₁,𝐵₂∈ℬ with 𝜇_𝐹(𝐾𝐵₁)=𝜇_𝐹(𝐾𝐵₂) and 𝜐_𝐹(𝐾𝐵₂)=0 such that 𝐵₁(𝑡)≤∇²_𝑧𝑅(𝑡,𝑧)≤𝐵₂(𝑡),∀𝑡∈ℝ,|𝑧|>𝑅₀. (11)Then, we have our first result.

Theorem 2. Assume that (𝐿₁), (𝑅₁), (𝑅₀), and (𝑅_∞) hold. If 𝜇_𝐹𝐾𝐵₁∉𝜇_𝐹𝐾𝐵₀,𝜇_𝐹𝐾𝐵₀+𝜐_𝐹𝐾𝐵₀, (12) then (HS) has at least one nontrivial homoclinic orbit. Moreover, if 𝜐_𝐹(𝐾𝐵₀)=0 and |𝜇_𝐹(𝐾𝐵₁)−𝜇_𝐹(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 some 𝑅₀>0 and a continuous symmetric matrix function 𝐵_∞∈ℬ with 𝜐_𝐹(𝐾𝐵_∞)=0 such that ±∇²_𝑧𝑅(𝑡,𝑧)≥±𝐵_∞(𝑡),∀𝑡∈ℝ,|𝑧|>𝑅₀. (13)Then, we have the following results.

Theorem 3. Assumes (𝐿₁), (𝑅₁), (𝑅₀), (𝑅⁺_∞) (or (𝑅⁻_∞)), and 𝜐_𝐹(𝐾𝐵₀)=0 hold. If 𝜇_𝐹(𝐾𝐵_∞)≥𝜇_𝐹(𝐾𝐵₀)+2 (or 𝜇_𝐹(𝐾𝐵_∞)≤𝜇_𝐹(𝐾𝐵₀)−2), then (HS) has at least one nontrivial homoclinic orbit.

Theorem 4. Suppose that (𝐿₁), (𝑅₁), (𝑅₀), (𝑅⁺_∞) (or (𝑅⁻_∞)), and 𝜐_𝐹(𝐾𝐵₀)=0 are satisfied. If in addition 𝑅 is even in 𝑧 and 𝜇_𝐹(𝐾𝐵_∞)≥𝜇_𝐹(𝐾𝐵₀)+2 (or 𝜇_𝐹(𝐾𝐵_∞)≤𝜇_𝐹(𝐾𝐵₀)−2), then (HS) has at least |𝜇_𝐹(𝐾𝐵_∞)−𝜇_𝐹(𝐾𝐵₀)|−1 pairs of nontrivial homoclinic orbits.

Remark 5. Lemma 1 shows that 𝜎(𝐴), the spectrum of 𝐴, consists of eigenvalues numbered by (counted in their multiplicities): ⋯≤𝜆₋₂≤𝜆₋₁≤0<𝜆₁≤𝜆₂≤⋯ (14) with 𝜆_±𝑘→±∞ as 𝑘→∞. Let 𝐵₀(𝑡)≡𝐵₀ and 𝐵_∞(𝑡)≡𝐵_∞, with the constants 𝐵₀, 𝐵_∞ satisfying 𝜆_𝑙<𝐵₀<𝜆_𝑙+1, and 𝜆_𝑙+𝑖<𝐵_∞<𝜆_𝑙+𝑖+1 for some 𝑙∈ℤ and 𝑖≥1 (or 𝑖≤−1). Define 𝑅()𝑧1𝑡,𝑧=𝛿(||)2𝐵₀|𝑧|²+(𝑧11−𝛿(||))2𝐵_∞|𝑧|², (15) 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 𝜆_𝑙+𝑖<𝐵_∞<𝜆_𝑙+𝑖+1 for some 𝑙∈ℤ and 𝑖≥2 (or 𝑖≤−2). Define 𝑅()𝑧1𝑡,𝑧=𝛿(||)2𝐵₀|𝑧|²+(𝑧11−𝛿(||))2𝐵_∞|𝑧|². (16) 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 assumption 𝜐_𝐹(𝐾𝐵_∞)=0 in (𝑅^±_∞) is not essential for our main results. For the case of (𝑅⁺_∞) with 𝜐_𝐹(𝐾𝐵_∞)≠0, let 𝐵_∞=𝐵_∞−𝜀𝐼_2𝑁 with 𝜀>0 small enough, where 𝐼_2𝑁 is the identity map on ℝ^2𝑁; then, 𝜇_𝐹(𝐾𝐵_∞)=𝜇_𝐹(𝐾𝐵_∞) and 𝜐_𝐹(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 𝜐_𝐹(𝐾𝐵_∞)≠0, if we replace 𝜇_𝐹(𝐾𝐵_∞) by 𝜇_𝐹(𝐾𝐵_∞)+𝜐_𝐹(𝐾𝐵_∞) in Theorems 3 and 4, then similar results hold. Indeed, let 𝐵_∞=𝐵_∞+𝜀𝐼_2𝑁 with 𝜀>0 small enough such that 𝜇_𝐹(𝐾𝐵_∞)=𝜇_𝐹(𝐾𝐵_∞)+𝜐_𝐹(𝐾𝐵_∞) and 𝜐_𝐹(𝐾𝐵_∞)=0, then this case is also reduced to the case of (𝑊⁻_∞) for 𝐵_∞ with 𝜐_𝐹(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

Let ℋ be a separable Hilbert space; for any self-adjoint operator 𝐴 on ℋ, there is a unique 𝐴-invariant orthogonal splitting ℋ=ℋ⁺(𝐴)⊕ℋ⁻(𝐴)⊕ℋ⁰(𝐴), (17) where ℋ⁰(𝐴) is the null space of 𝐴, 𝐴 is positive definite on ℋ⁺(𝐴) and negative definite on ℋ⁻(𝐴), and 𝑃_𝐴 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, 𝑃_ℱ−𝑃_ℱ−𝒯 is compact (see [19, Lemma  2.7]), where 𝑃_ℱ∶ℋ→ℋ⁻(ℱ) and 𝑃_ℱ−𝒯∶ℋ→ℋ⁻(ℱ−𝒯) are the respective projections. Then, by Fredholm operator theory, 𝑃_ℱ|_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 operator ℱ and a compact self-adjoint operator 𝒯 on ℋ, the relative Morse index pair (𝜇_ℱ(𝒯),𝜐_ℱ(𝒯)) is defined by 𝜇_ℱ(𝒯)=ind𝑃_ℱ|_ℋ⁻_(ℱ−𝒯),𝜐_ℱ(𝒯)=dimℋ⁰(ℱ−𝒯). (18)

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 numbers 𝛼<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 𝑑Φ^′(𝑢)−𝛼+𝛽2𝐼≤𝛽−𝛼2,∀𝑢∈ℋ, (19) 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 ‖𝑧‖_𝑉=|𝐴|^1/2𝑧²_ℋ+𝜀²‖𝑧‖²_ℋ^1/2, (20) where 𝜀>0 small and −𝜀∉𝜎(𝐴).

