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

Abstract

We study the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems 𝑞̈−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.

1. Introduction

Consider the following second-order non-autonomous Hamiltonian system𝑞̈−𝐿(𝑡)𝑞+∇_𝑞𝑊𝑡,𝑞=0, (1.1) where 𝐿(𝑡)∈𝐶(ℝ,ℝ^𝑁×𝑁) is a symmetric matrix-valued function, 𝑊∶ℝ×ℝ^𝑁→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 𝑞(̇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, (𝐿(𝑡)𝑢,𝑢)>0,∀𝑡∈ℝ,𝑢∈ℝ^𝑁⧵{0}. (1.2)

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 an 𝛼<2 such that 𝑙(𝑡)|𝑡|^𝛼−2⟶∞as|𝑡|⟶∞, (1.3) where 𝑙(𝑡) is the smallest eigenvalue of 𝐿(𝑡), that is, 𝑙(𝑡)≡inf_𝜉=1(𝐿(𝑡)𝜉,𝜉). (1.4)

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 by ℬ the set of all uniformly bounded symmetric 𝑁×𝑁 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^𝑁. 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 on 𝑊(𝑡,𝑞) and the nonlinearity ∇_𝑞𝑊(𝑡,𝑞) as follows. For notational simplicity, we set 𝐵₀(𝑡)=∇²_𝑞𝑊(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×ℝ^𝑁,ℝ), and there exists a constant 𝑐>0 such that∇²_𝑞𝑊𝑡,𝑞≤𝑐,∀𝑡,𝑞∈ℝ×ℝ^𝑁. (1.5)(𝑊₀)∇_𝑞𝑊(𝑡,0)≡0 and 𝐵₀∈ℬ,(𝑊_∞)there exists some 𝑅₀>0 and continuous symmetric matrix functions 𝐵₁,𝐵₂∈ℬ with 𝑖(𝐵₁)=𝑖(𝐵₂) and 𝜈(𝐵₂)=0 such that𝐵₁(𝑡)≤∇²_𝑞𝑊(𝑡,𝑧)≤𝐵₂(𝑡),∀𝑡∈ℝ,|𝑧|>𝑅₀. (1.6) Our first result reads as follows.

Theorem 1.1. Assume (𝐿₁), (𝑊₁), (𝑊₀), and (𝑊_∞) hold. If 𝑖𝐵₁∉𝑖𝐵₀,𝑖𝐵₀+𝜈𝐵₀, (1.7) 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 some 𝑅₀>0 and a continuous symmetric matrix function 𝐵_∞∈ℬ with 𝜈(𝐵_∞)=0 such that±∇²_𝑞𝑊𝑡,𝑞≥±𝐵_∞(𝑡),∀𝑡∈ℝ,𝑞>𝑅₀. (1.8)

The uniform boundary of ∇²_𝑞𝑊(𝑡,𝑞) displayed in condition (𝑊₁) can also be relaxed as(𝑊^∗₁)𝑊∈𝐶²(ℝ×ℝ^𝑁,ℝ), and there exists a constant 𝑐>0 such that ∇_𝑞𝑊𝑡,𝑞≤𝑐𝑞,∀𝑡,𝑞∈ℝ×ℝ^𝑁. (1.9)(𝑊₁^∗∗)For any 𝑀>0, ∇²_𝑞𝑊(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), (𝑊^∗₁), (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), (𝑊^∗₁), (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 assumption 𝜈(𝐵_∞)=0 in (𝑊^±_∞) is not essential for our main results. For the case of (𝑊⁺_∞) with 𝜈(𝐵_∞)≠0, let 𝐵_∞=𝐵_∞−𝜀𝐼_𝑛×𝑛 with 𝜀>0 small enough, where 𝐼_𝑛×𝑛 is the identity map on ℝ^𝑁, then 𝑖(𝐵_∞)=𝑖(𝐵_∞) and 𝜈(𝐵_∞)=0, and hence (𝑊⁺_∞) holds for 𝐵_∞. 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 𝐵_∞=𝐵_∞+𝜀𝐼_𝑛×𝑛 with 𝜀>0 small enough such that 𝑖(𝐵_∞)=𝑖(𝐵_∞)+𝜈(𝐵_∞) and 𝜈(𝐵_∞)=0, then this case is also reduced to the case of (𝑊⁻_∞) for 𝐵_∞ with 𝜈(𝐵_∞)=0.

Remark 1.5. Choose 𝑊(𝑡,𝑞)=𝑊(𝑡,𝑞)−𝑊(𝑡,0) instead of 𝑊 in (1.1), then conditions (𝑊^∗₁), (𝑊₁^∗∗), (𝑊₀), and (𝑊⁺_∞)(𝑜𝑟(𝑊⁻_∞)) still hold for 𝑊, so we can always assume 𝑊(𝑡,0)≡0.

