Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

Abstract

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

1. Introduction and Main Result

Consider the second-order nonautonomous Hamiltonian systems 𝑢̈(𝑡)−𝐿(𝑡)𝑢(𝑡)+∇𝑊(𝑡,𝑢(𝑡))=0,(1) where 𝐿∈𝐶(𝑅,𝑅^𝑁2) is a symmetric matrix valued function, 𝑊∈𝐶¹(𝑅×𝑅^𝑁,𝑅), and ∇𝑊(𝑡,𝑥)=(𝜕𝑊/𝜕𝑥)(𝑡,𝑥). We say that a nonzero solution 𝑢 of problem (1) is homoclinic (to 0) if 𝑢(𝑡)→0 and 𝑢̇(𝑡)→0 as |𝑡|→∞.

The existence of homoclinic orbits for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized by Poincaré [1]. Only during the last two decades such problem has been studied by using critical point theory.

If 𝐿(𝑡) and 𝑊(𝑡,𝑥) are independent of 𝑡 or periodic in 𝑡, many authors have studied the existence of homoclinic orbits for Hamiltonian systems, see, for instance, [2–9], and a more general case is considered in recent papers [10, 11]. In this case, the existence of homoclinic orbits is obtained by going to the limit of periodic solutions of approximating problems. In recent years, concentration compactness principle has also been widely used to deal with the perturbations of periodic or autonomous problems, for example, [12, 13].

If 𝐿(𝑡) and 𝑊(𝑡,𝑥) are neither autonomous nor periodic, the problem is quite different from the ones just described, because of the lack of compactness of the Sobolev embedding. Rabinowitz and Tanaka [14] study without any periodicity assumption and obtain the existence of homoclinic orbits of problem (1) by using a variant of the Mountain Pass Theorem without the Palais-Smale condition under the following condition.

(𝐿)  𝐿∈𝐶(𝑅,𝑅^𝑁2) is a symmetric and positively definite matrix for all 𝑡∈𝑅, and there exists a continuous function 𝑙∶𝑅→𝑅 such that 𝑙(𝑡)>0 for all 𝑡∈𝑅 and (𝐿(𝑡)𝑥,𝑥)≥𝑙(𝑡)|𝑥|²,𝑙(𝑡)→∞as|𝑡|→∞.(2) Assuming coercivity assumption (𝐿), Omana and Willem [15] obtain an improvement on the latter result by employing a new compact embedding theorem; in fact, they show that the (PS) condition is satisfied and obtain the existence and multiplicity of homoclinic orbits of problem (1) by using the usual Mountain Pass Theorem. After [14] and [15], many results [16–22] are obtained for the case where 𝐿(𝑡) is neither constant nor periodic in 𝑡.

Korman and Lazer [23] remove the technical coercivity in case that 𝐿(𝑡) and 𝑊(𝑡,𝑥) are even in 𝑡 and 𝐿(𝑡) is positively definite for all 𝑡∈𝑅, by approximating homoclinic orbits from solutions of boundary value problems, which is complemented by [24].

Most of the papers mentioned previously tackle the superquadratic case (see [2–10, 14–16, 18–21, 23, 24]) and the subquadratic case (see [17–19, 22, 25]). The following Ambrosetti-Rabinowitz condition is widely used in almost all papers tackling the superquadratic case.

(AR) There exists a constant 𝜇>2 such that, for every 𝑡∈𝑅 and 𝑥∈𝑅^𝑛⧵{0}, 0<𝜇𝑊(𝑡,𝑥)≤(∇𝑊(𝑡,𝑥),𝑥).(3) Many recent papers have complemented the (AR) condition, for example, [6, 16, 20–22, 24].

There are also many papers that tackle the multiplicity of homoclinic orbits, for example [20, 21, 23–25]. In particular, based on the variant Fountain Theorem of [26], Yang and Han [19] consider the multiplicity of homoclinic orbits for problem (1).

Theorem A (see [19, Theorem  1.2]). Suppose that 𝐿(𝑡) satisfies (𝐿) and (𝐿^′). For some 𝑎>0 and 𝑟>0, one of the following is true: (i)𝐿∈𝐶¹(𝑅,𝑅^𝑁2) and |𝐿^′(𝑡)|≤𝑎|𝐿(𝑡)|, for all |𝑡|≥𝑟, or(ii)𝐿∈𝐶²(𝑅,𝑅^𝑁2) and 𝐿^′′(𝑡)≤𝑎𝐿(𝑡), for all |𝑡|≥𝑟,where 𝐿^′(𝑡)=(𝑑/𝑑𝑡)𝐿(𝑡) and 𝐿^′′(𝑡)=(𝑑²/𝑑𝑡²)𝐿(𝑡) and 𝑊(𝑡,𝑥) satisfies the following.(W1)𝑊(𝑡,0)≡0 for all 𝑡∈𝑅.(W2)(∇𝑊(𝑡,𝑥),𝑥)≥0, for all (𝑡,𝑥)∈𝑅×𝑅^𝑁.(W3) There exist 𝑑₁>0,𝜇>1 such that |∇𝑊(𝑡,𝑥)|≤𝑑₁1+|𝑥|^𝜇,(4) for all (𝑡,𝑥)∈𝑅×𝑅^𝑁.(W4)lim_|𝑥|→0(∇𝑊(𝑡,𝑥)/|𝑥|)=0 uniformly for 𝑡∈𝑅.(W5) There exist 𝜈>2 and 𝑐>0 such that liminf_|𝑥|→∞(∇𝑊(𝑡,𝑥),𝑥)|𝑥|^𝜈≥𝑐>0(5) uniformly for 𝑡∈𝑅.(W6)𝑠⁻¹(∇𝑊(𝑡,𝑠𝑥),𝑥) is an increasing function of 𝑠∈(0,1], for all (𝑡,𝑥)∈𝑅×𝑅^𝑁.(W7)𝑊(𝑡,−𝑥)=𝑊(𝑡,𝑥), for all (𝑡,𝑥)∈𝑅×𝑅^𝑁.
Then system (1) has infinitely many homoclinic solutions 𝑢_𝑘 satisfying 12_𝑅𝑢̇_𝑘(𝑡)²+𝐿(𝑡)𝑢_𝑘(𝑡),𝑢_𝑘(𝑡−)𝑑𝑡_𝑅𝑊𝑡,𝑢_𝑘(𝑡)𝑑𝑡→+∞(6) as 𝑘→∞.

In the present paper, based on the Fountain Theorem, we can prove the same result under more generic conditions, which generalizes Theorem A. Our first result can be stated as follows.

Theorem 1. Assume that 𝐿 satisfies (𝐿) and (𝐿^′) and 𝑊 satisfies (W1), (W4), (W7), (W8), and (W9). (W8) For any 𝑇>0, lim_|𝑥|→∞𝑊(𝑡,𝑥)|𝑥|²=+∞,(7) uniformly in 𝑡∈[−𝑇,𝑇].(W9) there exists 𝜃≥1, such that 𝜃𝐹(𝑡,𝑥)≥𝐹(𝑡,𝑠𝑥)(8) for all (𝑡,𝑥)∈𝑅×𝑅^𝑁 and 𝑠∈[0,1], where 𝐹(𝑡,𝑥)=(∇𝑊(𝑡,𝑥),𝑥)−2𝑊(𝑡,𝑥).
Then problem (1) has infinitely many homoclinic orbits {𝑢_𝑘} satisfying 12_𝑅𝑢̇_𝑘(𝑡)²+𝐿(𝑡)𝑢_𝑘(𝑡),𝑢_𝑘(𝑡−)𝑑𝑡_𝑅𝑊𝑡,𝑢_𝑘(𝑡)𝑑𝑡→+∞(9)
as 𝑘→∞.