Consider the solutions of the following equation: 𝐴𝑧=Φ^′(𝑧),𝑧∈𝐷(𝐴). (21) Let 𝑃₀=^𝛽_−𝛽𝑑𝐸_𝜆,𝑃₊=_𝛽^+∞𝑑𝐸_𝜆,𝑃₋=^−𝛽_−∞𝑑𝐸_𝜆, (22) where {𝐸_𝜆} is the spectral resolution of 𝐴, and let ℋ_∗=𝑃_∗ℋ,∗=0,±. (23) Decompose the space 𝑉 as follows: 𝑉=𝑉₀⊕𝑉₋⊕𝑉₊, (24) where 𝑉_∗=|𝐴_𝜀|^−1/2ℋ_∗,∗=0,±, and 𝐴_𝜀=𝐴+𝜀𝐼.

For each 𝑢∈ℋ, we have the decomposition 𝑢=𝑢₊+𝑢₀+𝑢₋, (25) where 𝑢_∗∈ℋ_∗ and ∗=0,±; let 𝑧=𝑧₊+𝑧₀+𝑧₋, with 𝑧_∗=𝐴_𝜀^−1/2𝑢_∗,∗=0,±. (26) Define a functional 𝑓 on ℋ as follows: 𝑓(𝑢)=12𝑢₊²+𝑄₊𝑢₀²−𝑄₋𝑢₀²−‖𝑢‖²−Φ_𝜀(𝑧), (27) where 𝑄₊=∫^∞₀𝑑𝐸_𝜆,  𝑄₋=∫⁰_−∞𝑑𝐸_𝜆, and Φ_𝜀(𝑧)=(𝜀/2)‖𝑧‖_ℋ+Φ(𝑧).

The Euler equation of this functional is the system 𝑢_±=±𝐴_𝜀^−1/2𝑃_±Φ^′_𝜀(𝑧),𝑄_±𝑢₀=±𝐴_𝜀^−1/2𝑄_±𝑃₀Φ^′_𝜀(𝑧). (28)(29) 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 𝑢_±𝑧₀=𝐴_𝜀^1/2𝑧_±𝑧₀∈𝐶¹ℋ₀,ℋ, (30) which solves the system (28).

Let 𝑎𝑧₀=𝑓𝑢₊𝑧₀+𝑢₋𝑧₀+𝑢₀𝑧₀, (31) where 𝑢₀(𝑧₀)=|𝐴_𝜀|^1/2𝑧₀, and let 𝑧₀=𝑥 we have 𝑎(𝑥)=12(𝐴(𝑧(𝑥),𝑧(𝑥)))−Φ(𝑧(𝑥)), (32) 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 𝑥↦𝑧=𝑧(𝑥)=𝑧₊(𝑥)+𝑧₋(𝑥)+𝑥 (33) between the critical points of the 𝐶²-function 𝑎∈𝐶²(ℋ₀,ℝ) with the solutions of the operator equation 𝐴𝑧=Φ^′(𝑧),𝑧∈𝐷(𝐴). (34) Moreover, the functional 𝑎 satisfies 𝑎(𝑥)=12(𝐴(𝑧(𝑥)(𝑥(𝑧(𝑥,𝑎,𝑧)))−Φ))_′(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). (35)

Since ℋ₀ 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 (𝑚⁻_𝑎(𝑥),𝑚⁰_𝑎(𝑥)).

Now, let the Hilbert space ℋ be 𝐿²(ℝ,ℝ^2𝑁), and the operator 𝐴 be 𝐹=−𝐽(𝑑/𝑑𝑡)+𝐿, Φ(𝑢)=∫^∞_−∞𝑅(𝑡,𝑢). Then we have 𝑉=𝐸. For 𝑅∈𝐶²(ℝ×ℝ^2𝑁,ℝ) and |∇²_𝑧𝑅|≤𝐶_𝑅, for all (𝑡,𝑧)∈ℝ×ℝ^2𝑁, let −𝛼=𝛽≥2(𝐶_𝑅+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 that 𝑅∈𝐶²(ℝ×ℝ^2𝑁,ℝ), |∇²_𝑧𝑅|≤𝐶_𝑅, for all (𝑡,𝑧)∈ℝ×ℝ^2𝑁 and ∇_𝑧𝑅(𝑡,0)≡0; then one has 𝑢^±(𝑥)_𝐿2≤2𝛽𝐶_𝑅+1𝛽−2𝐶_𝑅‖𝑥‖−3𝜀^𝐿2,∀𝑥∈ℋ₀. (36) Moreover, one has 𝑢^±′(𝑥)_𝐿2⟶0,𝛽⟶∞. (37)

Proof. Note that 𝑢^±(𝑥)=±𝐴_𝜀^−1/2𝑃_±Φ^′_𝜀𝑧⁺+𝑧⁻+𝑥. (38) From ∇_𝑧𝑅(𝑡,0)=0, |∇²_𝑧𝑅|≤𝐶_𝑅, we have Φ^′(0)=0 and ‖Φ^′(𝑧)‖_𝐿²≤𝐶_𝑅‖𝑧‖_𝐿². Since ‖|𝐴_𝜀|^−1/2𝑃_±‖≤1/(𝛽−𝜀), we have 𝑢^±(𝑥)_𝐿2≤1Φ𝛽−𝜀^′(𝑧⁺+𝑧⁻+𝑥)+𝜀(𝑧⁺+𝑧⁻+𝑥)_𝐿2≤𝐶_𝑅+𝜀𝑧𝛽−𝜀⁺+𝑧⁻+𝑥_𝐿2≤𝐶_𝑅+𝜀𝑢𝛽−𝜀⁺(𝑥)_𝐿2𝛽+‖𝑢⁻(𝑥)‖_𝐿²𝛽+‖𝑥‖_𝐿². (39) Therefore, 𝑢⁺(𝑥)_𝐿2+‖𝑢⁻(𝑥)‖_𝐿²≤2𝛽𝐶_𝑅+𝜀𝛽−2𝐶_𝑅‖𝑥‖−3𝜀^𝐿2. (40) Next, since 𝑢^±′(𝑥)=±𝐴_𝜀^−1/2𝑃_±Φ^′_𝜀×𝑧⁺+𝑧⁻+𝑥𝑧^+′(𝑥)+(𝑧⁻)^′(𝑥)+𝐼, (41) where 𝐼 is the identity map on ℋ₀, we have 𝑢^+′(𝑥)_𝐿2+(𝑢⁻)^′(𝑥)_𝐿2≤2𝛽𝐶_𝑅+𝜀𝛽−2𝐶_𝑅−3𝜀,∀𝑥∈ℋ₀. (42)