2. Preliminaries

Denote by 𝐴 the self-adjoint extension of the operator −𝑑²/𝑑𝑡²+𝐿(𝑡) with domain 𝐷(𝐴)⊂𝐿²≡𝐿²(ℝ,ℝ^𝑁). Let {𝐸(𝜆)∶−∞<𝜆<∞} and |𝐴| be the spectral resolution and the absolute value of 𝐴, respectively, and let |𝐴|^1/2 be the square root of |𝐴| with domain 𝐷(|𝐴|). Set 𝑈=𝐼−𝐸(0)−𝐸(−0), where 𝐼 is the identity map on 𝐿². Then, 𝑈 commutes with 𝐴, |𝐴|, and |𝐴|^1/2, and 𝐴=𝑈|𝐴| is the polar decomposition of 𝐴. Let 𝐸=𝐷(|𝐴|^1/2), and define on 𝐸 the inner product and norm by(𝑢,𝑣)₀=𝐴^1/2𝑢,𝐴^1/2𝑣₂+(𝑢,𝑣)₂,‖𝑢‖₀=(𝑢,𝑢)₀^1/2, (2.1) where (⋅,⋅)_𝐿2 denotes the usual inner product on 𝐿²(ℝ,ℝ^𝑁). Then, 𝐸 is a Hilbert space. It is easy to see that 𝐸 is continuously embedded in 𝑊¹(ℝ,ℝ^𝑁). In fact, we further have the following lemmas.

Lemma 2.1 (see [16], Lemma 2.2). Suppose that 𝐿 satisfies (𝐿₁). Then, 𝐸 is compactly embedded in 𝐿^𝑝(ℝ,ℝ^𝑁) 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 𝐴 possesses a compact resolvent. Therefore, the spectrum 𝜎(𝐴) consists of only eigenvalues numbered in 𝜂₁≤𝜂₂≤⋯→∞ (counted with multiplicity), and the corresponding system of eigenfunctions {𝑒_𝑛∶𝑛∈ℕ}(𝐴𝑒_𝑛=𝜂_𝑛𝑒_𝑛) forms an orthogonal basis in 𝐿².

Let𝑛⁻=#𝑖∣𝜆_𝑖<0,𝑛⁰=#𝑖∣𝜆_𝑖=0,𝑛=𝑛⁻+𝑛⁰,𝐸⁻=span𝑒₁,…,𝑒_𝑛−,𝐸⁰=span𝑒_𝑛−+1,…,𝑒_𝑛,𝐸⁺=span{𝑒_𝑛+1,…}, (2.2) 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:(𝑢,𝑣)_𝐸=|𝐴|^1/2𝑢,𝐴^1/2𝑣_𝐿2+𝑢⁰,𝑣⁰_𝐿2,‖𝑢‖_𝐸=(𝑢,𝑢)_𝐸^1/2, (2.3) 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 decomposition 𝐸=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 Lemma 2.1 and the equivalence of the norms ‖⋅‖_𝐸 and ‖⋅‖₀, there exists a constant 𝑐_∞>0 such that‖𝑞‖_𝐿∞≤𝑐_∞‖𝑞‖_𝐸,∀𝑞∈𝐸. (2.4)

Define the quadratic form 𝑎 on 𝐸 by𝑎(𝑢,𝑣)=_ℝ((𝑢̇,𝑣)̇+(𝐿(𝑡)𝑢,𝑣))𝑑𝑡,∀𝑢,𝑣∈𝐸. (2.5) Then by definition, we have𝑎(𝑢,𝑢)=((𝑃⁺−𝑃⁻)𝑢,𝑢)_𝐸=‖𝑢⁺‖²_𝐸−‖𝑢⁻‖²_𝐸 (2.6) 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 by𝐴𝑢=𝑢⁺−𝑢⁻,∀𝑢∈𝐸,(𝐾𝑢,𝑣)_𝐸=(𝑢,𝑣)_𝐿2,∀𝑢∈𝐿²,𝑣∈𝐸, (2.7) 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 any 𝐵∈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(𝐾𝐵𝑢,𝑣)_𝐸=(𝐵𝑢,𝑣)_𝐿2,∀𝑢,𝑣∈𝐸. (2.8) 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. Denote𝑖(𝐵)=dim(𝐸⁻(𝐵,𝜈(𝐵)𝐸))=dim⁰(𝐵). (2.9)

Define the functionals Ψ and Φ on 𝐸 byΨ(𝑢)=_ℝ𝑊()Φ(𝑢)=1𝑡,𝑢𝑑𝑡,2𝑎()(𝑢)=1𝑢,𝑢−Ψ2_ℝ((𝑢̇,𝑢̇)+(𝐿(𝑡)𝑢,𝑢))𝑑𝑡−Ψ(𝑢). (2.10) 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)𝐻_𝑗(𝐸,Φ;ℝ)≅𝛿_𝑗,𝑟ℝ,𝑗=0,1,… for −𝑎∈ℝ large enough, where 𝑟=𝑖(𝐵₁).