Remark 2. Theorem 1 generalizes and improves Theorem A. Firstly, in Theorem 1 we remove the positiveness condition (W2) and the growth condition (W3), which are indispensable in Theorem A. Now we compare conditions (W5) and (W8), (W6), and (W9). Our condition (W8) is a local superquadratic condition and is really weaker than condition (W5). Under condition (W6), for all 𝑠∈[0,1] we have 𝐹()()=(())−(())[𝑊()(𝑡,𝑥−𝐹𝑡,𝑠𝑥∇𝑊𝑡,𝑥,𝑥∇𝑊𝑡,𝑠𝑥,𝑠𝑥−2𝑡,𝑥−𝑊𝑡,𝑠𝑥)]=2¹₀(∇𝑊(𝑡,𝑥),𝜏𝑥)𝑑𝜏−^𝑠₀𝑠⁻¹∇𝑊(𝑡,𝑠𝑥),𝜏𝑥𝑑𝜏−2¹_𝑠(∇𝑊(𝑡,𝜏𝑥),𝑥)𝑑𝜏=2¹_𝑠∇𝑊(𝑡,𝑥)−𝜏⁻¹∇𝑊(𝑡,𝜏𝑥),𝜏𝑥𝑑𝜏+2^𝑠₀∇𝑊(𝑡,𝑥)−𝑠⁻¹∇𝑊(𝑡,𝑠𝑥),𝜏𝑥𝑑𝜏≥0(10) for all (𝑡,𝑥)∈𝑅×𝑅^𝑁, which means that (W9) holds in the case that 𝜃=1. We consider the multiplicity of homoclinic orbits for problem (1) by using the Fountain Theorem in [27] which is simpler than the variant Fountain Theorem [26].
Moreover, under all conditions of Theorem 1 except (W7) we obtain an existence result.

Theorem 3. Assume that 𝐿 satisfies (𝐿) and (𝐿^′) and 𝑊 satisfies (W1), (W4), (W8) and (W9). Then problem (1) possesses a nontrivial homoclinic orbit.

Remark 4. In Theorem 3, we consider the existence of homoclinic orbits for problem (1) under a class of local superquadratic conditions without the (AR) condition and any periodicity assumptions on both 𝐿 and 𝑊. There are functions 𝐿 and 𝑊 which satisfy Theorem 3, but do not satisfy the corresponding results in [2–10, 14–16, 18–21, 23, 24]. For example, 𝐿(𝑡)=𝑡²+1𝐼_𝑁,𝑊(𝑡,𝑥)=2𝑎(𝑡)|𝑥|²ln1+|𝑥|²,(11) where 𝐼_𝑁 is the unit matrix of order 𝑁, 𝑎(𝑡)=1/(1+𝑡²).

2. Preliminary Results

In order to establish our results via critical point theory, we firstly describe some properties of the space 𝐸 on which the variational functional associated with problem (1) is defined. Let 𝐸=𝑢∈𝐻¹𝑅,𝑅^𝑁∶_𝑅|𝑢̇(𝑡)|²+(𝐿(𝑡)𝑢(𝑡),𝑢(𝑡))𝑑𝑡<+∞.(12) Then the space 𝐸 is a Hilbert space with the inner product (𝑢,𝑣)=_𝑅[(𝑢̇(𝑡),𝑣̇(𝑡))+(𝐿(𝑡)𝑢(𝑡),𝑣(𝑡))]𝑑𝑡(13) and the corresponding norm ‖𝑢‖²=(𝑢,𝑢).(14) Note that 𝐸⊂𝐻¹(𝑅,𝑅^𝑁)⊂𝐿^𝑝(𝑅,𝑅^𝑁) for all 𝑝∈[2,+∞] with the embedding being continuous. In particular, for 𝑝=2 and 𝑝=+∞, there exist constants 𝐶₂ and 𝐶_∞ such that ‖𝑢‖_𝐿²≤𝐶₂‖𝑢‖‖𝑢‖,∀𝑢∈𝐸,(15)_𝐿∞≤𝐶_∞‖𝑢‖,∀𝑢∈𝐸.(16) Here 𝐿^𝑝(𝑅,𝑅^𝑁)  (2≤𝑝≤+∞) and 𝐻¹(𝑅,𝑅^𝑁) denote the Banach spaces of functions on 𝑅 with values in 𝑅^𝑁 under the norms ‖𝑢‖_𝐿^𝑝=_𝑅|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝,‖𝑢‖(17)_𝐻1=‖𝑢‖²_𝐿2+‖𝑢̇‖²_𝐿2^1/2,(18) respectively. 𝐿^∞(𝑅,𝑅^𝑁) is the Banach space of essentially bounded functions from 𝑅 into 𝑅^𝑁 equipped with the norm ‖𝑢‖_∞=esssup𝑞(𝑡)∶𝑡∈𝑅.(19)

Lemma 5 (see [18]). Suppose that assumption (𝐿) holds. Then the embedding of 𝐸 into 𝐿^𝑝(𝑅,𝑅^𝑁) is compact for all 𝑝∈[2,+∞].

Denote by 𝐴 the self-adjoint extension of the operator −(𝑑²/𝑑𝑡²)+𝐿(𝑡) with the domain 𝐷(𝐴)⊂𝐿²≡𝐿²(𝑅,𝑅^𝑁).

Lemma 6 (see [18]). If L satisfies (𝐿) and (𝐿^′), then 𝐷(𝐴) is continuously embedded in 𝐻²(𝑅,𝑅^𝑁), and, consequently, one has 𝑢(𝑡)→0,𝑢̇(𝑡)→0(20) as |𝑡|→∞, for all 𝑢∈𝐷(𝐴).

Lemma 7. Suppose that assumptions (W1), (W4) and (W9) hold. Then 𝑊(𝑡,𝑥)≥0 for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁.

Proof. Given 𝑡∈𝑅 and 𝑥∈𝑅^𝑁, let 𝑓(𝑠)𝑊()∶=𝑡,𝑠𝑥𝑠²(21) for 𝑠>0; then 𝑓^′(𝑠)=(∇𝑊(𝑡,𝑠𝑥),𝑠𝑥)−2𝑊(𝑡,𝑠𝑥)𝑠³.(22) By (W1) and (W9), we have (∇𝑊(𝑡,𝑥),𝑥)−2𝑊(𝑡,𝑥)≥0(23) for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁. Hence, 𝑓^′(𝑠)≥0(24) for all 𝑠>0, which shows that 𝑓(𝑠) is nondecreasing in (0,+∞). It is clear that lim_𝑠→0𝑓(𝑠)=lim_𝑠→0𝑊(𝑡,𝑠𝑥)𝑠²=lim_𝑠→0(∇𝑊(𝑡,𝑠𝑥),𝑥).2𝑠(25) On the other hand, by (W4) one has lim_𝑠→0(∇𝑊(𝑡,𝑠𝑥),𝑥)2𝑠≤lim_𝑠→0|∇𝑊(𝑡,𝑠𝑥)|2|||𝑥|𝑠𝑥²→0.(26) Therefore lim_𝑠→0𝑓(𝑠)=0.(27) Now we get 𝑓(𝑠)≥0 for all 𝑠>0, which implies that 𝑊(𝑡,𝑥)=𝑓(1)≥0(28) for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁.