Remark 10. For 𝑧(𝑥), we also have that there is a constant 𝐶>0 dependent of 𝐶_𝑅, but independent of 𝛽, such that 𝑧^±(𝑥)_𝑉≤𝐶𝛽‖𝑥‖_𝐿²,𝑧^′±(𝑥)_𝑉≤𝐶𝛽,∀𝑥∈ℋ₀. (43)
If 𝑅 satisfies the condition (𝑅₁), then for any homoclinic orbit 𝑧 of (HS), ∇²_𝑧𝑅(⋅,𝑧)∈ℬ, and, hence, we have the associated index pair (𝜇_𝐹(𝐾𝐵), 𝜐_𝐹(𝐾𝐵)). For notation simplicity, in what follows, we set 𝜇_𝐹(𝑧)=𝜇_𝐹𝐾∇²_𝑧(𝑅(,𝜐𝑡,𝑧))_𝐹(𝑧)=𝜐_𝐹𝐾∇²_𝑧(𝑅(𝑡,𝑧)). (44)

Theorem 11. Let 𝑅∈𝐶²(ℝ×ℝ^2𝑁,ℝ) satisfying |∇²_𝑧𝑅|≤𝐶_𝑅, for all (𝑡,𝑧)∈ℝ×ℝ^2𝑁 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 𝑚⁻_𝑎(𝑥)=dim𝐸⁻ℋ₀+𝜇_𝐹𝐾∇²_𝑧𝑅((𝑥𝐸𝑡,𝑧))=dim⁻ℋ₀+𝜇_𝐹(𝑧(𝑥,𝑚))_0𝑎(𝑥)=𝜐_𝐹𝐾∇²_𝑧𝑅(𝑡,𝑧(𝑥))=𝜐_𝐹(𝑧(𝑥)), (45) where dim(𝐸⁻(ℋ₀)) is the dimension of the space ∫⁰⁻_−𝛽𝑑𝐸_𝜆(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 between 𝜇_𝐹(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 {ℱ_𝜃∣𝜃∈[0,1]} be a continuous path of self-adjoint Fredholm operators on the Hilbert space ℋ. The spectral flow of ℱ_𝜃 represents the net change in the number of negative eigenvalues of ℱ_𝜃 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 {ℱ_𝜃𝜃∣∈[0,1]} and let 𝒫_𝜃 be the projection from ℋ to ℋ⁰(ℱ_𝜃). When eigenvalue crossing occurs at ℱ_𝜃, the operator 𝒫_𝜃𝜕ℱ𝜕𝜃^𝜃𝒫_𝜃∶ℋ⁰ℱ_𝜃⟶ℋ⁰ℱ_𝜃 (46) is called a crossing operator, denoted by 𝐶_𝑟[ℱ_𝜃]. As mentioned in [28], an eigenvalue crossing at ℱ_𝜃 is said to be regular if the null space of 𝐶_𝑟[ℱ_𝜃] is trivial. In this case, we define sign𝐶_𝑟ℱ_𝜃=dimℋ⁺𝐶_𝑟ℱ_𝜃−dimℋ⁻𝐶_𝑟ℱ_𝜃. (47) A crossing occurring at ℱ_𝜃 is called simple crossing if dimℋ⁰(ℱ_𝜃)=1.

As indicated in [19], the spectral flow Sf(ℱ_𝜃) will remain the same after a small disturbance of ℱ_𝜃, that is, Sf(ℱ_𝜃)=Sf(ℱ_𝜃+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 ℱ_𝜃, 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 ℱ_𝜃 is Sfℱ_𝜃=,0≤𝜃≤1_{𝜃∈𝒟∗}sign𝐶_𝑟ℱ_𝜃−dimℋ⁻𝐶_𝑟ℱ₀+dimℋ⁺𝐶_𝑟ℱ₁, (48) where 𝒟^∗=𝒟∩(0,1). In what follows, the spectral flow of ℱ_𝜃 will be simply denoted by Sf(ℱ_𝜃) when the starting and end points of the flow are clear from the contents. And 𝑃_ℱ𝜃 will be simply denoted by 𝑃_𝜃.

Proposition 12 (see [18, Proposition  3]). Suppose that, for each 𝜃∈[0,1], ℱ_𝜃−ℱ₀ is a compact operator on ℋ then ind𝑃₀|_ℋ⁻_(ℱ1)=−𝑆𝑓ℱ_𝜃. (49)

Thus, from Definition 7, 𝜇_ℱ0(𝒯)=−Sfℱ_𝜃,0≤𝜃≤1, (50) where ℱ_𝜃=ℱ−𝜃𝒯,  𝒯 is a compact operator. Moreover, if 𝜎(𝒯)⊂[0,∞) and 0∉𝜎_𝑃(𝒯), from the definition of spectral flow, we have 𝜇_ℱ0(𝒯)=−Sfℱ_𝜃,0≤𝜃≤1=_[)𝜃∈0,1𝜐_ℱ(𝜃𝒯)=_[)𝜃∈0,1dimℋ⁰(ℱ−𝜃𝒯). (51) 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 of 𝑅⁻_∞ 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 𝐹𝑧(𝑡)=∇_𝑧𝑅(𝑡,𝑧(𝑡)), let 𝑧(𝑡)=𝑧(−𝑡), it is easy to check that 𝑧(𝑡) is a homoclinic solution of 𝐹𝑧(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 𝜇_𝐹(−𝐵_∞(−𝑡))−𝜇_𝐹(−𝐵₀(−𝑡))=𝜇_𝐹(𝐵₀(𝑡))−𝜇_𝐹(𝐵_∞(𝑡)). Thus, we only consider the case of 𝑅⁺_∞ from now on.

