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

Abstract

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

1. Introduction

Consider the following second-order Hamiltonian system: 𝑢̈(𝑡)+∇𝑉(𝑡,𝑢(𝑡))=0,𝑡∈ℝ, (HS) where 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑁)∈ℝ^𝑁, 𝑉(𝑡,𝑢)=−𝐾(𝑡,𝑢)+𝑊(𝑡,𝑢), 𝐾,𝑊∶ℝ×ℝ^𝑁→ℝ are 𝐶¹ maps. We will say that a solution 𝑢∶ℝ→ℝ^𝑁 of (HS) is homoclinic (to 0), if 𝑢(𝑡)→0, as |𝑡|→∞. In addition, if 𝑢≢0, then 𝑢 is called a nontrivial homoclinic solution.

Inspired by the excellent monographs [1, 2], by now, the existence and multiplicity of homoclinic solutions for Hamiltonian systems have been extensively investigated in many papers via variational methods; see [3–7] for the first order systems and [8–19] for the second systems, and most of them treat the following system: 𝑢̈(𝑡)−𝐿(𝑡)𝑢(𝑡)+∇𝑊(𝑡,𝑢(𝑡))=0,𝑡∈ℝ, (1) where 𝐿(𝑡) is a symmetric matrix-valued function and 𝑊∈𝐶¹(ℝ,ℝ^𝑁).

For the periodic case, the periodicity is used to control the lack of compactness due to the fact that (1) is set on all ℝ. In 1990, Rabinowitz [12] first proved that (1) has a 2𝑘𝑇-periodic solution 𝑢_𝑘, which is bounded uniformly for 𝑘, and obtained a homoclinic solution for (1) as a limit of 2𝑘𝑇-periodic solution.

For the nonperiodic case, the problem is quite different from the one described in nature. Rabinowitz and Tanaka [13] introduced a type of coercive condition on the matrix 𝐿:(L₁)  𝑙(𝑡)∶=inf_|𝑥|=1𝐿(𝑡)𝑥⋅𝑥→+∞, as |𝑡|→∞.

 They first obtained the existence of homoclinic solution for the nonperiodic system (1) under the well-known (AR) growth condition by using Ekeland’s variational principle.

 In 1995, Ding [8] strengthened condition (L₁) by(L₂) there exists a constant 𝛼>0 such that 𝑙(𝑡)|𝑡|^−𝛼→+∞as|𝑡|→∞. (2)Under the condition (L₂) and some subquadratic conditions on 𝑊(𝑡,𝑢), Ding proved the existence and multiplicity of homoclinic solutions for the system (1). From then on, the condition (L₁) or (L₂) is extensively used in nonperiodic second-order Hamiltonian systems. However, the assumption (L₁) or (L₂) is a rather restrictive and not very natural condition as it excludes, for example, the case of constant matrices 𝐼_𝑁.

In 2005, Izydorek and Janczewska [9] first presented the “pinching” condition (see the following (V₂)) and relaxed the conditions (L₁) and (L₂). They studied the general periodic Hamiltonian system 𝑢̈(𝑡)+∇𝑉(𝑡,𝑢(𝑡))=𝑓(𝑡),𝑡∈ℝ, (3) where 𝑉(𝑡,𝑢)=−𝐾(𝑡,𝑢)+𝑊(𝑡,𝑢) and obtained the following result.

Theorem A (see [9]). Let the following conditions hold: (V₁) 𝑉(𝑡,𝑢)=−𝐾(𝑡,𝑢)+𝑊(𝑡,𝑢), where 𝑉 is continuous and 𝑇 periodic with respect to 𝑡, 𝑇>0;(V₂) there exist 𝑏₁,𝑏₂>0 such that 𝑏₁|𝑢|²≤𝐾(𝑡,𝑢)≤𝑏₂|𝑢|²,∀(𝑡,𝑢)∈ℝ×ℝ^𝑁; (4)(V₃) 𝐾(𝑡,𝑢)≤(𝑢,∇𝐾(𝑡,𝑢))≤2𝐾(𝑡,𝑢) for all (𝑡,𝑢)∈ℝ×ℝ^𝑁;(V₄) ∇𝑊(𝑡,𝑢)=𝑜(|𝑢|) as 𝑢→0 uniformly in 𝑡;(V₅) there is a constant 𝜇>2 such that 0<𝜇𝑊(𝑡,𝑢)≤(∇𝑊(𝑡,𝑢),𝑢),∀(𝑡,𝑢)∈ℝ×ℝ^𝑁; (5)(V₆) 𝑏₁∶=min{1,2𝑏₁}>2𝑀 and ‖𝑓‖_𝐿²<(𝑏₁−2𝑀)/2𝐶^∗, where 𝑀=sup{𝑊(𝑡,𝑢)∶𝑡∈[0,𝑇],|𝑢|=1} and 𝐶^∗ is a positive constant depending on 𝑇. Then the system (3) possesses a nontrivial homoclinic solution 𝑢∈𝑊^1,2(ℝ,ℝ^𝑁) such that 𝑢̇(𝑡)→0 as 𝑡→±∞.