Lemma 8. Assume that assumptions (𝐿) and (W4) hold and 𝑢_𝑛⇀𝑢 (weakly) in 𝐸. Then ∇𝑊(𝑡,𝑢_𝑛)→∇𝑊(𝑡,𝑢) in 𝐿²(𝑅,𝑅^𝑁).

Proof. Assume that 𝑢_𝑛⇀𝑢 in 𝐸. Then there exists a constant 𝑑>0 such that sup_𝑛∈𝑁𝑢_𝑛𝐿∞≤𝑑,‖𝑢‖_𝐿^∞≤𝑑.(29) By (W4), for every 𝜀>0, there exists 𝛿>0 such that |∇𝑊(𝑡,𝑥)|<𝜀|𝑥|(30) for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁 with |𝑥|<𝛿. Now we claim that given 𝑑>0, for any 𝛿>0, there exists 𝑇₀>0 such that |𝑢(𝑡)|<𝛿(31) for all |𝑡|>𝑇₀ and all 𝑢∈𝐸 with ‖𝑢‖≤𝑑. If not, there exists 𝛿₀>0, for all 𝑛∈𝑁, and there exists 𝑢_𝑛∈𝐸 with ‖𝑢_𝑛‖≤𝑑 and 𝑡_𝑛>𝑛 such that 𝑢_𝑛𝑡_𝑛≥𝛿₀.(32) On the other hand, by Lemma 5𝑢_𝑛→𝑢(33) as 𝑛→∞ in 𝐿^∞. In view of (32) and (33), we have 𝑢𝑡_𝑛≥𝑢_𝑛𝑡_𝑛−𝑢_𝑛−𝑢_𝐿∞≥𝛿₀−𝑢_𝑛−𝑢_𝐿∞≥12𝛿₀(34) when 𝑛 is large enough, which is a contradiction to the fact that lim_|𝑡|→∞|𝑢(𝑡)|=0.(35) Hence, (31) holds. It follows from (29), (30), and (31) that ∇𝑊𝑡,𝑢_𝑛(𝑡)≤𝜀𝑢_𝑛(𝑡,|((𝑡|𝑢(𝑡)∇𝑊𝑡,𝑢))|≤𝜀)|(36) for all 𝑛∈𝑁 and |𝑡|>𝑇₀. By Lemma 5, 𝑢_𝑛→𝑢 in 𝐿²(𝑅,𝑅^𝑁), and 𝑢_𝑛(𝑡)→𝑢(𝑡) for almost every 𝑡∈𝑅 and passing to a subsequence if necessary: ^∞_𝑛=1𝑢_𝑛−𝑢_𝐿2<∞,(37) which implies 𝑣_𝑘(𝑡)∶=^𝑘_𝑛=1𝑢_𝑛(𝑡)−𝑢(𝑡)∈𝐿²(𝑅,𝑅)(38) for 𝑘∈𝑁⁺ and 𝑣_𝑘1−𝑣_𝑘2𝐿2≤^𝑘2_𝑛=𝑘1𝑢_𝑛−𝑢_𝐿2(39) for 𝑘₂>𝑘₁∈𝑁⁺. Since {𝑢_𝑛} is a Cauchy sequence in 𝐿²(𝑅,𝑅^𝑁), so by (39) we know that {𝑣_𝑘} is also a Cauchy sequence in 𝐿²(𝑅,𝑅), which together with (38) and the completeness of 𝐿²(𝑅,𝑅) shows that 𝜈(𝑡)∶=^∞_𝑛=1𝑢_𝑛(𝑡)−𝑢(𝑡)(40) is well defined and 𝜈∈𝐿²(𝑅,𝑅).(41) In consequence, ∇𝑊𝑡,𝑢_𝑛(𝑡)((𝑡𝑢−∇𝑊𝑡,𝑢))≤𝜀_𝑛(𝑡+|𝑢(𝑡𝑢))|≤𝜀_𝑛(𝑡)(𝑡|𝑢(𝑡(𝜈(𝑡)|𝑢(𝑡−𝑢)+2)|≤𝜀+2)|)(42) for all 𝑛∈𝑁 and |𝑡|>𝑇₀. Consequently, _𝑅∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))²=𝑑𝑡_{|𝑡|≤𝑇0}∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))²+𝑑𝑡_|𝑡|>𝑇0∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))²≤𝑑𝑡_{|𝑡|≤𝑇0}∇𝑊𝑡,𝑢_𝑛(𝑡)+|∇𝑊(𝑡,𝑢(𝑡))|²+𝑑𝑡_|𝑡|>𝑇0𝜀²|𝜈(𝑡)|²+4|𝜈(𝑡)||𝑢(𝑡)|+4|𝑢(𝑡)|²≤𝑑𝑡_{|𝑡|≤𝑇0}max_{|𝑥|≤𝑑}|∇𝑊(𝑡,𝑥)|+max_{|𝑥|≤𝑑}|∇𝑊(𝑡,𝑥)|²𝑑𝑡+𝜀²‖𝜈‖²_𝐿2+4‖𝜈‖_𝐿²‖𝑢‖_𝐿²+4‖𝑢‖²_𝐿2<∞(43) for all 𝑛∈𝑁. Then using Lebesgue’s convergence theorem, the lemma is proved.

In our paper we will also use the following lemma which is a special case of Lemma  1.1 in [28], due to Arioli and Szulkin [29].

Lemma 9 (see [28, 29]). Let {𝑢_𝑛} be a bounded sequence in 𝐿^𝑠(𝑅,𝑅^𝑁), 1≤𝑠<∞ such that {𝑢̇_𝑛} is bounded in 𝐿^𝑞(𝑅,𝑅^𝑁), 1≤𝑞<∞. If, in addition, there exists 𝑙>0 such that sup_𝑦∈𝑅^𝑦+𝑙_𝑦−𝑙𝑢_𝑛(𝑡)^𝑠𝑑𝑡→0(44) as 𝑛→∞, then 𝑢_𝑛→0(45) in 𝐿^𝑝(𝑅,𝑅^𝑁) for all 𝑝∈(𝑠,∞).

Now we introduce some notations and some necessary definitions which will be used later. Let 𝐵 be a real Banach space, 𝐼∈𝐶¹(𝐵,𝑅), which means that 𝐼 is continuously Frechet-differentiable functional defined on B. Recall that 𝐼∈𝐶¹(𝐵,𝑅) is said to satisfy the (PS) condition if any sequence {𝑢_𝑛}_𝑛∈𝑁⊂𝐵, for which {𝐼(𝑢_𝑛)} is bounded and 𝐼^′(𝑢_𝑛)→0 as 𝑛→+∞ possesses a convergent subsequence in 𝐵.

Moreover, let 𝐵_𝑟 be the open ball in 𝐵 with the radius 𝑟 and centered at 0, and 𝜕𝐵_𝑟 denotes its boundary; we obtain the existence of homoclinic orbits of problem (1) by the use of the following well-known Mountain Pass Theorem [30].