3. Proof of Our Main Results

Proof of Lemma 1. Recall the operator 𝐹=−𝒥(𝑑/𝑑𝑡)+𝐿(𝑡), with domain 𝐷(𝐹)=𝐻¹(ℝ,ℝ^2𝑁) if 𝐿(𝑡) is bounded and 𝐷(𝐹)⊂𝐻¹(ℝ,ℝ^2𝑁) 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 (𝑢,𝑣)_𝐸=𝐹^1/2𝑢,𝐹^1/2𝑣₂+(𝑢,𝑣)₂,‖𝑢‖_𝐸=(𝑢,𝑢)_𝐸^1/2, (52) where (⋅,⋅)_𝐿² denotes the usual inner product on 𝐿²(ℝ,ℝ^2𝑁). From (𝐿₁), there is a matrix 𝐿₀ such that 𝑃(𝐿(𝑡)−𝐿₀)≥0 for all 𝑡∈ℝ. We have 𝐿(𝑡)=𝐹−𝐹₀+𝐿₀, with 𝐹₀=−𝒥(𝑑/𝑑𝑡)+𝐿₀ and 𝐷(𝐹₀)=𝐻¹. Thus, for any 𝑧∈𝐸, 𝐿(𝑡)𝑧,𝑃^𝑇𝑧_𝐿2≤𝐹𝑧,𝑃^𝑇𝑧_𝐿2+𝐹₀𝑧,𝑃^𝑇𝑧_𝐿2+𝐿₀𝑧,𝑃^𝑇𝑧_𝐿2≤𝑐‖𝑧‖²_𝐸. (53) Let 𝐾⊂𝐸 be a bounded set. We will show that 𝐾 is precompact in 𝐿^𝑝 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 _|𝑡|>𝑅|𝑧|²𝑑𝑡≤𝑐|𝑅|^𝛼−2_|𝑡|>𝑅𝐿(𝑡)𝑧,𝑃^𝑇𝑧_ℝ2𝑁|𝑅|𝑑𝑡≤𝑐^𝛼−2‖𝑧‖²_𝐸. (54) For any 𝜀>0, from (54), we can choose 𝑅₀ large enough, such that _|𝑡|>𝑅0|𝑧|²𝜀𝑑𝑡<²4,∀𝑧∈𝐾. (55) On the other hand, by the definition of ‖⋅‖_𝐸, we have _{|𝑡|≤𝑅0}|𝑧|²𝑑𝑡≤‖𝑧‖_𝐸≤𝐶,∀𝑧∈𝐾. (56) Thus, by the Sobolev compact embedding theorem, there exist 𝑧₁,𝑧₂,…,𝑧_𝑚∈𝐾, such that for any 𝑧∈𝐾, there is 𝑧_𝑖 satisfying 𝑧−𝑧_𝑖^𝑝_{𝐿𝑝((−𝑅0,𝑅0),ℝ2𝑁)}<𝜀²2. (57) From (55) and (57), we have ‖𝑧−𝑧_𝑖‖_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 𝐻^1/2, hence, by the Sobolev embedding theorem, 𝐸 is continuously embedded in 𝐿^𝑝, for all 𝑝>2. For any 𝑝>2, by the Hölder inequality we have _ℝ|𝑧|^𝑝𝑑𝑡≤‖𝑧‖_𝐿²‖𝑧‖_𝐿^𝑝−1_2(𝑝−1)≤𝐶‖𝑧‖_𝐿²‖𝑧‖_𝐸^𝑝−1, (58) thus, the embedding 𝐸↪𝐿^𝑝 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 𝛼_𝑝 satisfying 𝛼_𝑝∈(0,𝛼) and (𝛼_𝑝/(2−𝑝)⋅𝑝)>1. Denote that 𝑟=𝛼_𝑝/(2−𝑝). For 𝑅>0 and 𝑧∈𝐸, denote that 𝐸¹_𝑅(𝑧)={𝑡;|𝑡|≥𝑅and|𝑡|^𝑟|𝑧(𝑡)|>1} and 𝐸²_𝑅(𝑧)={𝑡;|𝑡|≥𝑅and|𝑡|^𝑟|𝑧(𝑡)|≤1}. Then, from (53), _{𝐸1𝑅(𝑧)}|𝑧|^𝑝𝑑𝑡=_{𝐸1𝑅(𝑧)}|𝑡|^𝑟|𝑧|^𝑝|𝑡|^−𝑟𝑝≤𝑑𝑡_{𝐸1𝑅(𝑧)}|𝑧|²|𝑡|^𝛼𝑝≤𝑐𝑑𝑡|𝑅|^{𝛼−𝛼𝑝}𝐿(𝑡)𝑧,𝑃^𝑇𝑧_𝐿2≤𝑐|𝑅|^{𝛼−𝛼𝑝}‖𝑧‖²_𝐸, (59) and so _{|𝑡|≥𝑅}|𝑧|^𝑝=𝑑𝑡_{𝐸1𝑅(𝑧)}|𝑧|^𝑝𝑑𝑡+_{𝐸2𝑅(𝑧)}|𝑧|^𝑝≤𝑐𝑑𝑡|𝑅|^{𝛼−𝛼𝑝}‖𝑧‖²_𝐸+2𝑟𝑝−1𝑅^𝑟𝑝−1,∀𝑧∈𝐸. (60) Let 𝐾⊂𝐸 be a bounded set. For any 𝜀>0, from (60), choose 𝑅₀>0 large enough, such that _{|𝑡|≥𝑅0}|𝑧|^𝑝𝜀𝑑𝑡<^𝑝4,∀𝑧∈𝐾. (61) On the other hand, by the Sobolev compact embedding theorem, there are 𝑧₁,𝑧₂,…,𝑧_𝑚∈𝐾, such that for any 𝑧∈𝐾, there exists 𝑧_𝑖 satisfying 𝑧−𝑧_𝑖^𝑝_{𝐿𝑝((−𝑅0,𝑅0),ℝ2𝑁)}<𝜀^𝑝2. (62) From (61) and (62), we have 𝑧−𝑧_𝑖𝐿𝑝<𝜀; (63) that is to say, 𝐾 has a finite 𝜀-net in 𝐿^𝑝, and the embedding 𝐸↪𝐿^𝑝 is compact. The proof of the lemma is complete.