Proof. Assume {𝑢_𝑛}⊂𝐸 with Φ′(𝑢_𝑛)→0 as 𝑛→∞. That is, ‖𝐴𝑢_𝑛−𝐾∇_𝑞𝑊(𝑡,𝑢_𝑛)‖_𝐸⟶0. (2.11) First, we prove {𝑢_𝑛} is bounded in 𝐸. For each 𝜀∈(0,1), define 𝐶_𝑛∈ℬ by 𝐶_𝑛(𝑡)=¹⁰∇²_𝑞𝑊𝑡,𝑠𝑢_𝑛𝑑𝑠,𝑢_𝑛(𝑡≥𝑅)⁰𝜀,𝐵₁(𝑡),𝑢_𝑛(𝑡<𝑅)⁰𝜀. (2.12) 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, (2.13) where 𝑐 is the constant in condition (𝑊₁) and 𝐼 is the identity map on ℝ^𝑁. 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(𝐶_𝑛)=𝑖(𝐵₁) and 𝜈(𝐶_𝑛)=0. Thus 𝐴−𝐾𝐶_𝑛 is reversible on 𝐸 and there is a constant 𝛿>0, such that ‖(𝐴−𝐾𝐶_𝑛)𝑢‖_𝐸≥𝛿‖𝑢‖_𝐸,∀𝑢∈𝐸,𝑛∈ℕ⁺. (2.14) On the other hand, for 𝑏∈(0,1), there is a constant 𝑐>0 depending on 𝑏, such that for each 𝑛∈ℕ⁺, ∇_𝑞𝑊𝑡,𝑢_𝑛(𝑡)−𝐶_𝑛𝑢_𝑛(𝑡)≤𝑐|𝑢_𝑛(𝑡)|^𝑏,∀𝑡∈ℝ. (2.15) Choose 𝑏>(𝛼−1)/(3−𝛼) in (2.15), we have (𝐴𝑢_𝑛−𝐾∇_𝑞𝑊(𝑡,𝑢_𝑛))−(𝐴−𝐾𝐶_𝑛)𝑢_𝑛²_𝐸=𝐾(∇_𝑢𝑊(𝑡,𝑢_𝑛)−𝐶_𝑛𝑢_𝑛)²_𝐸≤∇_𝑢𝑊(𝑡,𝑢_𝑛)−𝐶_𝑛𝑢_𝑛²_𝐿2≤𝑐_ℝ∇_𝑢𝑊𝑡,𝑢_𝑛−𝐶_𝑛𝑢_𝑛|𝑢_𝑛|^𝑏|𝑢_𝑛|^1+𝑏𝑢𝑑𝑡≤𝑐_𝑛𝐿^1+𝑏_1+𝑏. (2.16) 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 {𝑢_𝑛} in bounded in 𝐸. Thus, there exists a subsequence {𝑢_𝑛𝑘} such that {𝐾∇_𝑞𝑊(𝑡,𝑢_𝑛𝑘)} is convergent in 𝐸. By the definition of Φ, we have 𝑢⁺_𝑛𝑘−𝑢⁻_𝑛𝑘−𝐾∇_𝑞𝑊𝑡,𝑢_𝑛𝑘=Φ^′𝑢_𝑛𝑘⟶0in𝐸, (2.17) which yields the convergence of {𝑢⁺_𝑛𝑘} and {𝑢⁻_𝑛𝑘} in 𝐸. Additionally, passing to a subsequence, if necessary, {𝑢⁰_𝑛𝑘} is convergent in 𝐸 since dim(𝐸⁰)<∞. Thus, {𝑢_𝑛𝑘} is convergent and the (PS) condition is verified. By Lemma 5.1 in Chapter II of [25], we have 𝐻_𝑞𝑋,(𝑎)_𝛼;ℝ≅𝛿_𝑞,𝑟ℝ,𝑞=0,1,…, (2.18) for −𝛼∈ℝ large enough.