Lemma 10 (see [30]). Let 𝐵 be a real Banach space and let 𝐼∈𝐶¹(𝐵,𝑅) satisfying the (PS) condition. Suppose that 𝐼(0)=0 and that(A1) there are constants 𝜌,𝛼>0 such that 𝐼|_𝜕𝐵𝜌≥𝛼,(A2) there is an 𝑒∈𝐵⧵𝐵_𝜌 such that 𝐼(𝑒)<0.Then 𝐼 possesses a critical value 𝑐≥𝛼. Moreover 𝑐 can be characterized as 𝑐=inf_𝑓∈Γmax_[]𝑡∈0,1𝐼𝑔(𝑡),(46) where Γ=𝑓∈𝐶([0,1],𝐵)∶𝑓(0)=0,𝑓(1)=𝑒.(47) As shown in [31], a deformation lemma can be proved with the (𝐶)_𝑐 condition replacing the usual (PS) condition, and it turns out that Lemma 10 holds true under the (𝐶)_𝑐 condition.

In order to prove the multiplicity of homoclinic orbits, we will use the Fountain Theorem. Since 𝐸 is a Hilbert space, then there exists a basis {𝑣_𝑛}⊂𝑋 such that 𝑋=⊕_𝑗≥1𝑋_𝑗, where 𝑋_𝑗=span{𝑣_𝑗}. Letting 𝑌_𝑘=⊕^𝑘_𝑗=1𝑋_𝑗,𝑍_𝑘=⊕_𝑗≥𝑘𝑋_𝑗, now we show the following Fountain Theorem.

Lemma 11 (see [27]). If 𝐼∈𝐶¹(𝑋,𝑅) satisfies the (𝐶)_𝑐 condition, 𝐼(−𝑢)=𝐼(𝑢), and for every 𝑘∈𝑁, there exists 𝜌_𝑘>𝑟_𝑘>0 such that(i)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)→+∞, as 𝑘→+∞;(ii)𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≤0.Then 𝐼 has a sequence of critical points {𝑢_𝑘} such that 𝐼(𝑢_𝑘)→+∞ as 𝑘→∞.

In the proof of Theorem 1, the following lemma will also be used. A similar result with respect to elliptic problem has been proved in [27].

Lemma 12. Suppose that 2≤𝑝≤+∞; then one has 𝛽_𝑘𝑝∶=sup_{𝑢∈𝑍𝑘,‖𝑢‖=1}‖𝑢‖_𝐿^𝑝→0(48) as 𝑘→∞.

Proof. It is clear that 0<𝛽_𝑘+1(𝑝)≤𝛽_𝑘(𝑝), so there exists 𝛽(𝑝) such that 𝛽_𝑘𝑝→𝛽𝑝≥0(49) as 𝑘→∞ for every 2≤𝑝≤+∞. By the definition of 𝛽_𝑘(𝑝), there exists 𝑢_𝑘(𝑝)∈𝑍_𝑘 with ‖𝑢_𝑘(𝑝)‖=1 such that 𝑢_𝑘𝑝_𝐿𝑝>𝛽_𝑘𝑝2(50) for every 2≤𝑝≤+∞ and 𝑘∈𝑁⁺. Since {𝑢_𝑘(𝑝)}_𝑘∈𝑁 is bounded, then there exists 𝑢(𝑝)∈𝐸 such that 𝑢_𝑘𝑝⇀𝑢𝑝(51) as 𝑘→∞. Now since {𝑣_𝑗} is a basis of 𝐸, it follows that for all 𝑗∈𝑁0=𝑢_𝑘𝑝,𝑣_𝑗→𝑢𝑝∀𝑘>𝑗,𝑣_𝑗(52) as 𝑘→∞, which shows that 𝑢(𝑝)=0. By Lemma 5 we have 𝑢_𝑘𝑝→0(53) in 𝐿^𝑝 for all 2≤𝑝≤+∞, which together with (49) and (50) implies that 𝛽(𝑝)=0.

3. Proof of Theorems

Define the functional 𝐼∶𝐸→𝑅 by 𝐼(𝑢)=_𝑅12|𝑢̇(𝑡)|²+12(𝐿(𝑡)𝑢(𝑡)(𝑡((𝑡=1,𝑢))−𝑊𝑡,𝑢))𝑑𝑡2‖𝑢‖²−_𝑅𝑊(𝑡,𝑢(𝑡))𝑑𝑡.(54)

Lemma 13. Under the conditions (𝐿), (𝐿^′), and (W4), 𝐼∈𝐶¹(𝐸,𝑅), and for all 𝑢,𝑣∈𝐸 one has 𝐼^′(𝑢),𝑣=_𝑅𝑢̇(𝑡),𝑣̇(𝑡+(𝐿(𝑡)𝑢(𝑡)(𝑡−(((𝑡(𝑡[()),𝑣))∇𝑊𝑡,𝑢)),𝑣))]𝑑𝑡.(55) Moreover, any critical point of 𝐼 on 𝐸 is a solution of problem (1) with 𝑢(±∞)=0 and 𝑢̇(±∞)=0.