From then on, following the idea of [9], some researchers are devoted to relaxing the conditions (L₁) and (L₂) and studying the existence of homoclinic solutions of system (HS) or (3) under the periodicity assumption of the potential, such as [10, 11, 16, 19].

Very recently, Daouas [3] removed the periodicity condition and studied the existence of homoclinic solutions for the nonperiodic system (3), when 𝑊 is superquadratic at the infinity. Motivated by [3], in this work, we will study the existence of homoclinic solutions of the nonperiodic system (HS), when 𝑊 satisfies the asymptotically quadratic condition at the infinity. It is worth noticing that there are few works concerning this case for system (HS) or (3) up to now.

Our result is presented as follows.

Theorem 1. Let 𝐴∶=sup{𝐾(𝑡,𝑢)∶𝑡∈ℝ,|𝑢|≤1}<+∞ hold. Moreover, assume that the following conditions hold:(H₁) 𝐾(𝑡,0)≡0, and there exists a constant 𝑎>0 such that 𝐾(𝑡,𝑢)≥𝑎|𝑢|²,∀(𝑡,𝑢)∈ℝ×ℝ^𝑁; (6)(H₂) there exists 𝛽∈(1,2] such that 𝐾(𝑡,𝑢)≤(𝑢,∇𝐾(𝑡,𝑢))≤𝛽𝐾(𝑡,𝑢),∀(𝑡,𝑢)∈ℝ×ℝ^𝑁; (7)(H₃) 𝑊(𝑡,0)≡0 and ∇𝑊(𝑡,𝑢)=𝑜(|𝑢|) as 𝑢→0 uniformly in 𝑡, and there exist, 𝑀₀>0 such that |∇𝑊(𝑡,𝑢)||𝑢|≤𝑀₀, (8) for any 𝑡∈ℝ and 𝑢∈ℝ^𝑁;(H₄) 𝑊(𝑡,𝑢)−𝑤(𝑡)|𝑢|²=𝑜(|𝑢|²) as |𝑢|→∞ uniformly in 𝑡, where 𝑤∈𝐿^∞(ℝ,ℝ) with 𝑤_∞∶=inf_𝑡∈ℝ𝑤(𝑡)>2𝐴;(H₅) 𝑊(𝑡,𝑢)∶=(1/2)(∇𝑊(𝑡,𝑢),𝑢)−𝑊(𝑡,𝑢)→+∞ as |𝑢|→+∞, and 𝑊()inf𝑡,𝑢|𝑢|²∶𝑡∈ℝwith𝑐≤|𝑢|<𝑑>0, (9) for any 𝑐,𝑑>0. Then the system (HS) possesses a nontrivial homoclinic solution 𝑢∈𝑊^1,2(ℝ,ℝ^𝑁) such that 𝑢̇(𝑡)→0 as 𝑡→±∞.

Remark 2. Theorem 1 treats the asymptotically quadratic case on 𝑊. Consider the functions 𝐾(𝑡,𝑢)=1+𝑒^−|𝑡||𝑢|²,𝑊(𝑡,𝑢)=𝑑(𝑡)|𝑢|²11−(|𝑢,ln𝑒+|) (10) where 𝑑∈𝐿^∞(ℝ,ℝ) and inf_𝑡∈ℝ𝑑(𝑡)>4+32𝜋².
A straightforward computation shows that 𝐾 and 𝑊 satisfy the assumptions of Theorem 1, but 𝐾 does not satisfy the conditions (L₁) and (L₂). Hence, Theorem 1 also extends the results in [8, 13].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.