Consider the homoclinic orbits of the linear Hamiltonian systems 𝑧̇(𝑡)(𝑡)𝑧(𝑡)𝑧(𝑡)|𝑡|=𝐽𝐵,∀𝑡∈ℝ,⟶0,⟶∞, (64) where 𝑧(𝑡)∶ℝ→ℝ^2𝑁,  𝐽=^0−𝐼𝑁_𝐼𝑁0, 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,ℝ^2𝑁) and we have the following lemma.

Lemma 14. The dimension of the solution space 𝑆 will be less than or equal to 𝑁. Thus for any homoclinic orbit 𝑧(𝑡) of (HS), if 𝑅 satisfies (𝑅₁), one has 0≤𝜐_𝐹(𝑧)≤𝑁. (65)

Proof. As usual, we define the symplectic groups on ℝ^2𝑁 by 𝑆𝑝(2𝑁)=𝑀∈ℒℝ^2𝑁,∣𝑀^𝑇𝐽𝑀=𝐽, (66) where ℒ(ℝ^2𝑁) is the set of all 2𝑁×2𝑁 real matrices, and 𝑀^𝑇 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 𝑧(𝑡)=𝑊(𝑡)𝑧(0),lim_𝑡→∞𝑊(𝑡)𝑧(0)=0. (67) Denote by 𝑆₀ the subset of ℝ^2𝑁 satisfying 𝑆₀=𝑧∈ℝ^2𝑁∣lim_𝑡→∞𝑊(𝑡)𝑧=0, (68) 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, lim_𝑡→∞𝑊(𝑡)𝑧₀=0,lim_𝑡→∞𝑊(𝑡)𝐽𝑧₀=0. (69) Since 𝑊(𝑡) is a path in 𝑆𝑝(2𝑁), 𝑊^𝑇(𝑡)𝐽𝑊(𝑡)=𝐽, for all 𝑡∈ℝ, thus 0=lim_𝑡→∞𝐽𝑊(𝑡)𝑧₀,𝑊(𝑡)𝐽𝑧_0ℝ2𝑁=−lim_𝑡→∞𝑧₀,𝑊^𝑇(𝑡)𝐽𝑊(𝑡)𝐽𝑧_0ℝ2𝑁=𝑧₀,𝑧_0ℝ2𝑁, (70) which contradicts 𝑧₀≠0. Since 𝐽 is an isomorphism on ℝ^2𝑁, we have dim𝑆₀≤𝑁. And from the definition of 𝜐_𝐹(𝑧) in the last part of Section 2.2, we have completed the proof.

Before the proof of Theorem 2, 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 𝑃=^𝛽_−𝛽𝑑𝐸_𝜆,𝑋=𝑃𝐿²ℝ,ℝ^2𝑁. (71) 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)  𝐻_𝑞(𝑋,𝑎;ℝ)≅𝛿_𝑞,𝑟ℝ,𝑞=0,1,… for −𝑎∈ℝ large enough, where 𝑟=dim(𝐸⁻(𝑋))+𝜇_𝐹(𝐾𝐵₁).

Proof. Assume that there is a sequence {𝑥_𝑛}⊂𝑋, satisfying 𝑎^′(𝑥_𝑛)→0(𝑛→∞). That is, 𝐹𝑧_𝑛−𝐾∇_𝑧𝑅𝑡,𝑧_𝑛𝐸⟶0, (72) where 𝑧_𝑛=𝑧(𝑥_𝑛) defined in Section 2.1. Since 𝑋 is a finite dimensional space, and from the definition of 𝑧_𝑛, it is enough to prove that {𝑧_𝑛} is bounded in 𝐸. For each 𝜀∈(0,1), define 𝐶_𝑛∈ℬ by 𝐶_𝑛(𝑡)=∫¹₀∇²_𝑧𝑅𝑡,𝑠𝑧_𝑛𝑑𝑠,𝑧_𝑛(𝑡≥𝑅)⁰𝜀,𝐵₁(𝑡),𝑧_𝑛(𝑡<𝑅)⁰𝜀. (73) It is easy to verify that {𝐶_𝑛} satisfies 𝐵₁(𝑡)−𝜀𝐵₁(𝑡)+𝑐⋅𝐼≤𝐶_𝑛(𝑡)≤𝐵₂(𝑡)+𝜀𝑐⋅𝐼−𝐵₂(𝑡),∀𝑡∈ℝ, (74) where 𝑐 is the constant in condition (𝑅₁) and 𝐼 is the identity map on ℝ^2𝑁. Since 𝐵₁≤𝐵₂,  𝜇_𝐹(𝐵₁)=𝜇_𝐹(𝐵₂) and 𝜐_𝐹(𝐾𝐵₁)=𝜐_𝐹(𝐾𝐵₂)=0, we can choose 𝜀 small enough, such that for each 𝑛∈ℕ⁺, 𝜇_𝐹(𝐾𝐶_𝑛)=𝜇_𝐹(𝐾𝐵₁) and 𝜐_𝐹(𝐾𝐶_𝑛)=0. Thus 𝐹−𝐾𝐶_𝑛 is reversible on 𝐸, and there is a constant 𝛿>0, such that (𝐹−𝐾𝐶_𝑛)𝑧_𝐸≥𝛿‖𝑧‖_𝐸,∀𝑧∈𝐸,𝑛∈ℕ⁺. (75) On the other hand, for 𝑏∈(0,1), there is a constant 𝑐>0 depending on 𝑏, such that for each 𝑛∈ℕ⁺, ∇_𝑧𝑅𝑡,𝑧_𝑛(𝑡)−𝐶_𝑛𝑧_𝑛(𝑡)≤𝑐𝑧_𝑛(𝑡)^𝑏,∀𝑡∈ℝ. (76) Choose 𝑏>(1−𝛼)/(1+𝛼) in (76); that is, 1+𝑏∈(2/(1+𝛼),2), we have 𝐹𝑧_𝑛−𝐾∇_𝑧𝑅𝑡,𝑧_𝑛−𝐹−𝐾𝐶_𝑛𝑧_𝑛²_𝐸=𝐾∇_𝑧𝑅𝑡,𝑧_𝑛−𝐶_𝑛𝑧_𝑛²_𝐸≤∇_𝑧𝑅𝑡,𝑧_𝑛−𝐶_𝑛𝑧_𝑛²_𝐿2≤𝑐_ℝ∇_𝑧𝑅𝑡,𝑧_𝑛−𝐶_𝑛𝑧_𝑛𝑧_𝑛^𝑏𝑧_𝑛^1+𝑏𝑧𝑑𝑡≤𝑐_𝑛𝐿^1+𝑏_1+𝑏. (77) 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 {𝑧_𝑛} is bounded in 𝐸, and 𝑎 satisfies the (PS) conditions. And by Lemma  5.1 in Chapter II of [21], we have 𝐻_𝑞𝑋_,(𝑎)_𝛼;ℝ≅𝛿_𝑞,𝑟ℝ,𝑞=0,1,…, (78) for −𝛼∈ℝ large enough.