In order to prove Theorems 1.2 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 (𝑊^∗₁), (𝑊₁^∗∗), and (𝑊⁺_∞) hold, then there exists a sequence of functions: 𝑊_𝑚∈𝐶²ℝ×ℝ^𝑁,ℝ,𝑚∈ℕ, (2.19) satisfying the following properties:(1)there exists an increasing sequence of real numbers 𝑅_𝑚→∞(𝑚→∞) such that 𝑊_𝑚𝑡,𝑞≡𝑊𝑡,𝑞,∀𝑡∈ℝ,𝑞≤𝑅_𝑚; (2.20)(2)for each 𝑚∈ℕ, there is a 𝑐_𝑚>0 such that ∇²_𝑞𝑊_𝑚𝑡,𝑞≤𝑐_𝑚,∀𝑡∈ℝ,𝑞∈ℝ^𝑁,∇²_𝑞𝑊_𝑚𝑡,𝑞≥𝐵_−𝜀∶=𝐵_∞−𝜀𝐼_𝑛×𝑛,∀𝑡∈ℝ,𝑞≥𝑅₀ (2.21)(2.22) for 𝜀>0 small enough;(3)there exist some 𝑐>0 such that ∇_𝑞𝑊_𝑚𝑡,𝑞≤𝑐𝑞,∀𝑡,𝑞∈ℝ×ℝ^𝑁,𝑚∈ℕ; (2.23)(4)for each 𝑚∈ℕ, there exists some 𝐶_𝑚>0 and a constant 𝛾 with 𝛾𝐼_𝑛×𝑛>𝐵_∞, 𝜈(𝛾𝐼_𝑛×𝑛)=0 such that ∇_𝑞𝑊_𝑚𝑡,𝑞−𝛾𝑞<𝐶_𝑚,∀𝑡,𝑞∈ℝ×ℝ^𝑁, (2.24) where 𝐼_𝑛×𝑛 is the identity map on ℝ^𝑁.

Proof. Define 𝜂∶[0,∞)→ℝ by 𝜂(𝑡)=20,0≤𝑡<1,9(𝑡−1)³−19(𝑡−1)⁴,1≤𝑡<2,1−128912+𝑡²,2≤𝑡<∞. (2.25) It is easy to see that 𝜂∈𝐶²([0,∞),ℝ). Choose a sequence {𝑅_𝑚} of positive numbers such that 𝑅₀<𝑅₁<𝑅₂<⋯<𝑅_𝑚<⋯→∞ as 𝑚→∞. For each 𝑚∈ℕ, let 𝜂_𝑚(𝑡)=𝜂(𝑡/𝑅_𝑚) and 𝑊_𝑚𝑡,𝑞=1−𝜂_𝑚𝑞𝑊+𝛾𝑡,𝑞2𝜂_𝑚𝑞𝑞²,𝑚∈ℕ. (2.26) As in [23, 24], we can easily check that 𝑊_𝑚 satisfies (2.20), (2.22) (with 𝜀=0), (2.23) and (2.24) for each 𝑚∈ℕ.
Define𝑊_𝑚𝑞𝑡,𝑞=𝜌−𝑇_𝑚𝑇_𝑚𝑊_𝑚+𝛾𝑡,𝑞2𝑞1−𝜌−𝑇_𝑚𝑇_𝑚|𝑞|², (2.27) 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 𝑇_𝑚 large enough, then 𝑊_𝑚 will satisfy (2.20)–(2.24).

Remark 2.6. Similar to Remark 1.4, 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 {𝑊_𝑚} constructed in (2.26) will also satisfy (2.20), (2.21), (2.23), (2.24) (with 𝛾𝐼_𝑛×𝑛<𝐵_∞ and 𝜈(𝛾𝐼_𝑛×𝑛)=0) and ∇²_𝑞𝑊_𝑚𝑡,𝑞≤𝐵_𝜀∶=𝐵_∞+𝜀𝐼_𝑛×𝑛,∀𝑡∈ℝ,𝑞>𝑅₀, (2.28) with 𝜀>0 small enough, therefore, 𝑖(𝐵_𝜀)=𝑖(𝐵_∞) and 𝜈(𝐵_𝜀)=𝜈(𝐵_∞)=0.

DefineΨ_𝑚(𝑢)=_ℝ𝑊_𝑚()Φ𝑡,𝑢𝑑𝑡,∀𝑢∈𝐸,_𝑚(𝑢)=12𝑎(𝑢,𝑢)−Ψ_𝑚(𝑢),∀𝑢∈𝐸. (2.29) Form Lemma 2.1 and (2.21) in Lemma 2.5, we have Φ_𝑚∈𝐶²(𝐸,ℝ). Similarly, we can define Ψ_𝛾(𝑢)=1/2(𝛾𝐾𝑢,𝑢)_𝐸 and Φ_𝛾(𝑢)=1/2𝑎(𝑢,𝑢)−Ψ_𝛾(𝑢) for all 𝑢 in 𝐸, and it follows that Φ_𝛾∈𝐶²(𝐸,ℝ).

Lemma 2.8. For each m∈ℕ, Φ_𝑚 satisfies the (PS) condition and the critical-point set 𝒦_𝑚={𝑧∈𝐸∣Φ^′_𝑚(𝑧)=0} is a compact set.