Proof. We firstly show that 𝐼 is well defined. It follows from (30) that for any 𝜀>0, there exists 𝛿>0 such that ()≤10≤𝑊𝑡,𝑥2𝜀|𝑥|²(56) for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁 with |𝑥|<𝛿. Letting 𝑢∈𝐸, then 𝑢∈𝐶⁰(𝑅,𝑅^𝑁), the space of continuous function 𝑢 on 𝑅, such that 𝑢(𝑡)→0 as |𝑡|→∞. Therefore there exists 𝑇₁>0 such that |𝑢(𝑡)|<𝛿(57) for all |𝑡|>𝑇₁. Hence, one has _𝑅𝑊(𝑡,𝑢(𝑡))𝑑𝑡≤^𝑇1_−𝑇1𝑊((𝑡1𝑡,𝑢))𝑑𝑡+2𝜀_|𝑡|>𝑇1|𝑢(𝑡)|²≤𝑑𝑡_{𝑇1−𝑇1}𝑊((𝑡1𝑡,𝑢))𝑑𝑡+2𝜀‖𝑢‖²_𝐿2<∞,(58) so 𝐼 is well defined.
Next we prove that 𝐼∈𝐶¹(𝐸,𝑅). Rewrite 𝐼 as follows 𝐼=𝐼₁−𝐼₂,(59) where 𝐼₁=_𝑅12|𝑢̇(𝑡)|²+12(𝐿(𝑡)𝑢(𝑡)(𝑡𝐼,𝑢))𝑑𝑡,₂=_𝑅𝑊(𝑡,𝑢(𝑡))𝑑𝑡.(60) It is easy to check that 𝐼₁∈𝐶¹(𝐸,𝑅) and 𝐼^′₁(𝑢),𝑣=_𝑅[(𝑢̇(𝑡),𝑣̇(𝑡))+(𝐿(𝑡)𝑢(𝑡),𝑣(𝑡))]𝑑𝑡.(61) It remains to show that 𝐼₂∈𝐶¹(𝐸,𝑅). By the mean value theorem, for any 𝑢,𝑣∈𝐸 and ℎ∈[0,1] we have 𝑊((𝑡)(𝑡((𝑡=(((𝑡)(𝑡)𝑣(𝑡(𝑡,𝑡,𝑢+ℎ𝑣))−𝑊𝑡,𝑢))∇𝑊𝑡,𝑢+ℎ𝜃)),𝑣))(62) where 𝜃(𝑡)∈(0,1). For any 𝑢,𝑣∈𝐸, there exists 𝑇₂>0 such that |𝑢(𝑡)|+|𝑣(𝑡)|<𝛿(63) for all |𝑡|>𝑇₂, so that |𝑢(𝑡)+ℎ𝜃(𝑡)𝑣(𝑡)|<𝛿(64) for all |𝑡|>𝑇₂, which together with (15) and (30) implies _𝑅max_[]ℎ∈0,1|(∇𝑊(𝑡,𝑢(𝑡)+ℎ𝜃(𝑡)𝑣(𝑡)),𝑣(𝑡))|𝑑𝑡=_{|𝑡|≤𝑇2}max_[]ℎ∈0,1((𝑡)(𝑡)𝑣(𝑡(𝑡+|(∇𝑊𝑡,𝑢+ℎ𝜃)),𝑣))|𝑑𝑡_|𝑡|>𝑇2max_[]ℎ∈0,1((𝑡)(𝑡)𝑣(𝑡(𝑡≤|(∇𝑊𝑡,𝑢+ℎ𝜃)),𝑣))|𝑑𝑡_{|𝑡|≤𝑇2}max_{|𝑥|≤‖𝑢‖𝐿∞+‖𝑣‖𝐿∞}|(𝑣(𝑡+∇𝑊𝑡,𝑥)||)|𝑑𝑡_|𝑡|>𝑇2𝜀|𝑢(𝑡)||𝑣(𝑡)||𝑣(𝑡)|²≤𝑑𝑡_{|𝑡|≤𝑇2}max_{|𝑥|≤‖𝑢‖𝐿∞+‖𝑣‖𝐿∞}|(𝑣(𝑡‖𝑢‖∇𝑊𝑡,𝑥)||)|𝑑𝑡+𝜀_𝐿²‖𝑣‖_𝐿²+‖𝑣‖²_𝐿2<∞.(65)
Then by Lebesgue’s convergence theorem, we have 𝐼^′₂(𝑢),𝑣=lim_ℎ→0+𝐼₂(𝑢+ℎ𝑣)−𝐼₂(𝑢)ℎ=lim_ℎ→0+𝑅[𝑊(𝑡,𝑢(𝑡)+ℎ𝑣(𝑡))−𝑊(𝑡,𝑢(𝑡))]𝑑𝑡=lim_ℎ→0+𝑅(((𝑡)(𝑡)𝑣(𝑡(𝑡=∇𝑊𝑡,𝑢+ℎ𝜃)),𝑣))𝑑𝑡_𝑅(∇𝑊(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡.(66) Now we show that 𝐼^′₂ is continuous. Supposing that 𝑢_𝑛→𝑢 in 𝐸, by an easy computation, one has sup_‖𝑣‖=1𝐼^′₂𝑢_𝑛−𝐼^′₂(𝑢),𝑣=sup_{‖𝑣‖=1𝑅}∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡≤sup_‖𝑣‖=1∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2‖𝑣‖_𝐿²≤𝐶₂∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2.(67) Hence by Lemma 8, we obtain 𝐼^′₂𝑢_𝑛−𝐼^′₂(𝑢),𝑣→0(68) as 𝑛→∞ uniformly with respect to 𝑣, which implies the continuity of 𝐼^′₂. Now we have proved 𝐼∈𝐶¹(𝐸,𝑅).(69)
Finally, we show that any critical point 𝑢 of 𝐼 is a solution of problem (1) satisfying 𝑢(±∞)=0 and 𝑢̇(±∞)=0. If 𝑢∈𝐸 is a critical point of 𝐼, a standard argument shows that 𝑢∈𝐶²(𝑅,𝑅^𝑁) satisfies (1). By Lemma 6, we only need to show that 𝑢 is an element of 𝐷(𝐴). It follows from (30) and (57) that |∇𝑊(𝑡,𝑢(𝑡))|<𝜀|𝑢(𝑡)|(70) for all |𝑡|>𝑇₁. Hence, one has _𝑅|∇𝑊(𝑡,𝑢(𝑡))|²𝑑𝑡≤^𝑇1_−𝑇1|∇𝑊(𝑡,𝑢(𝑡))|²𝑑𝑡+𝜀²_|𝑡|>𝑇1|𝑢(𝑡)|²≤𝑑𝑡_{𝑇1−𝑇1}|∇𝑊(𝑡,𝑢(𝑡))|²𝑑𝑡+𝜀²‖𝑢‖²_𝐿2<∞,(71) so ∇𝑊(𝑡,𝑢)∈𝐿²(𝑅,𝑅^𝑁), which together with (1) implies that 𝐴𝑢∈𝐿²(𝑅,𝑅^𝑁). This means; 𝑢∈𝐷(𝐴), and the proof is completed.

Lemma 14. Under conditions (𝐿), (W4), and (W9), 𝐼 satisfies the (𝐶)_𝑐 condition.