From Theorem 11, 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 Theorems 3 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 𝑅_𝑘∈𝐶²(ℝ×ℝ^2𝑁,ℝ),𝑘∈ℕ, satisfying the following properties. (1)There exists an increasing sequence of real numbers 𝑀_𝑘→∞(𝑘→∞) such that 𝑅_𝑘(𝑡,𝑧)≡𝑅(𝑡,𝑧),∀𝑡∈ℝ,|𝑧|≤𝑀_𝑘. (79)(2)For each 𝑘∈ℕ, there is a 𝐶>0 independent of 𝑘, such that ∇²_𝑧𝑅_𝑘(𝑡,𝑧)≤𝐶,∀𝑡∈ℝ,𝑧∈ℝ^2𝑁,∇²_𝑧𝑅_𝑘(𝑡,𝑧)≥𝐵_∞,∀𝑡∈ℝ,|𝑧|≥𝑅₀. (80)(3)For each 𝑘∈ℕ, there exist some 𝐶_𝑘>0 and a constant 𝛾 with 𝛾𝐼_2𝑁>𝐵_∞, 𝜈_𝐹(𝐾𝛾𝐼_2𝑁)=0, such that ∇_𝑧𝑅_𝑘(𝑡,𝑧)−𝛾𝑧<𝐶_𝑘,∀(𝑡,𝑧)∈ℝ×ℝ^2𝑁, (81) where 𝐼_2𝑁 is the identity map on ℝ^2𝑁.

Proof. Define 𝜂∶[0,∞)→ℝ by 𝜂(𝑠)=20,0≤𝑠<1,9(𝑠−1)³−19(𝑠−1)⁴,1≤𝑠<2,1−128912+𝑠²,2≤𝑠<∞. (82) It is easy to see that 𝜂∈𝐶²([0,∞),ℝ). Choose a sequence {𝑀_𝑘} of positive numbers such that 𝑅₀<𝑀₁<𝑀₂<⋯<𝑀_𝑘<⋯→∞ as 𝑘→∞. For each 𝑘∈ℕ, let 𝜂_𝑘(𝑠)=𝜂(𝑠/𝑀_𝑘) and 𝑅_𝑘(𝑡,𝑧)=1−𝜂_𝑘𝑧𝑅()+𝛾(||)𝑡,𝑧2𝜂_𝑘(|𝑧|)|𝑧|²,𝑘∈ℕ. (83) As in [23, 29], we can check that 𝑅_𝑘 satisfies (79)–(81) for each 𝑘∈ℕ.

For each 𝑘∈ℕ, we consider the following problem 𝐹𝑧=∇_𝑧𝑅_𝑘(),𝑧(𝑡)𝑡,𝑧⟶0,𝑧^′(𝑡)⟶0,𝑡⟶∞, (HS)_𝑘 where 𝑅_𝑘 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 𝛽>max2(𝐶+1),2𝛾+1,𝛽∉𝜎𝐴₀. (84) Let 𝑃_𝛽=^𝛽_−𝛽𝑑𝐸_𝜆,𝑋_𝛽=𝑃_𝛽𝐿²ℝ,ℝ^2𝑁. (85) Thus, for each 𝑘 and such a 𝛽 fixed, by Theorem 8, we have a functional 𝑎_𝑘,𝛽(𝑥),𝑥∈𝑋_𝛽, (86) whose critical points give rise to solutions of (HS)_𝑘. Similarly we have a functional 𝑎_𝛾,𝛽(𝑥),𝑥∈𝑋_𝛽, (87) whose critical points give rise to solutions of the following systems (HS)_𝛾𝑧(𝑡)𝐹𝑧=𝛾𝑧,⟶0,𝑧^′(𝑡)⟶0,𝑡⟶∞. (HS)_𝛾 For notational simplicity, we denote 𝑎_𝑘,𝑎_𝛾 for 𝑎_𝑘,𝛽 and 𝑎_𝛾,𝛽. Define Φ_𝑘(𝑧)=^∞_−∞𝑅_𝑘(𝑡,𝑧)𝑑𝑡. (88) For the functional 𝑎_𝑘, similar to Lemma 15, we have the following lemma.

Lemma 17. (1)  𝑎_𝑘 satisfies (PS) condition, and the critical point set of 𝑎_𝑘 is compact;
(2)  𝐻_𝑞(𝑋_𝛽,(𝑎_𝑘)_𝛼𝑘;ℝ)≅𝛿_{𝑞,𝑟𝛽}ℝ,𝑞=0,1,… for −𝛼_𝑘∈ℝ large enough, where 𝑟_𝛽=dim(𝐸⁻(𝑋_𝛽))+𝜇_𝐹(𝐾𝛾𝐼_2𝑁).