Proof. For any 𝑚∈ℕ, assume {𝑢_𝑛}⊂𝐸 with Φ^′_𝑚(𝑢_𝑛)→0 as 𝑛→∞: ‖Φ^′_𝑚(𝑢_𝑛)−Φ_𝛾^′′(0)𝑢_𝑛‖_𝐸=‖𝐾(∇_𝑞𝑊_𝑚(𝑡,𝑢_𝑛)−𝛾𝑢_𝑛)‖_𝐸≤‖∇_𝑞𝑊_𝑚(𝑡,𝑢_𝑛)−𝛾𝑢_𝑛‖_𝐿2. (2.30) By (2.23) and (2.27), there hold ∇_𝑞𝑊_𝑚𝑡,𝑞−𝛾𝑞≡0for𝑡∈ℝ,𝑞>2𝑇_𝑚,∇_𝑞𝑊_𝑚𝑡,𝑞−𝛾𝑞<𝑐𝑞,∀𝑡,𝑞∈ℝ×ℝ^𝑁, (2.31) for some 𝑐>0. Choose 𝑏∈(0,1) with 𝑏>(𝛼−1)/(3−𝛼), then there exists a constant 𝑐>0 such that ∇_𝑞𝑊_𝑚(𝑡,𝑢_𝑛)−𝛾𝑢_𝑛²_𝐿2=∇_𝑞𝑊_𝑚𝑡,𝑢_𝑛−𝛾𝑢_𝑛,∇_𝑞𝑊_𝑚𝑡,𝑢_𝑛−𝛾𝑢_𝑛∇𝑑𝑡≤𝑐^𝑞𝑊_𝑚𝑡,𝑢_𝑛−𝛾𝑢_𝑛|𝑢_𝑛|^𝑏|𝑢_𝑛|^1+𝑏𝑢𝑑𝑡≤𝑐_𝑛𝐿^1+𝑏_1+𝑏. (2.32) Since 𝛼<2, 𝜈(𝛾𝐼_𝑛×𝑛)=0 and Ψ_𝛾^′′(0) has bounded inverse, from Lemma 2.1 and (2.30), (2.32), we have {𝑢_𝑛} is bounded in 𝐸. From the similar argument in Lemma 2.4, {𝑢_𝑛} has a convergent subsequence and the (PS) condition is verified. From the same reason, we can also prove that 𝒦_m is compact set.

From Lemma II.5.1 in [25], by standard argument, we have the following.

Lemma 2.9. For any 𝑚∈ℕ, there is an 𝑎_𝑚∈ℝ with −𝑎_𝑚 large enough such that 𝐻_𝑞𝐸,(Φ_𝑚)_𝑎𝑚;ℝ=𝛿_𝑞𝑟ℝ, (2.33) where 𝑟=dim(𝐸⁻_𝛾)=𝑖(𝛾𝐼_𝑛×𝑛).

Note that 𝜃 is an isolated critical point of Φ_𝑚 since 𝜈(𝐵₀)=0. For each 𝑚∈ℕ, let 𝒦^∗_𝑚=𝒦_𝑚⧵{𝜃}, then 𝒦^∗_𝑚 is also compact since 𝒦_𝑚 is compact. Then we have

Lemma 2.10. For any 𝜖,𝜇>0 small enough, for each 𝑚∈ℕ, there exists a functional Φ_𝑚 such that(1)‖Φ_𝑚−Φ_𝑚‖_𝐶2<𝜖,(2)Φ(𝑧)=Φ_𝑚, 𝑧∉𝑁_2𝜇(𝒦^∗_𝑚),(3)Φ_𝑚^′′(𝑧)=Φ_𝑚^′′(𝑧),𝑧∈𝑁_𝜇(𝒦^∗_𝑚),where 𝑁_𝜇(𝒦^∗_𝑚)={𝑧∈𝑋∣dist(𝑧,𝒦^∗_𝑚)<𝜇}. Moreover, Φ_𝑚 satisfies the (PS) condition and has only a finite number of critical points, all nontrivial critical points of Φ_𝑚 lie in 𝑁_𝜇(𝒦^∗_𝑚) and are nondegenerate.