2. Preliminaries

Following the similar idea of [20], consider the following nil-boundary value problems: 𝑢̈(𝑡)((𝑡[],𝑢()(𝑇)+∇𝑉𝑡,𝑢))=0,∀𝑡∈−𝑇,𝑇−𝑇=𝑢=0. (11) For each 𝑇>0, let 𝐸_𝑇=𝑊^1,2([−𝑇,𝑇],ℝ^𝑁), where 𝑊^1,2[−𝑇,𝑇],ℝ^𝑁=𝑢∶[−𝑇,𝑇]→ℝ^𝑁𝑢()(𝑇)̇isanabsolutelycontinuousfunction,−𝑇=𝑢=0and𝑢∈𝐿²[−𝑇,𝑇],ℝ^𝑁, (12) equipped with the norm ‖𝑢‖=^𝑇_−𝑇|𝑢̇(𝑡)|²+|𝑢(𝑡)|²𝑑𝑡^1/2. (13) Furthermore, for 𝑝>1, let 𝐿^𝑝_𝑇=𝐿^𝑝([−𝑇,𝑇],ℝ^𝑁) and 𝐿^∞_𝑇=𝐿^∞([−𝑇,𝑇],ℝ^𝑁) under their habitual norms. We need the following result.

Proposition 3 (see [9]). There is a positive constant 𝐶 such that for each 𝑇>0 and 𝑢∈𝐸_𝑇 the following inequality holds: ‖𝑢‖_𝐿^∞_𝑇≤𝐶‖𝑢‖. (14)

Note that the inequality (14) holds true with constant 𝐶=√2 if 𝑇≥1/2 (see [9]). Subsequently, we may assume this condition is fulfilled.

Consider a functional 𝐼∶𝐸_𝑇→ℝ defined by 𝐼(𝑢)=^𝑇_−𝑇12|𝑢̇(𝑡)|²((𝑡=1−𝑉𝑡,𝑢))𝑑𝑡2^𝑇_−𝑇|𝑢̇(𝑡)|²𝑑𝑡+^𝑇_−𝑇𝐾((𝑡−𝑡,𝑢))𝑑𝑡_𝑇−𝑇𝑊(𝑡,𝑢(𝑡))𝑑𝑡. (15) Then 𝐼∈𝐶¹(𝐸_𝑇,ℝ), and it is easy to show that for all 𝑢,𝑣∈𝐸_𝑇, we have 𝐼^′(𝑢)𝑣=^𝑇_−𝑇𝑢̇(𝑡),𝑣̇(𝑡−(𝑊((𝑡((𝑡(𝑡=[())𝑡,𝑢)),∇𝑉𝑡,𝑢)),𝑣))]𝑑𝑡_𝑇−𝑇𝑢̇(𝑡),𝑣̇(𝑡+(((𝑡(𝑡−(((𝑡(𝑡[())∇𝐾𝑡,𝑢)),𝑣))∇𝑊𝑡,𝑢)),𝑣))]𝑑𝑡. (16) It is well known that critical points of 𝐼 are classical solutions of the problem (11). We will obtain a critical point of 𝐼 by using an improved version of the Mountain Pass Theorem. For completeness, we give this theorem.

Recall that a sequence {𝑢_𝑗} is a (𝐶)-sequence for the functional 𝜑 if 𝜑(𝑢_𝑗) is bounded and (1+‖𝑢_𝑗‖)𝜑^′(𝑢_𝑗)→0. A functional 𝜑 satisfies the (𝐶)-condition if and only if any (𝐶)-sequence for 𝜑 contains a convergent subsequence.

Theorem 4 (see [21]). Let 𝐸 be a real Banach space, and let 𝜑∈𝐶¹(𝐸,ℝ) satisfy the (C)-condition and 𝜑(0)=0. If 𝜑 satisfies the following conditions:(A₁) there exist constants 𝜌,𝛼>0 such that 𝜑|_{𝜕𝐵𝜌(0)}≥𝛼;(A₂)  there exists 𝑒∈𝐸⧵𝐵_𝜌(0) such that 𝜑(𝑒)≤0, then 𝜑 possesses a critical value 𝑐≥𝛼 given by 𝑐=inf_𝑔∈Γmax_[]𝑠∈0,1𝜑𝑓(𝑠), (17) where 𝐵_𝜌(0) is an open ball in 𝐸 of radius 𝜌 at about 0, and Γ=𝑓∈𝐶([0,1],𝐸)∶𝑓(0)=0,𝑓(1)=𝑒. (18)

Proof. As shown in Bartolo et al. [22], a deformation lemma can be proved with the (𝐶)-condition replacing the usual (𝑃𝑆)-condition, and it turns out that the standard version Mountain Pass Theorem (see Rabinowitz [21]) holds true under the (𝐶)-condition.