Proof. The proof is similar to Lemma 15. From Theorem 11, we have 𝑎^′_𝑘(𝑥)−𝑎_𝛾^′′(0)𝑥_𝐿2=𝑃_𝛽𝐾(∇_𝑧Φ_𝑘(𝑧_𝑘(𝑥))−𝛾𝑥)_𝐿2≤𝑃_𝛽∇_𝑧𝑅_𝑘𝑧_𝑘(𝑥)−𝛾𝑧_𝑘(𝑥)_𝐿2≤∇_𝑧𝑅_𝑘𝑧_𝑘(𝑥)−𝛾𝑧_𝑘(𝑥)_𝐿2. (89) Similar to (76), for 𝑏∈(0,1), there is some 𝑐>0, such that ∇_𝑧𝑅_𝑘𝑡,𝑧_𝑘−𝛾𝑧_𝑘≤𝑐𝑧_𝑘(𝑡),∀𝑡∈ℝ. (90) Choose 𝑏∈((1−𝛼)/(1+𝛼),1), similar to (77), and we have ∇_𝑧𝑅_𝑘(𝑡,𝑧_𝑘)−𝛾𝑧_𝑘²_𝐿2≤𝑐𝑧_𝑘𝐿^1+𝑏_1+𝑏. (91) From Lemma 1, Remark 10, and (91), ∇_𝑧𝑅_𝑘𝑡,𝑧_𝑘−𝛾𝑧_𝑘²_𝐿2≤𝑐𝑧_𝑘𝐸^1+𝑏≤𝑐𝛽‖𝑥‖_𝐿^1+𝑏₂. (92) From (89) and (92), we have 𝑎^′_𝑘(𝑥)−𝑎_𝛾^′′(0)𝑥²_𝐿2≤𝑐𝛽‖𝑥‖_𝐿^1+𝑏₂. (93) Now, for each 𝑘∈ℕ, we assume that {𝑥_𝑚}⊂𝑋_𝛽 satisfying ‖𝑎^′_𝑘(𝑥_𝑚)‖→0. By 𝜈_𝐹(𝐾𝛾𝐼_2𝑁)=0, we have that 𝑎_𝛾^′′(0) is invertible on 𝑋_𝛽, since 𝑏<1 the sequence {𝑥_𝑚} must be bounded. Thus the (PS) condition for 𝑎_𝑘 holds. From the same reason, we have the compactness of the critical point set of 𝑎_𝑘. And by Lemma  5.1 in Chapter II of [21], we have 𝐻_𝑞𝑋_𝛽,𝑎_𝑘𝛼𝑘;ℝ≅𝛿_{𝑞,𝑟𝛽}ℝ,𝑞=0,1,…, (94) for −𝛼_𝑘∈ℝ large enough.

Lemma 18. There exist 𝑐>0, such that for any 𝑘∈ℕ, and 𝑧∈𝐿² satisfies the systems (HS)_𝑘, if 𝜇_𝐹(𝑧)≤𝜇_𝐹(𝐾𝐵_∞)−1, one has ‖𝑧‖_𝐿^∞≤𝑐.

Proof. We prove it indirectly. Assume there exist 𝑅_𝑘 and 𝑧_𝑘 satisfying the conditions, and ‖𝑧_𝑘‖_𝐿∞→∞; that is, ‖𝑧‖_𝐹→∞. Since |∇_𝑧𝑅_𝑘(𝑡,𝑧)|<𝑐|𝑧|, for all 𝑡∈ℝ,𝑧∈ℝ^2𝑁, we have ‖𝑧_𝑘‖_𝐹≤𝑐‖𝑧_𝑘‖_𝐿2. Denote that 𝑦_𝑘=𝑧_𝑘/‖𝑧_𝑘‖_𝐹; then, we have 𝑦_𝑘→𝑦 in 𝐿² for some 𝑦∈𝐿² with ‖𝑦‖_𝐿2>0, and 𝐹𝑦_𝑘=𝑅^′_𝑘𝑡,𝑧_𝑘𝑧_𝑘𝐹. (95) Then, for any 𝑟>0, there exists 𝐶_𝑟>0, satisfying 𝑦̇_𝑘(𝑡)≤𝐶_𝑟𝑦_𝑘(𝑡),𝑡∈𝐼_𝑟, (96) where 𝐼_𝑟=[−𝑟,𝑟]. Since ‖𝑦‖_𝐿2>0, there is a 𝑟₀>0, such that 𝑦_{𝐿2(−𝑟,𝑟)}>12𝑦_𝐿2>0,∀𝑟>𝑟₀. (97) Then, from the similar argument in [23], there is a subsequence throught which we may assume that {𝑦_𝑘} converges in uniform norm to 𝑦, and 𝑦(𝑡)≠0, for all 𝑡∈𝐼_𝑟. Therefore |𝑧_𝑘(𝑡)|→∞ uniformly on 𝐼_𝑟, and there is 𝐾(𝑟) depending on 𝑟, such that |𝑧_𝑘(𝑡)|≥𝑅₀, for any 𝑡∈𝐼_𝑟 and 𝑘≥𝐾(𝑟).
Performing the saddle point reduction on the following systems: 𝐹𝑧=𝐵_∞𝑧(𝑡)𝑧,⟶0,𝑧^′(𝑡)⟶0,𝑡⟶∞, (HS)_∞ for 𝛽 large enough, we have the functional 𝑎_∞,𝛽 (denote by 𝑎_∞ for simplicity) and the function 𝑧(𝑥); since 𝜐_𝐹(𝐾𝐵_∞)=0, we have the following decomposition: 𝑋_𝛽=𝑋⁺_𝛽+𝑋⁻_𝛽, (98) where 𝑎_∞^′′(0) is positive definite on 𝑋⁺_𝛽 and negative definite on 𝑋⁻_𝛽. From Remark 10, and 𝜐_𝐹(𝐾𝐵_∞)=0, there exists 𝛼>0, such that for 𝛽 large enough, 𝐹−𝐵_∞(𝑡)𝑥,𝑥_𝐿2≤−𝛼‖𝑥‖²_𝐿2,∀𝑥∈𝑋⁻_𝛽. (99) From the uniform boundary of ∇²_𝑧𝑅_𝑘(𝑡,𝑧) and Remark 10, we can choose 𝛽 large enough, such that ∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘𝑧_𝑘^′±𝑥_𝑘𝑥,𝑥_𝐿2≤𝛼4‖𝑥‖_𝐿²,∀𝑥∈𝐿², (100) where 𝑥_𝑘=𝑃_𝛽𝑧_𝑘, 𝑧_𝑘(𝑥_𝑘)=𝑧_𝑘(𝑡) defined in Theorem 8. Choose 𝜀>0 small enough and 𝜀<𝛼/4, such that 𝜇_𝐹(𝐾𝐵_∞)=𝜇_𝐹(𝐾(𝐵_∞−𝜀⋅𝐼𝑑)). Since 𝑋⁻_𝛽 is finite dimensional space, choose 𝑟 large enough, such that ∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘−𝐵_∞𝑥,𝑥_{𝐿2(𝐼𝑐𝑟)}≥−𝜀(𝑥,𝑥)_𝐿²,∀𝑥∈𝑋⁻_𝛽, (101) where 𝐼^𝑐_𝑟=ℝ⧵𝐼_𝑟, and from the definition of 𝑅_𝑘, ∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘(𝑡)≥𝐵_∞(𝑡),𝑡∈𝐼_𝑟,𝑘≥𝐾(𝑟); (102) that is, ∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘−𝐵_∞𝑥,𝑥_{𝐿2(𝐼𝑟)}≥0,∀𝑥∈𝑋⁻_𝛽,𝑘≥𝐾(𝑟). (103) From (101) and (103), ∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘𝑥,𝑥_𝐿2≥𝐵_∞𝑥,𝑥_𝐿2−𝜀(𝑥,𝑥)_𝐿², (104) for 𝑘 large enough. Thus we have 𝑎_𝑘^′′𝑥_𝑘𝑥,𝑥_𝐿2=𝐹−∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘𝑥,𝑥_𝐿2−∇²_𝑧𝑅_𝑘𝑡,𝑧_𝑘𝑧_𝑘^′++𝑧_𝑘^′−𝑥,𝑥_𝐿2≤𝐹−𝐵_∞𝑥,𝑥_𝐿2+𝛼2‖𝑥‖²_𝐿2+𝜀‖𝑥‖²_𝐿2𝛼≤−4‖𝑥‖²_𝐿2. (105) That is, 𝑚⁻_𝑎𝑘(𝑥)≥𝑚⁻_𝑎∞(0), from Theorem 11, 𝑚⁻_𝑎𝑘(𝑥_𝑘)=dim(𝐸⁻(ℋ₀))+𝜇_𝐹(𝑧_𝑥(𝑥_𝑘)), 𝑚⁻_𝑎∞(0)=dim(𝐸⁻(ℋ₀))+𝜇_𝐹(𝐾𝐵_∞), and thus 𝜇_𝐹(𝑧_𝑥)≥𝜇_𝐹(𝐾𝐵_∞), which contradicts the assumption.