Proof. We follow the idea of [26], since 𝒦^∗_𝑚 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 𝑙(𝑧)=1,∀𝑧∈𝑁_𝜇𝒦^∗_𝑚,𝑙(𝑧)=0,∀𝑧∈𝐸⧵𝑁_2𝜇𝒦^∗_𝑚. (2.34) Let 𝑀=sup_{𝑧∈𝑁2𝜇(𝒦∗𝑚)}{‖𝑧‖},𝐶_𝑙=‖𝑙(𝑧)‖_𝐶2, 𝛿=inf_{𝑧∈𝑁2𝜇(𝒦∗𝑚)⧵𝑁𝜇(𝒦∗𝑚)}{‖Φ^′_𝑚(𝑧)‖}>0. We use the Sard-Smale Theorem to find 𝑦∈𝐸, with ‖𝑦‖<min{𝜖/𝐶_𝑙(2+2𝑀),𝛿/2(𝐶_𝑙(1+2𝑀))}, and −𝑦 is a regular value for Φ^′_𝑚. For any 𝑧₀∈𝑁_2𝜇(𝒦^∗_𝑚), the functional is defined by Φ_𝑚(𝑧)=Φ_𝑚(𝑧)+𝑙(𝑧)⟨𝑦,𝑧−𝑧₀⟩. (2.35) By the fact ‖𝑦‖<𝜖/𝐶_𝑙(2+2𝑀) and the definition of 𝑙(𝑧), it is easy to check that conclusions (1), (2), and (3) hold. Since ‖𝑦‖<𝛿/2𝐶_𝑙(1+2𝑀) and −𝑦 is a regular value for Φ_𝑚, then all nontrivial critical points of Φ_𝑚 are nondegenerate and lie in 𝑁_𝜇(𝒦^∗_𝑚).
In order to prove that Φ_𝑚 satisfies the (PS) condition for each 𝑚∈ℕ, assume there is a sequence {𝑧_𝑛}⊂𝐸 such that Φ^′_𝑚(𝑧_𝑛)→0, (𝑛→∞). From the definition of Φ_𝑚, we have ‖Φ^′_𝑚(𝑧)‖>𝛿/2 for all 𝑧∈𝑁_2𝜇(𝒦^∗_𝑚)⧵𝑁_𝜇(𝒦^∗_𝑚). So 𝑧_𝑛∈(𝐸⧵𝑁_2𝜇(𝒦^∗_𝑚))∪𝑁_𝜇(𝒦^∗_𝑚), when 𝑛 is large enough. From the definition of Φ_𝑚 and the proof in Lemma 2.8, we know that Φ_𝑚 satisfies the (PS) condition and hence has a finite number of critical points.

3. Proof of the Main Results

From Lemma 2.4, Theorem 1.1 is a direct consequence of Theorem 5.1 and Corollary 5.2 in chapter II of [25].