Lemma 5. Assume that (𝐻₂) holds, then 𝐾()𝑢𝑡,𝑢≤𝐾𝑡,|𝑢||𝑢|^𝛽,∀𝑡∈ℝ,|𝑢|≥1. (19)

Proof. From (H₂) it follows that for 𝑢≠0 a map given by (0,∞)∋𝜈↦𝑊𝑡,𝜈⁻¹𝑢 (20) is nondecreasing. Similar to the proof in [12], we can get the conclusion.

Lemma 6 (see [9]). Let 𝑢∶ℝ→ℝ^𝑁 be a continuous map such that 𝑢̇ is locally square integrable. Then, for all 𝑡∈ℝ, one has |𝑢(𝑡)|≤√2^𝑡+1/2_𝑡−1/2|𝑢(𝑠)|²+|𝑢̇(𝑠)|²𝑑𝑠^1/2. (21)

3. Proof of Theorem 1

Lemma 7. Under the assumptions of Theorem 1, the problem (11) possesses a nontrivial solution.

Proof. It suffices to prove that the functional 𝐼 satisfies all the assumptions of Theorem 4.
Step 1. We show that the functional 𝐼 satisfies the (𝐶)-condition. Let 𝐼𝑢_𝑗beboundedand1+𝑢_𝑗𝐼^′𝑢_𝑗→0. (22)
Observe that, for 𝑗 large, it follows from (H₁) and (H₂) that there exists a constant 𝐶₀ such that 𝐶₀≥𝐼𝑢_𝑗−12𝐼^′𝑢_𝑗𝑢_𝑗=^𝑇_−𝑇12∇𝑊𝑡,𝑢_𝑗,𝑢_𝑗−𝑊𝑡,𝑢_𝑗+𝑑𝑡_𝑇−𝑇𝐾𝑡,𝑢_𝑗−12∇𝐾𝑡,𝑢_𝑗,𝑢_𝑗≥𝑑𝑡_𝑇−𝑇𝑊(𝑡,𝑢)𝑑𝑡. (23) Arguing indirectly, assume as a contradiction that ‖𝑢_𝑗‖→∞. Setting 𝑣_𝑗=𝑢_𝑗/‖𝑢_𝑗‖, then ‖𝑣_𝑗‖=1, and by Proposition 3, one has 𝑣_{𝑗𝐿∞𝑇}≤√2𝑣_𝑗=√2. (24) Note that 𝐼^′𝑢_𝑗𝑢_𝑗=𝑢̇_𝑗²_𝐿2𝑇+^𝑇_−𝑇∇𝐾𝑡,𝑢_𝑗,𝑢_𝑗−𝑑𝑡_𝑇−𝑇∇𝑊𝑡,𝑢_𝑗,𝑢_𝑗≥𝑢̇𝑑𝑡_{𝑗2𝐿2𝑇}+^𝑇_−𝑇𝐾𝑡,𝑢_𝑗−𝑑𝑡_𝑇−𝑇∇𝑊𝑡,𝑢_𝑗,𝑢_𝑗𝑑𝑡≥𝐶₁𝑢_𝑗²−^𝑇_−𝑇∇𝑊𝑡,𝑢_𝑗𝑢_𝑗𝑑𝑡, (25) where 𝐶₁=min{1,𝑎}>0. This implies that ^𝑇_−𝑇∇𝑊𝑡,𝑢_𝑗𝑢_𝑗𝑢_𝑗²𝑑𝑡=^𝑇_−𝑇∇𝑊𝑡,𝑢_𝑗𝑣_𝑗²𝑢_𝑗𝑑𝑡→𝐶₁. (26) Set for 𝑠≥0ℎ(𝑠)∶=inf𝑊(𝑡,𝑢)∶𝑡∈[−𝑇,𝑇],𝑢∈ℝ^𝑁with|𝑢|≥𝑠. (27) By (H₅), ℎ(𝑠)→∞ as 𝑠→∞.