Proof of Theorem 3. As claimed in Remark 13, we can only consider the case of (𝑅⁺_∞). Note that 𝑧=0 is a critical point of 𝑎_𝑘, and the morse index of 0 for 𝑎_𝑘 is 𝑚⁻_𝑎𝑘(0)=dim(𝐸⁻(ℋ₀))+𝜇_𝐹(𝐾𝐵₀), since 𝛾⋅𝐼_2𝑁>𝐵_∞, we have 𝜇_𝐹𝐾𝛾⋅𝐼_2𝑁≥𝜇_𝐹𝐾𝐵_∞≥𝜇_𝐹𝐾𝐵₀. (106) From proposition (2) in Lemma 17, use the (𝑚⁻_𝑎𝑘(0))th and (𝑚⁻_𝑎𝑘(0)+1)th Morse inequalities, and 𝑎_𝑘 has a nontrivial critical point 𝑥_𝑘 with its morse index 𝑚⁻_𝑎𝑘(𝑥_𝑘)≤𝑚⁻_𝑎𝑘(0)+1; that is, 𝜇_𝐹(𝑧_𝑘)≤𝜇_𝐹(𝐾𝐵₀)+1≤𝜇_𝐹(𝐾𝐵_∞)−1, then from Lemma 18, we have that {𝑧_𝑘} is bounded in 𝐿^∞. Thus 𝑧_𝑘 is a nontrivial solution of (HS) for 𝑘 large enough.

The proof of Theorem 4 is similar to the proof of Theorem 3. Instead of Morse theory we make use of minimax arguments for multiplicity of critical points.

Let 𝑋 be a Hilbert space and assume 𝜙∈𝐶²(𝑋,ℝ) is an even functional, satisfying the (PS) condition and 𝜙(0)=0. Denote 𝑆_𝑐={𝑢∈∣𝑋‖𝑢‖=𝑐}.

Lemma 19 (see [30, Corollary  10.19]). Assume that 𝑌 and 𝑍 are subspaces of 𝑋 satisfying dim𝑌=𝑗>𝑘=codim𝑍. If there exist 𝑅>𝑟>0 and 𝛼>0 such that inf𝜙𝑆_𝑟∩𝑍≥𝛼,sup𝜙𝑆_𝑅∩𝑌≤0, (107) then 𝜙 has 𝑗−𝑘 pairs of nontrivial critical points {±𝑥₁,±𝑥₂,…,±𝑥_𝑗−𝑘}, so that 𝜇(𝑥_𝑖)≤𝑘+𝑖, for 𝑖=1,2,…,𝑗−𝑘.

First, we consider the case of (𝑅⁺_∞), since 𝑅 is even, we have that 𝑅_𝑘 is also even and satisfies Lemma 16. Let 𝑌=𝑋⁻_𝛽, and 𝑍 the positive space of 𝑎_𝑘^′′(0) in 𝑋_𝛽, and we have and dim𝑌=𝐸⁻(𝑋_𝛽)+𝜇_𝐹(𝐾𝐵_∞), codim𝑍=𝐸⁻(𝑋_𝛽)+𝜇_𝐹(𝐾𝐵₀), dim𝑌>codim𝑍. So, 𝑎_𝑘 has 𝑙∶=𝜇_𝐹(𝐾𝐵_∞)−𝜇_𝐹(𝐾𝐵₀) pairs of nontrivial critical points ±𝑥₁,±𝑥₂,…,±𝑥_𝑙, (108) and 𝑙−1 pairs of them satisfy 𝑚⁻𝑥_𝑖≤𝜇_𝐹𝐾𝐵₀+𝑖<𝜇_𝐹𝐾𝐵_∞,𝑖=1,2,…,𝑙−1. (109) 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 that 𝑌 and 𝑍 are subspaces of 𝑋 satisfying dim𝑌=𝑗>𝑘=codim𝑍. If there exist 𝑟>0, and 𝛼>0 such that inf𝜙(𝑍)>−∞,sup𝜙𝑆_𝑟∩𝑌≤−𝛼, (110) then 𝜙 has 𝑗−𝑘 pairs of nontrivial critical points ±𝑢₁,±𝑢₂,…,±𝑢_𝑗−𝑘 so that 𝜇(𝑢_𝑖)+𝜈(𝑢_𝑖)≥𝑘+𝑖−1 for 𝑖=1,2,…,𝑗−𝑘.

The proof is similar to the case of (𝑅⁺_∞), and we omit it here.

Acknowledgment

This work is partially supported by NFSC (11126154, 11126146, and 11201196).