Proof. 𝐼 satisfies the (𝐶)_𝑐 condition; that is, for every {𝑢_𝑛}⊂𝐸, {𝑢_𝑛} has a convergent subsequence if 𝐼(𝑢_𝑛) is bounded and (1+‖𝑢_𝑛‖)‖𝐼^′(𝑢_𝑛)‖→0 as 𝑛→∞. Assume that {𝑢_𝑛}⊂𝐸 is a sequence such that 𝐼𝑢_𝑛→𝑐(72) is bounded and 1+𝑢_𝑛𝐼^′𝑢_𝑛→0(73) as 𝑛→∞. Hence, we have lim_{𝑛→∞𝑅}12∇𝑊𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)−𝑊𝑡,𝑢_𝑛(𝑡)𝑑𝑡=lim_𝑛→∞𝐼𝑢_𝑛−12𝐼^′𝑢_𝑛,𝑢_𝑛=𝑐.(74)
Firstly, we show that {𝑢_𝑛} is bounded; if not, up to a subsequence we have 𝑢_𝑛→∞(75) as 𝑛→∞. Letting 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, then {𝑣_𝑛} is bounded in 𝐸. By Lemma 5, we have 𝑣_𝑛⇀𝑣in𝐸,𝑣_𝑛→𝑣in𝐿²𝑅,𝑅^𝑁(76) as 𝑛→∞. We claim the following.
Claim 1. consider lim_𝑛→∞sup_𝑦∈𝑅^𝑦+1_𝑦−1𝑣_𝑛(𝑡)²𝑑𝑡=0.(77) Otherwise, for some 𝜎>0, up to a subsequence we have sup_𝑦∈𝑅^𝑦+1_𝑦−1𝑣_𝑛(𝑡)²𝑑𝑡≥𝜎>0.(78) We can choose {𝑦_𝑛}⊂𝑅 such that ^𝑦𝑛_𝑦⁺¹_𝑛−1𝑣_𝑛(𝑡)²𝜎𝑑𝑡≥2.(79) In view of 𝑣_𝑛→𝑣 in 𝐿²(𝑅,𝑅^𝑁) and (79), we have ‖𝑣‖²_𝐿2+𝜎4≥_𝑅|𝑣(𝑡)|²𝑑𝑡+_𝑅𝑣_𝑛(𝑡)−𝑣(𝑡)²𝑑𝑡≥^𝑦𝑛_𝑦⁺¹_𝑛−1|𝑣(𝑡)|²𝑑𝑡+^𝑦𝑛_𝑦⁺¹_𝑛−1𝑣_𝑛(𝑡)−𝑣(𝑡)²≥𝑑𝑡_{𝑦𝑛𝑦+1𝑛−1}𝑣_𝑛(𝑡)²≥𝜎𝑑𝑡2(80) when 𝑛 is large enough. By (80), there exists 𝜀₀>0, such that the set Θ={𝑡∈𝑅∶|𝑣(𝑡)|≥𝜀₀} has a positive Lebesgue measure. Moreover similar to (57), there exists 𝑇₃>0 such that |𝑣(𝑡)|<𝜀₀ for all |𝑡|>𝑇₃, which implies that Θ⊂[−𝑇₃,𝑇₃]. For all 𝑡∈Θ, one has |𝑢_𝑛(𝑡)|→∞ as 𝑛→∞, which together with (W8) shows 𝑊𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)²=𝑊𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)²𝑣_𝑛(𝑡)²→+∞(81) as 𝑛→∞ uniformly for all 𝑡∈Θ. Hence by Lemma 6 and the fact that 𝑊(𝑡,𝑥)≥0 for all 𝑡∈𝑅 and 𝑥∈𝑅^𝑁, we have 12−𝑐+𝑜(1)𝑢_𝑛²=(1/2)𝑢_𝑛²−𝐼𝑢_𝑛𝑢_𝑛²=_𝑅𝑊𝑡,𝑢_𝑛(𝑡)𝑢_𝑛²𝑑𝑡=_𝑅𝑊𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)²𝑣_𝑛(𝑡)²≥𝑑𝑡_Θ𝑊𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)²𝑣_𝑛(𝑡)²𝑑𝑡→∞(82) as 𝑛→∞, which is a contradiction. Therefore we have proved Claim 1. Since ‖𝑣_𝑛‖ is bounded, by Lemma 9, we have 𝑣_𝑛→0(83) in 𝐿^𝑞(𝑅,𝑅^𝑁) for all 𝑞>2. Next, we will derive a contradiction. For any given 𝑟>0, ‖𝑟𝑣_𝑛‖=𝑟. Similar to (31), for 𝛿>0 defined in (56), there exists 𝑇₄>0 such that 𝑟𝑣_𝑛(𝑡)<𝛿(84) for all |𝑡|>𝑇₄ and all 𝑛∈𝑁, which together with (56) shows that 0≤𝑊𝑡,𝑟𝑣_𝑛(𝑡)≤𝜀2𝑟²𝑣_𝑛(𝑡)²(85) for all 𝑛∈𝑁 and |𝑡|>𝑇₄. In view of (83), 𝑣_𝑛→0 in 𝐿³(𝑅,𝑅^𝑁), which implies that _{|𝑡|≤𝑇4}𝑣_𝑛(𝑡)²=𝑑𝑡_{|𝑡|≤𝑇4}1𝑑𝑡^1/3_{|𝑡|≤𝑇4}𝑣_𝑛(𝑡)³𝑑𝑡^2/3→0(86) as 𝑛→∞. We can derive from (W4) that max_{|𝑥|≤𝑟𝐶∞}|∇𝑊(𝑡,𝑥)||𝑥|(87) is bounded for all |𝑡|≤𝑇₄. Combining (86) and (87), we have _{|𝑡|≤𝑇4}max_{|𝑥|≤𝑟𝐶∞}|∇𝑊(𝑡,𝑥)||𝑥|𝑣_𝑛(𝑡)²𝑑𝑡<𝜀(88) when 𝑛 is large enough. It follows from (85) and (88) that 0≤_𝑅𝑊𝑡,𝑟𝑣_𝑛(𝑡)=𝑑𝑡_|𝑡|>𝑇4𝑊𝑡,𝑟𝑣_𝑛(𝑡)𝑑𝑡+_{|𝑡|≤𝑇4}𝑊𝑡,𝑟𝑣_𝑛(𝑡)𝑑𝑡=_|𝑡|>𝑇4𝜀2𝑟²𝑣_𝑛(𝑡)²+𝑑𝑡_{|𝑡|≤𝑇410}∇𝑊𝑡,𝑠𝑟𝑣_𝑛(𝑡),𝑟𝑣_𝑛(𝑡=)𝑑𝑠𝑑𝑡_|𝑡|>𝑇4𝜀2𝑟²𝑣_𝑛(𝑡)²+𝑑𝑡_{|𝑡|≤𝑇410}∇𝑊𝑡,𝑠𝑟𝑣_𝑛(𝑡)𝑠𝑟𝑣_𝑛(𝑡)𝑟𝑣_𝑛(𝑡)²≤𝑠𝑑𝑠𝑑𝑡_𝑅𝜀2𝑟²𝑣_𝑛(𝑡)²+𝑑𝑡_{|𝑡|≤𝑇4}𝑟²2max_{|𝑥|≤𝑟𝐶∞}|∇𝑊(𝑡,𝑥)||𝑥|𝑣_𝑛(𝑡)²≤𝜀𝑑𝑡2𝑟²𝐶²₂+𝑟²2𝜀(89) when 𝑛 is large enough, which implies that lim_{𝑛→∞𝑅}𝑊𝑡,𝑟𝑣_𝑛(𝑡)𝑑𝑡=0(90) for any given 𝑟>0. Choose a sequence {𝑟_𝑛}⊂[0,1], such that 𝐼𝑟_𝑛𝑢_𝑛=max_[]𝑟∈0,1𝐼𝑟𝑢_𝑛.(91) Given 𝑀>0, since 𝑛 is large enough, we have 2√𝑀‖𝑢_𝑛‖⁻¹∈[0,1]; using (90) with 𝑟=2√𝑀, we obtain 𝐼𝑟_𝑛𝑢_𝑛2√≥𝐼𝑀𝑢_𝑛𝑢_𝑛=𝐼2√𝑀𝑣_𝑛=2𝑀−_𝑅𝑊𝑡,2√𝑀𝑣_𝑛𝑑𝑡≥𝑀(92) for 𝑛 large enough, which together with the arbitrary of 𝑀 implies that 𝐼𝑟_𝑛𝑢_𝑛→+∞(93) as 𝑛→∞. In view of (91) and the fact that 𝑟_𝑛∈(0,1), we have _𝑅𝑟_𝑛𝑢̇_𝑛²+𝐿(𝑡)𝑟_𝑛𝑢_𝑛,𝑟_𝑛𝑢_𝑛−∇𝑊𝑡,𝑟_𝑛𝑢_𝑛,𝑟_𝑛𝑢_𝑛=𝐼𝑑𝑡_′𝑟_𝑛𝑢_𝑛,𝑟_𝑛𝑢_𝑛=𝑟_𝑛𝑑𝑑𝑡_{𝑟=𝑟𝑛}𝐼𝑟𝑢_𝑛=0.(94) By (W9), we get _𝑅𝐹𝑡,𝑢_𝑛1𝑑𝑡≥𝜃_𝑅𝐹𝑡,𝑟_𝑛𝑢_𝑛𝑑𝑡(95) for all 𝑡∈𝑅 and 𝑛∈𝑁. It follows from (93) and (94) that _𝑅12∇𝑊𝑡,𝑢_𝑛,𝑢_𝑛−𝑊𝑡,𝑢_𝑛≥1𝑑𝑡𝜃_𝑅12∇𝑊𝑡,𝑟_𝑛𝑢_𝑛,𝑟_𝑛𝑢_𝑛−𝑊𝑡,𝑟_𝑛𝑢_𝑛=1𝑑𝑡𝜃_𝑅12𝑟_𝑛𝑢̇_𝑛²+𝐿(𝑡)𝑟_𝑛𝑢_𝑛,𝑟_𝑛𝑢_𝑛−𝑊𝑡,𝑟_𝑛𝑢_𝑛=1𝑑𝑡𝜃𝐼𝑟_𝑛𝑢_𝑛→+∞(96) as 𝑛→∞, which contradicts (74). Therefore we have proved that {𝑢_𝑛} is bounded.
By Lemma 5 and the fact that {𝑢_𝑛} is bounded in 𝐸, there exist 𝑢∈𝐸, 𝑑>0, and a subsequence of {𝑢_𝑛} again denoted by {𝑢_𝑛} such that sup_𝑛∈𝑁𝑢_𝑛𝐿∞‖𝑢‖𝑢≤𝑑≤𝑑,(97)_𝑛⇀𝑢in𝐸,𝑢_𝑛→𝑢in𝐿²𝑅,𝑅^𝑁(98) as 𝑛→∞. Arguing as in Lemma 8, we can also define 𝜈(𝑡)∶=^∞_𝑛=1𝑢_𝑛(𝑡)−𝑢(𝑡)(99) and 𝜈∈𝐿²(𝑅,𝑅). It is obvious that 𝐼^′𝑢_𝑛−𝐼^′(𝑢),𝑢→0(100) as 𝑛→∞. By (𝐿), (97), and Lemma 8 one has _𝑅∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡)),𝑢_𝑛(𝑡)≤𝑑𝑡∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2𝑢_𝑛(𝑡)_𝐿2≤𝐶₂∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2𝑢_𝑛≤𝐶₂𝑑∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2→0(101) as 𝑛→∞, which implies that 𝐼^′𝑢_𝑛−𝐼^′(𝑢),𝑢_𝑛→0(102) as 𝑛→∞. Summing up (100) and (102), we have 𝐼^′𝑢_𝑛−𝐼^′(𝑢),𝑢_𝑛−𝑢→0(103) as 𝑛→∞. On the other hand, by Lemma 7 and (97) we get _𝑅∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡)),𝑢_𝑛(𝑡)(𝑡)≤−𝑢𝑑𝑡(∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2×𝑢_𝑛(𝑡)−𝑢(𝑡)_𝐿2≤𝐶₂∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2𝑢_𝑛−𝑢≤2𝐶₂𝑑∇𝑊𝑡,𝑢_𝑛(𝑡)−∇𝑊(𝑡,𝑢(𝑡))_𝐿2→0(104) as 𝑛→∞. An easy computation shows that 𝐼^′𝑢_𝑛−𝐼^′(𝑢),𝑢_𝑛=𝑢−𝑢_𝑛−𝑢²−_𝑅∇𝑊𝑡,𝑢_𝑛(𝑡)((𝑡,𝑢−∇𝑊𝑡,𝑢))_𝑛(𝑡)−𝑢(𝑡)𝑑𝑡.(105) Consequently, ‖𝑢_𝑛−𝑢‖→0 as 𝑛→∞.