For 0≤𝑙<𝑚, let Ω_𝑗(𝑙,𝑚)=𝑡∈[−𝑇,𝑇]∶𝑙≤𝑢_𝑗(𝑡,𝐶)<𝑚_𝑚𝑙𝑊()=inf𝑡,𝑢|𝑢|²∶𝑡∈[−𝑇,𝑇]with𝑙≤|𝑢(𝑡)|<𝑚. (28) Then by (H₅), 𝐶^𝑚_𝑙>0. One has 𝑊𝑡,𝑢_𝑗≥𝐶^𝑚_𝑙𝑢_𝑗²,∀𝑡∈Ω_𝑗(𝑙,𝑚). (29) It follows from (23) that 𝐶₀≥_{Ω𝑗(0,𝑙)}𝑊𝑡,𝑢_𝑗𝑑𝑡+_{Ω𝑗(𝑙,𝑚)}𝑊𝑡,𝑢_𝑗+𝑑𝑡_{Ω𝑗(𝑚,∞)}𝑊𝑡,𝑢_𝑗≥𝑑𝑡_{Ω𝑗(0,𝑙)}𝑊𝑡,𝑢_𝑗𝑑𝑡+𝐶^𝑚_{𝑙Ω𝑗(𝑙,𝑚)}𝑢_𝑗²(𝑚Ω𝑑𝑡+ℎ)_𝑗(𝑚,∞) (30) which implies that Ω_𝑗(≤𝐶𝑚,∞)⁰ℎ(𝑚)→0 (31) as 𝑚→∞ uniformly in 𝑗, and for any fixed 0<𝑙<𝑚_{Ω𝑗(𝑙,𝑚)}𝑣_𝑗²1𝑑𝑡=𝑢_𝑗^{2Ω𝑗(𝑙,𝑚)}𝑢_𝑗²𝐶𝑑𝑡≤⁰𝐶^𝑚_𝑙𝑢_𝑗²→0 (32) as 𝑗→∞. Using (14) and (31), we have _{Ω𝑗(𝑚,∞)}𝑣_𝑗²𝑑𝑡≤𝑣_𝑗²_𝐿∞𝑇⋅Ω_𝑗(𝑚,∞)≤2Ω_𝑗(𝑚,∞)→0 (33) as 𝑚→∞ uniformly in 𝑗.
Let 0<𝜖<𝐶₁/3. By (H₃) there is 𝑙_𝜖>0 such that |(<𝜖∇𝑊𝑡,𝑢)||𝑢|4𝑇 (34) for all |𝑡|≤𝑙_𝜖. Consequently, _{Ω𝑗(0,𝑙𝜖)}∇𝑊𝑡,𝑢_𝑗𝑣_𝑗²𝑢_𝑗𝜖𝑑𝑡≤4𝑇_{Ω𝑗(0,𝑙𝜖)}𝑣_𝑗²≤𝜖𝑑𝑡𝑣4𝑇^{𝑗2𝐿∞𝑇}2𝑇<𝜖 (35) for all 𝑗.
By (31), we can take 𝑚_𝜖 large such that _{Ω𝑗(𝑚𝜖,∞)}𝑣_𝑗²𝜖𝑑𝑡<𝑀₀. (36) Hence, by (H₃) one has _{Ω𝑗(𝑚𝜖,∞)}∇𝑊𝑡,𝑢_𝑗𝑣_𝑗²𝑢_𝑗𝑑𝑡≤𝑀_{0Ω𝑗(𝑚𝜖,∞)}𝑣_𝑗²𝑑𝑡<𝜖 (37) for all 𝑗. By (32) there exists 𝑗₀ such that _{Ω𝑗(𝑙𝜖,𝑚𝜖)}∇𝑊𝑡,𝑢_𝑗𝑣_𝑗²𝑢_𝑗𝑑𝑡≤𝑀_{0Ω𝑗(𝑙𝜖,𝑚𝜖)}𝑣_𝑗²𝑑𝑡<𝜖 (38)
for all 𝑗≥𝑗₀. By (35)–(38), one has limsup_𝑗→∞^𝑇_−𝑇∇𝑊𝑡,𝑢_𝑗𝑣_𝑗²𝑢_𝑗𝑑𝑡≤3𝜖<𝐶₁ (39) which contradicts with (26). So {𝑢_𝑗} is bounded in 𝐸_𝑇. In a similar way to Proposition B. 35 in [21], we can prove that {𝑢_𝑗} has a convergent subsequence. Hence 𝐼 satisfies the (𝐶)-condition.