Proof of Theorem 1.2. We first consider the case of (𝑊⁺_∞). We divide the proof into two steps and follow the ideas of [24].Step 1. We claim that Φ_𝑚 has a nontrivial critical point 𝑧_𝑚 with its Morse index satisfying 𝑚⁻𝑧_𝑚≤𝑖𝐵₀+1. (3.1) Note that 𝑧=0 is a critical point of Φ_𝑚. The Morse index of 0 for Φ_𝑚 is 𝑖(𝐵₀), since 𝛾𝐼_𝑛×𝑛≥𝐵_∞, 𝑖𝛾𝐼_𝑛×𝑛≥𝑖𝐵_∞>𝑖𝐵₀+1. (3.2)
If Φ_𝑚 has only finite critical points, consider (𝑖(𝐵₀)+1)th Morse inequality:^𝑞_𝑝=0(−1)^𝑞−𝑝𝑀_𝑝𝑎_𝑚,𝑏_𝑚,Φ_𝑚≥^𝑞_𝑝=0(−1)^𝑞−𝑝𝛽_𝑝𝑎_𝑚,𝑏_𝑚,Φ_𝑚, (3.3) where 𝑞=𝑖(𝐵₀)+1, and 𝑏_𝑚 is large enough such that 𝒦_𝑚⊂Φ_𝑚⁻¹[𝑎_𝑚,𝑏_𝑚].
By Lemmas 2.8 and 2.9, we have𝛽_𝑝𝑎_𝑚,𝑏_𝑚,Φ_𝑚=rank𝐻_𝑝𝐸,(Φ_𝑚)_𝑎𝑚=𝛿_𝑝𝑟, (3.4) where 𝑟=𝑖(𝛾𝐼_𝑛×𝑛). Since 𝑖(𝛾𝐼_𝑛×𝑛)>𝑖(𝐵₀)+1, the right side of the inequality is equal to 0. If Φ_𝑚 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.
If Φ_𝑚 has infinitely many critical points. Assume for any 𝑧∈𝒦^∗_𝑚,𝑚⁻(𝑧)>𝑖𝐵₀+1, (3.5) then from Lemma 2.10, we can choose 𝜇 small enough such that(1)0∉𝑁_2𝜇(𝒦^∗_𝑚), so 0 is also an isolated critical point of Φ_𝑚 and has the same Morse index 𝑖(𝐵₀);(2)for any 𝑧∈𝑁_𝜇(𝒦^∗_𝑚), 𝑚⁻(𝑧) is the dimension of the negative subspace of Φ^′′(𝑧), satisfying 𝑚⁻(𝑧)>𝑖(𝐵₀)+1. (Because Φ_𝑚 is 𝐶² continuous, we can assume this.)
From (3) in Lemma 2.10, if 𝑧 is a nontrivial critical point of Φ_𝑚, the Morse index 𝑚⁻Φ_𝑚(𝑧) satisfies𝑚⁻Φ_𝑚(𝑧)>𝑖𝐵₀+1. (3.6) Then choose 𝑎_𝑚 satisfying 𝑁_2𝜇(𝒦^∗_𝑚)∩(Φ_𝑚)𝑎_𝑚=∅, that is, (Φ_𝑚)𝑎_𝑚=(Φ_𝑚)𝑎_𝑚. So, 𝐻_𝑞𝐸,(Φ_𝑚)𝑎_𝑚;ℝ=𝐻_𝑞𝐸,(Φ_𝑚)𝑎_𝑚;ℝ=𝛿_𝑞𝑟ℝ. (3.7) Then, Φ_𝑚 will not satisfy the (𝑖(𝐵₀)+1)th Morse inequality. It is a contradiction.Step 2. We show that {𝑧_𝑚} is bounded in 𝐿^∞(ℝ,ℝ^𝑁), so from the definition of Φ_𝑚, 𝑧_𝑚 is a nontrivial critical point of Φ for 𝑚 large enough.
We prove it indirectly, assume ‖𝑧_𝑚‖_𝐿∞→∞, from (2.4), we have ‖𝑧_𝑚‖_𝐸→∞. Denote 𝑣_𝑚=𝑧_𝑚/‖𝑧_𝑚‖_𝐸. Passing to a subsequence, we assume that for some 𝑣∈𝐸,𝑣_𝑚⇀𝑣in𝐸,𝑣_𝑚⟶𝑣in𝐿²,𝑣_𝑚⟶𝑣a.e.inℝ. (3.8) Since 𝑧_𝑚 satisfies 𝑧_𝑚̈(𝑡)−𝐿(𝑡)𝑧_𝑚(𝑡)+∇_𝑞𝑊_𝑚(𝑡,𝑧_𝑚)=0, we have 𝑧_𝑚𝐸≤𝑐𝑧_𝑚𝐿2,forsome𝑐>0, (3.9) which implies ‖𝑣_𝑚‖_𝐿2≥1/𝑐 for each 𝑚∈ℕ and thus ‖𝑣‖_𝐿2≥1/𝑐.
For each 𝑚∈ℕ, let ℎ_𝑚(𝑡)=∇_𝑞𝑊_𝑚(𝑡,𝑧_𝑚)/‖𝑧_𝑚‖_𝐸. By (𝑊₀), (2.20), and (2.23), there holds |ℎ_𝑚(𝑡)|≤𝑐|𝑣_𝑚(𝑡)| for all 𝑡∈ℝ. Assume ℎ_𝑚(𝑡)⇀ℎ(𝑡) in 𝐿², then |ℎ(𝑡)|≤𝑐|𝑣(𝑡)|, almost everywhere in ℝ. By standard argument, we have𝑣̈−𝐿(𝑡)𝑣+ℎ(𝑡)=0,∀𝑡∈ℝ. (3.10) By Lemma  3.1 in [23], we have 𝑣(𝑡)≠0 almost everywhere in ℝ, which implies 𝑧_𝑚(𝑡)→∞almost everywhere in ℝ. For any 𝑢∈𝐸⁻(𝐵_−𝜀)=𝐸⁻(𝐵_∞), Φ_𝑚^′′𝑧_𝑚𝑢,𝑢=_ℝ|𝑢̇|²−(𝐿(𝑡)𝑢,𝑢)−∇²_𝑞𝑊_𝑚𝑡,𝑧_𝑚(𝑡)𝑢²𝑑𝑡. (3.11) By (𝑊₁^∗∗), (2.20), and (2.22), {∇²_𝑞𝑊_𝑚} is uniformly bounded from below. Using Fatou’s Lemma, we have liminf_{𝑚→∞ℝ}∇²_𝑞𝑊_𝑚𝑡,𝑧_𝑚𝑢²𝑑𝑡≥_ℝliminf_𝑚→∞∇²_𝑞𝑊_𝑚𝑡,𝑧_𝑚𝑢²𝑑𝑡≥_ℝ𝐵_−𝜀(𝑡)𝑢²𝑑𝑡, (3.12) which implies limsup_𝑚→∞Φ_𝑚^′′𝑧_𝑚𝑢,𝑢≤_ℝ|𝑢̇|²−(𝐿(𝑡)𝑢,𝑢)𝑑𝑡−_ℝ𝐵_−𝜀(𝑡)𝑢²𝑑𝑡<0, (3.13) which contradicts to 𝑚⁻(𝑧_𝑚)≤𝑖(𝐵₀)+1<𝑖(𝐵_−𝜀). So {𝑧_𝑚} is bounded in 𝐿^∞(ℝ,ℝ^𝑁), and Φ has a nontrivial critical point.