Proof of Theorem 1. By Lemma 13 and Lemma 14, 𝐼∈𝐶¹(𝐸,𝑅) satisfies the (𝐶)_𝑐 condition and 𝐼(𝑢)=𝐼(−𝑢); hence to prove Theorem 1 we should just show that 𝐼 has the geometric properties (i) and (ii) of Lemma 11.
(i) By Lemma 12𝛽_𝑘𝑝=sup_{𝑢∈𝑍𝑘,‖𝑢‖=1}‖𝑢‖_𝐿^𝑝→0(106) as 𝑘→∞ for 𝑝∈[2,+∞]. We choose 𝑟_𝑘=min{1/𝛽_𝑘(∞),1/𝛽_𝑘(4)}; then 𝑟_𝑘→∞ as 𝑘→∞, and for every 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝑟_𝑘, we have ‖𝑢‖_𝐿^∞≤𝛽_𝑘(∞)𝑟_𝑘<1,‖𝑢‖_𝐿⁴≤𝛽_𝑘(4)𝑟_𝑘<1.(107) Similar to (31), there exists 𝑇₅>0 such that |𝑢(𝑡)|<𝛿(108) for all |𝑡|>𝑇₅ and all 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝑟_𝑘, where 𝛿 is defined in (56). Consequently, by (56), for any 𝜀>0((𝑡≤10≤𝑊𝑡,𝑢))2𝜀|𝑢(𝑡)|²(109) for all |𝑡|>𝑇₅ and all 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝑟_𝑘. Hence, we have for all 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝑟_𝑘𝐼(𝑢)=12‖𝑢‖²−_|𝑡|>𝑇5𝑊(𝑡,𝑢(𝑡))𝑑𝑡−_{|𝑡|≤𝑇5}𝑊((𝑡≥1𝑡,𝑢))𝑑𝑡2‖𝑢‖²−_|𝑡|>𝑇512𝜀|𝑢(𝑡)|²𝑑𝑡−_{|𝑡|≤𝑇5}max_|𝑥|≤1𝑊()≥1𝑡,𝑥𝑑𝑡2‖𝑢‖²−12𝜀𝛽_𝑘(2)²‖𝑢‖²−_{|𝑡|≤𝑇5}max_|𝑥|≤1𝑊()≥1𝑡,𝑥𝑑𝑡4‖𝑢‖²−_{|𝑡|≤𝑇5}max_|𝑥|≤1𝑊(𝑡,𝑥)𝑑𝑡(110) when 𝜀 is small enough. Therefore, one has 𝑏_𝑘=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝐼(𝑢)≥inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}14‖𝑢‖²−_{|𝑡|≤𝑇5}max_|𝑥|≤1𝑊()=1𝑡,𝑥𝑑𝑡4𝑟²_𝑘−_{|𝑡|≤𝑇5}max_|𝑥|≤1𝑊(𝑡,𝑥)𝑑𝑡→∞(111) as 𝑘→∞.
(ii) Firstly, we claim that there exists a constant 𝜀>0 such that meas{𝑡∈𝑅∶|𝑢(𝑡)|≥𝜀‖𝑢‖}≥𝜀(112) for all 𝑢∈𝑌_𝑘⧵{0}. Otherwise, for every 𝑛∈𝑁, there exists 𝑢_𝑛∈𝑌_𝑘⧵{0} such that meas𝑡∈𝑅∶𝑢_𝑛(𝑡≥1)𝑛𝑢_𝑛<1𝑛.(113) Without loss of generality, we suppose that ‖𝑢_𝑛‖=1; then there is meas𝑡∈𝑅∶𝑢_𝑛(𝑡≥1)𝑛<1𝑛.(114) In view of the compactness of the unit sphere of 𝑌_𝑘, there exists a subsequence which is still denoted by {𝑢_𝑛} such that 𝑢_𝑛 converges to some 𝑢₀∈𝑌_𝑘 as 𝑛→∞. It is clear that ‖𝑢₀‖=1. Since all the norms in 𝑌_𝑘 are equivalent, we have 𝑢_𝑛→𝑢₀ in 𝐿² as 𝑛→∞; that is, _𝑅𝑢_𝑛(𝑡)−𝑢₀(𝑡)²𝑑𝑡→0(115) as 𝑛→∞. Thus there exist constants 𝜏₁ and 𝜏₂ such that meas𝑡∈𝑅∶𝑢₀(𝑡)≥𝜏₁≥𝜏₂.(116) If not, we have meas𝑡∈𝑅∶𝑢₀(𝑡≥1)𝑛=0(117) for all 𝑛∈𝑁, which implies that 0≤_𝑅𝑢₀(𝑡)⁴𝑑𝑡≤𝑢₀²_𝐿∞𝑢₀²_𝐿2<1𝑛²𝐶²₂𝑢₀²=1𝑛²𝐶²₂→0(118) as 𝑛→∞. Hence 𝑢₀=0, which contradicts ‖𝑢₀‖=1. Therefore (116) holds. Let Ω₀={𝑡∈𝑅∶|𝑢₀(𝑡)|≥𝜏₁}, Ω_𝑛={𝑡∈𝑅∶|𝑢_𝑛(𝑡)|<1/𝑛}, and Ω^𝑐_𝑛=𝑅⧵Ω_𝑛. It follows from (114) and (116) that measΩ_𝑛⧵Ω₀=measΩ₀⧵Ω^𝑐_𝑛∩Ω₀≥measΩ₀−measΩ^𝑐_𝑛∩Ω₀=𝜏₂−1𝑛.(119) We can choose 𝑛 large enough, such that 𝜏₁−1𝑛≥12𝜏₁,𝜏₂−1𝑛≥12𝜏₂.(120) Then we have 𝑢_𝑛(𝑡)−𝑢₀(𝑡)²≥𝑢_𝑛(𝑡)−𝑢₀(𝑡)²≥𝜏₁−1𝑛²≥14𝜏²₁(121) for all 𝑡∈Ω_𝑛∩Ω₀. In consequence, _𝑅𝑢_𝑛(𝑡)−𝑢₀(𝑡)²≥𝑑𝑡_Ω𝑛∩Ω0𝑢_𝑛(𝑡)−𝑢₀(𝑡)²≥1𝑑𝑡4𝜏²₁⋅measΩ_𝑛∩Ω₀≥14𝜏²₁𝜏₂−1𝑛≥18𝜏²₁𝜏₂>0(122) for all 𝑛 being large enough, which is a contradiction to (115). Hence (112) holds.
Similar to (31), for any 𝜀>0 there exists 𝑇₆>0 such that |𝑣(𝑡)|<𝜀(123) for all |𝑡|>𝑇₆ and all 𝑣∈𝑌_𝑘 with ‖𝑣‖≤1. Consequently, for all 𝑢∈𝑌_𝑘, |𝑢(𝑡)|‖𝑢‖<𝜀(124) for all |𝑡|>𝑇₆, which implies that Ω_𝑢∶={𝑡∈𝑅∶|𝑢(𝑡)|≥𝜀‖𝑢‖}⊂[−𝑇₆,𝑇₆] for all 𝑢∈𝑌_𝑘. By (W8), there exists 𝐺>0 such that 𝑊()≥1𝑡,𝑥𝜀³|𝑥|²(125) for all 𝑥∈𝑅^𝑁 with |𝑥|≥𝐺 and |𝑡|<𝑇₆. Hence we have 𝑊((𝑡≥1𝑡,𝑢))𝜀³|𝑢(𝑡)|²≥1𝜀‖𝑢‖²(126) for all 𝑡∈Ω_𝑢 and 𝑢∈𝑌_𝑘 with ‖𝑢‖≥𝐺/𝜀, which together with Lemma 7 implies that 𝐼(𝑢)‖𝑢‖²=12−_𝑅𝑊(𝑡,𝑢(𝑡))‖𝑢‖²≤1𝑑𝑡2−_Ω𝑢𝑊(𝑡,𝑢(𝑡))‖𝑢‖²=1𝑑𝑡2−1𝜀⋅measΩ_𝑢≤12−1<0(127) for all 𝑢∈𝑌_𝑘 and ‖𝑢‖≥𝐺/𝜀. So we can choose 𝜌_𝑘>max{𝐺/𝜀,𝑟_𝑘}; then 𝑎_𝑘=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐼(𝑢)≤0.(128)
Hence by Lemma 11, we obtain that problem (1) has infinitely many homoclinic solutions {𝑢_𝑘} satisfying 12_𝑅𝑢̇_𝑘(𝑡)²+𝐿(𝑡)𝑢_𝑘(𝑡),𝑢_𝑘(𝑡−)𝑑𝑡_𝑅𝑊𝑡,𝑢_𝑘(𝑡)𝑑𝑡→+∞(129) as 𝑘→∞.