Step 2. We show that the functional 𝐼 satisfies the condition (A₁) of Theorem 4.
Observe that, by (H₃) and (H₄), given 0<𝜖<𝑎, there exists some 𝐶_𝜖>0 such that |𝑊(𝑡,𝑢)|≤𝜀|𝑢|²+𝐶_𝜀|𝑢|^𝑝 (40) for all 𝑢∈ℝ^𝑁 and 𝑡∈[−𝑇,𝑇], where 𝑝>2. It follows from (H₁), (40), and Proposition 3 that 𝐼(𝑢)=12^𝑇_−𝑇|𝑢̇(𝑡)|²𝑑𝑡+^𝑇_−𝑇𝐾((𝑡−𝑡,𝑢))𝑑𝑡_𝑇−𝑇𝑊((𝑡≥1𝑡,𝑢))𝑑𝑡2‖𝑢̇‖²_𝐿2𝑇+𝑎‖𝑢‖²_𝐿2𝑇−𝜖‖𝑢‖²_𝐿2𝑇−𝐶_𝜖^𝑇_−𝑇|𝑢(𝑡)|^𝑝≥1𝑑𝑡2‖𝑢̇‖²_𝐿2𝑇+𝑎‖𝑢‖²_𝐿2𝑇−𝜖‖𝑢‖²_𝐿2𝑇−2𝑇𝐶_𝜖‖𝑢‖^𝑝_𝐿∞𝑇1≥min2,𝑎−𝜖‖𝑢‖²−2^𝑝/2+1𝑇𝐶_𝜖‖𝑢‖^𝑝. (41) Hence there exist 𝛼>0 and 𝜌>0 such that 𝐼(𝑢)≥𝛼 for all 𝑢∈𝐸_𝑇 with ‖𝑢‖=𝜌.
Step 3. We show that the functional 𝐼 satisfies the condition (A₂) of Theorem 4.
By (H₄), there exists 𝐵>0 such that 𝑊(𝑡,𝑢)≥𝑤_∞|𝑢|²−𝐵,∀𝑡∈[−𝑇,𝑇],𝑢∈ℝ^𝑁. (42) Let 𝑒(𝑡)=𝜁|sin(𝜔𝑡)|𝑒₀,𝑡∈[−𝑇,𝑇], (43) where 𝜔=2𝜋/𝑇 and 𝑒₀=(1,0,…,0). Clearly, 𝑒∈𝐸_𝑇. By (15), (42), and Lemma 5, one has 𝐼(𝑒)=𝜁²𝜔²2^𝑇_−𝑇|cos(𝜔𝑡)|²𝑑𝑡+_{{𝑡∈[−𝑇,𝑇];|𝑒(𝑡)|≤1}}𝐾((𝑡+𝑡,𝑒))𝑑𝑡_{{𝑡∈[−𝑇,𝑇];|𝑒(𝑡)|≥1}}𝐾(𝑡,𝑒(𝑡))𝑑𝑡−^𝑇_−𝑇𝑊((𝑡≤𝑡,𝑒))𝑑𝑡𝑇𝜁²𝜔²2+2𝑇𝐴+𝐴_{{𝑡∈[−𝑇,𝑇];|𝑒(𝑡)|≥1}}|𝑒(𝑡)|^𝛽𝑑𝑡−𝑤_∞𝜁^2𝑇_−𝑇|sin(𝜔𝑡)|²=𝑑𝑡+2𝑇𝐵𝑇𝜁²𝜔²2+2𝑇𝐴+𝐴𝜁^2𝑇_−𝑇|sin(𝜔𝑡)|²𝑑𝑡−𝑇𝑤_∞𝜁²𝜔+2𝑇𝐵=𝑇²2+𝐴−𝑤_∞𝜁²+2𝑇(𝐴+𝐵). (44) Since 𝑤_∞>2𝐴+32𝜋² and 𝑇>2/𝐴𝜋, then 𝜔²/2+𝐴−𝑤_∞<0. So 𝐼(𝑒)→−∞ as |𝜁|→∞. So, we can choose large enough 𝜁∈ℝ such that ‖𝑒‖>𝜌 and 𝐼(𝑒)<0.
Clearly 𝐼(0)=0; then, by application of Theorem 4, there exists a critical point 𝑢_𝑇∈𝐸_𝑇 of 𝐼 such that 𝐼(𝑢_𝑇)≥𝛼 for all 𝑇>2/𝐴𝜋.