In order to prove the case of (𝑊⁻_∞), we choose 𝛾 with 𝛾𝐼_𝑛×𝑛≤𝐵_∞ and 𝜈(𝛾𝐼_𝑛×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 𝑧_𝑚 of Φ_𝑚 satisfying𝑚⁻𝑧_𝑚+𝜈𝑧_𝑚≥𝑖𝐵_∞+1. (3.14) As in Step 2 for the case of (𝑊⁺_∞), we will show that {𝑧_𝑚} is bounded in 𝐿^∞(ℝ,ℝ^𝑁). We prove it indirectly, assume ‖𝑧_𝑚‖_𝐿∞→∞, 𝑧_𝑚(𝑡)→∞ almost everywhere in ℝ. We claim there exists 𝑚₀∈ℕ such that if 𝑚>𝑚₀, then for any 𝑢∈𝐸⁺(𝐵_∞)⧵{0}, (Φ_𝑚^′′(𝑧_𝑚)𝑢,𝑢)_𝐸>0, (3.15) which implies 𝑚⁻(𝑧_𝑚)+𝜈(𝑧_𝑚)≤𝑖(𝐵_∞). If (3.15) is false, then there exist 𝑚_𝑗→∞ and 𝑢_𝑗∈𝐸⁺(𝐵_∞) with ‖𝑢_𝑗‖_𝐸=1 such that (Φ_𝑚^′′(𝑧_𝑚𝑗)𝑢_𝑗,𝑢_𝑗)_𝐸≤0, which can be rewritten as (𝐴𝑢_𝑗,𝑢_𝑗)_𝐸−(𝐾∇²_𝑞𝑊_𝑚𝑗(𝑡,𝑧_𝑚𝑗)𝑢_𝑗,𝑢_𝑗)_𝐸≤0,1−2‖𝑢⁻_𝑗‖²_𝐸≤(𝐾∇²_𝑞𝑊_𝑚𝑗(𝑡,𝑧_𝑚𝑗)𝑢_𝑗,𝑢_𝑗)_𝐸. (3.16)(3.17) Passing to a subsequence if necessary, we assume that 𝑢_𝑗→𝑢 in 𝐿² and almost everywhere in ℝ for some 𝑢∈𝐸⁺(𝐵_∞)=𝐸⁺(𝐵_𝜀) with ‖𝑢‖_𝐸≤1. Since liminf_{𝑗→∞ℝ}−∇²_𝑞𝑊_𝑚𝑗𝑡,𝑧_𝑚𝑗𝑢²_𝑗𝑑𝑡≥_ℝliminf_𝑗→∞−∇²_𝑞𝑊_𝑚𝑗𝑡,𝑧_𝑚𝑗𝑢²_𝑗𝑑𝑡≥−(𝐵_𝜀𝑢,𝑢)_𝐿2, (3.18) we see that limsup_𝑗→∞(𝐾∇²_𝑞𝑊_𝑚𝑗(𝑡,𝑧_𝑚𝑗)𝑢_𝑗,𝑢_𝑗)_𝐸≤(𝐵_𝜀𝑢,𝑢)_𝐿2. (3.19) From (3.17) and (3.19), we have 1−2‖𝑢⁻‖²_𝐸≤(𝐵_𝜀𝑢,𝑢)_𝐿2, (3.20) If 𝑢=0, it is a contradiction, or we have 1−2‖𝑢⁻‖²_𝐸≤(𝐵_𝜀𝑢,𝑢)_𝐿2<(𝐴𝑢,𝑢)_𝐸=𝑢⁺²_𝐸−‖𝑢⁻‖²_𝐸≤1−2‖𝑢⁻‖²_𝐸, (3.21) which is also a contradiction. Thus, {𝑧_𝑚} is bounded in 𝐿^∞(ℝ,ℝ^𝑁) 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 Theorem 1.3 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.

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 3.1 (see [27], Corollary 10.19). Assume 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, (3.22) then 𝜙 has 𝑗−𝑘 pairs of nontrivial critical points {±𝑥₁,±𝑥₂,…,±𝑥_𝑗−𝑘}, so that 𝜇(𝑢_𝑖)≤𝑘+𝑖, for 𝑖=1,2,…,𝑗−𝑘.

First, we consider the case of (𝑊⁺_∞), since 𝑊 is even, we have 𝑊_𝑚 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 Φ_𝑚 satisfies Lemma 3.1 for 𝑅 and 1/𝑟 large enough. So, Φ_𝑚 has 𝑙∶=𝑖_𝐴(𝐵_∞)−𝑖_𝐴(𝐵₀) pairs of nontrivial critical points:±𝑥₁,±𝑥₂,…,±𝑥_𝑙, (3.23) and 𝑙−1 pairs of them satisfying𝑚⁻𝑥_𝑖≤𝑖𝐵₀+𝑖<𝑖𝐵_∞,𝑖=1,2,…,𝑙−1. (3.24) 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). Assume 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, (3.25) 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 (𝑊⁺_∞), we omit it here.

Acknowledgment

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