Lemma 8. 𝑢_𝑇 is bounded uniformly in 𝑇>2/𝐴𝜋.

Proof. Define the set of paths Γ_𝑇=𝑓∈𝐶[0,1],𝐸_𝑇∣𝑓(0)=0,𝑓(1)=𝑒. (45) It follows from Lemma 7 that there exists a solution 𝑢_𝑇 of problem (11) at which inf_{𝑓∈Γ𝑇}max_[]𝑠∈0,1𝐼𝑓(𝑠)≡𝐷_𝑇 (46) is achieved. Let 𝑇>𝑇. Since any function in 𝐸_𝑇 can be regarded as belonging to 𝐸_𝑇 if one extends it by zero in [−𝑇,𝑇]⧵[−𝑇,𝑇], then Γ_𝑇⊂Γ_𝑇. Therefore, for any solution 𝑢_𝑇 of problem (11), we obtain 𝐼𝑢_𝑇=𝐷_𝑇≤𝐷_1/2uniformlyin𝑇>2𝐴𝜋. (47) Notice that 𝐼^′(𝑢_𝑇)=0, and together with (47), one has 𝐼𝑢_𝑇≤𝐷_1/2,1+𝑢_𝑇𝐼^′𝑢_𝑇=0. (48) The rest of the proof is similar to that of Step 1 in Lemma 7. Hence there exists a constant 𝑀₁>0, independent of 𝑇 such that 𝑢_𝑇≤𝑀₁,∀𝑇>2𝐴𝜋. (49) The proof is complete.

Take a sequence 𝑇_𝑛→∞, and consider the problem (11) on the interval [−𝑇_𝑛,𝑇_𝑛]. By Lemma 7, there exists a nontrivial solution 𝑢_𝑛∶=𝑢_𝑇𝑛 of problem (11).

Lemma 9. Let {𝑢_𝑛}_𝑛∈ℕ be the sequence given above. Then there exists a subsequence {𝑢_𝑛𝑗}_𝑗∈ℕ convergent to 𝑢₀ in 𝐶¹_loc(ℝ,ℝ^𝑁).

Proof. First we prove that the sequences ‖𝑢_𝑛‖_𝐿^∞_𝑇𝑛, ‖𝑢̇_𝑛‖_𝐿^∞_𝑇𝑛, and ‖𝑢̈_𝑛‖_𝐿^∞_𝑇𝑛 are bounded. From (14) and (49), for 𝑛 large enough, one has 𝑢_{𝑛𝐿∞𝑇𝑛}≤𝐶𝑀₁∶=𝑀₂. (50) By (11) and (50), for all 𝑡∈[−𝑇_𝑛,𝑇_𝑛], there exists 𝑀₃>0 independent of 𝑛 such that 𝑢̈_{𝑛𝐿∞𝑇𝑛}≤𝑀₃. (51) It follows from the Mean Value Theorem that for every 𝑛∈𝑁 and 𝑡∈ℝ, there exists 𝜏_𝑛∈[𝑡−1,𝑡] such that 𝑢̇_𝑛𝜏_𝑛=^𝑡_𝑡−1𝑢̇_𝑛(𝑠)𝑑𝑠=𝑢_𝑛(𝑡)−𝑢_𝑛(𝑡−1). (52) Combining the above with (50), and (51) we get 𝑢̇_𝑛(𝑡)=^𝑡_𝜏𝑛𝑢̈_𝑛(𝑠)𝑑𝑠+𝑢̇_𝑛𝜏_𝑛≤^𝑡_𝑡−1𝑢̈_𝑛(𝑠)𝑑𝑠+𝑢_𝑛(𝑡)−𝑢_𝑛(𝑡−1)≤𝑀₃+2𝑀₂∶=𝑀₄ (53) and hence for 𝑛 large enough 𝑢̇_{𝑛𝐿∞𝑇𝑛}≤𝑀₄. (54) Second we show that the sequences {𝑢_𝑛}_𝑛∈ℕ and {𝑢̇_𝑛}_𝑛∈ℕ are equicontinuous. Indeed, for any 𝑛∈ℕ and 𝑡₁,𝑡₂∈ℝ, by (54), we have 𝑢_𝑛𝑡₁−𝑢_𝑛𝑡₂=^𝑡1_𝑡2𝑢̇_𝑛(𝑠)𝑑𝑠≤^𝑡1_𝑡2𝑢̇_𝑛(𝑠)𝑑𝑠≤𝑀₃𝑡₁−𝑡₂. (55) Similarly, by (51), one gets 𝑢̇_𝑛𝑡₁−𝑢̇_𝑛𝑡₂≤𝑀₂𝑡₁−𝑡₂. (56) By using the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence {𝑢_𝑛𝑗}_𝑗∈ℕ and a function 𝑢₀ such that 𝑢_𝑛𝑗→𝑢₀,as𝑗→∞in𝐶¹_locℝ,ℝ^𝑁. (57) The proof is complete.

Lemma 10. Let 𝑢₀∶ℝ→ℝ^𝑁 be the function given by (57). Then 𝑢₀ is the homoclinic solution of (HS).

Proof. First we show that 𝑢₀ is a solution of (HS). Let {𝑢_𝑛𝑗}_𝑗∈ℕ be the sequence given by Lemma 9, then 𝑢̈_𝑛𝑗(𝑡)+∇𝑉𝑡,𝑢_𝑛𝑗(𝑡)=0 (58) for every 𝑗∈ℕ and 𝑡∈[−𝑇_𝑛𝑗,𝑇_𝑛𝑗]. Take 𝑏,𝑐∈ℝ with 𝑏<𝑐. There exists 𝑗₀∈ℕ such that for all 𝑗>𝑗₀; we get [𝑏,𝑐]⊂[−𝑇_𝑛𝑗,𝑇_𝑛𝑗] and 𝑢̈_𝑛𝑗(𝑡)=−∇𝑉𝑡,𝑢_𝑛𝑗(𝑡),∀𝑡∈[𝑏,𝑐]. (59) Integrating (59) from 𝑏 to 𝑡∈[𝑏,𝑐], we have 𝑢̇_𝑛𝑗(𝑡)−𝑢̇_𝑛𝑗(𝑏)=−^𝑡_𝑏∇𝑉𝑠,𝑢_𝑛𝑗(𝑠)𝑑𝑠,∀𝑡∈[𝑏,𝑐]. (60) Since 𝑢_𝑛𝑗→𝑢₀ uniformly on [𝑏,𝑐] and 𝑢̇_𝑛𝑗→𝑢̇₀ uniformly on [𝑏,𝑐] as 𝑗→∞, then, from (60), we obtain 𝑢̇₀(𝑡)−𝑢̇₀(𝑏)=−^𝑡_𝑏∇𝑉𝑠,𝑢₀(𝑠)𝑑𝑠,∀𝑡∈[𝑏,𝑐]. (61) Because of the arbitrariness of 𝑏 and 𝑐, we conclude that 𝑢₀ satisfies (HS).
Second we prove that 𝑢₀(𝑡)→0, as |𝑡|→∞. Note that, by (49), for 𝑘∈ℕ, there exists 𝑗₀∈ℕ such that, for all 𝑗>𝑗₀, one has ^𝑇𝑛𝑗_{−𝑇𝑛𝑗}𝑢_𝑛𝑗(𝑡)²+𝑢̇_𝑛𝑗(𝑡)²𝑑𝑡≤𝑢_𝑛𝑗²≤𝑀²₁. (62) Letting 𝑗→∞, one gets ^𝑇𝑛𝑗_{−𝑇𝑛𝑗}𝑢₀(𝑡)²+𝑢̇₀(𝑡)²𝑑𝑡≤𝑀²₁ (63) and letting 𝑗→∞, we have ^+∞_−∞𝑢₀(𝑡)²+𝑢̇₀(𝑡)²𝑑𝑡≤𝑀²₁ (64) and so _{|𝑡|≥𝑟}𝑢₀(𝑡)²+𝑢̇₀(𝑡)²𝑑𝑡→0. (65) From Lemma 6 and (65), we obtain 𝑢₀(𝑡)→0 as |𝑡|→∞.
Next we show that 𝑢̇₀(𝑡)→0 as |𝑡|→∞. Indeed, applying again Lemma 6 to 𝑢̇₀, we obtain 𝑢̇₀(𝑡)≤√2^𝑡+1/2_𝑡−1/2𝑢̇₀(𝑠)²+𝑢̈₀(𝑠)²𝑑𝑠^1/2. (66)
Also, from (65), we get 𝑢̇^𝑡+1/2_𝑡−1/20(𝑠)²𝑑𝑠→0,as|𝑡|→∞. (67) Hence, it is enough to prove that 𝑢̈^𝑡+1/2_𝑡−1/20(𝑠)²𝑑𝑠→0,as|𝑡|→∞. (68) Since 𝑢₀ is a solution of (HS), one has 𝑢̈^𝑡+1/2_𝑡−1/20(𝑠)²𝑑𝑠=^𝑡+1/2_𝑡−1/2∇𝑉𝑠,𝑢₀(𝑠)²𝑑𝑠. (69) Since ∇𝑉(𝑡,0)=0 for all 𝑡∈ℝ and 𝑢₀(𝑡)→0, as |𝑡|→∞, (68) follows from (69).
Finally, similar to the proof in [12], we can prove that 𝑢₀ is nontrivial, and we omit it here. The proof of Theorem 1 is complete.