Proof of Theorem 3. We divide the proof of Theorem 3 into the following three steps.
Step 1. It is clear that 𝐼(0)=0, and we have proved that 𝐼∈𝐶¹(𝐸,𝑅) satisfies the (𝐶)_𝑐 condition in Lemma 13 and Lemma 14.
Step 2. Letting 𝜌=𝛿/𝐶_∞ and ‖𝑢‖=𝜌, we have ‖𝑢‖_𝐿^∞≤𝛿, where 𝐶_∞>0 is defined in (16) and 𝛿 is defined in (56). Hence, by (56), for any 𝜀>0((𝑡≤10≤𝑊𝑡,𝑢))2𝜀|𝑢(𝑡)|²(130) for all 𝑡∈𝑅 and 𝑢∈𝐸 with ‖𝑢‖=𝜌. In consequence, combining this with (54), we obtain 𝐼(𝑢)=12‖𝑢‖²−_𝑅𝑊()≥1𝑡,𝑢𝑑𝑡2‖𝑢‖²−𝜀2‖𝑢‖²_𝐿2≥12−𝜀2𝐶²₂‖𝑢‖²(131) for all 𝑡∈𝑅 and 𝑢∈𝐸 with ‖𝑢‖=𝜌. Setting 𝜀=1/2𝐶²₂, the inequality (131) implies that 𝐼|_𝜕𝐵𝜌≥𝜌²4∶=𝛼>0.(132)
Step 3. It remains to prove that there exists an 𝑒∈𝐸 such that ‖𝑒‖>𝜌 and 𝐼(𝑒)≤0, where 𝜌 is defined in Step 2. By (W8), for any 𝑇,𝑀>0, there exists 𝐺>0 such that 𝑊(𝑡,𝑥)|𝑥|²≥𝑀(133) for all |𝑥|>𝐺 and 𝑡∈[−𝑇,𝑇]. Letting 𝑢₀(𝑡)=𝑒^−𝑡2𝑒₁, where 𝑒₁=(1,0,…,0), hence |𝑢₀(𝑡)|>𝑒⁻⁴ for all 𝑡∈[1,2]. It is clear that when 𝑟>𝐺𝑒⁴, 𝑟𝑢₀(𝑡)>𝐺(134) for all 𝑡∈[1,2], which together with (133) shows that when 𝑟>𝐺𝑒⁴𝑊𝑡,𝑟𝑢₀(𝑡)𝑟𝑢₀(𝑡)²≥𝑀(135) for all 𝑡∈[1,2]. Combining (135), Lemma 7, and the fact that ∫²₁|𝑢₀(𝑡)|²𝑑𝑡>0, we have 𝐼𝑟𝑢₀𝑟²=12𝑢₀²−_𝑅𝑊𝑡,𝑟𝑢₀(𝑡)𝑟𝑢₀(𝑡)²𝑢₀(𝑡)²=1𝑑𝑡2𝑢₀²−²₁𝑊𝑡,𝑟𝑢₀(𝑡)𝑟𝑢₀(𝑡)²𝑢₀(𝑡)²𝑑𝑡−_𝑅⧵[1,2]𝑊𝑡,𝑟𝑢₀(𝑡)𝑟𝑢₀(𝑡)²𝑢₀(𝑡)²≤1𝑑𝑡2𝑢₀²−²₁𝑊𝑡,𝑟𝑢₀(𝑡)𝑟𝑢₀(𝑡)²𝑢₀(𝑡)²≤1𝑑𝑡2𝑢₀²−𝑀²₁𝑢₀(𝑡)²𝑑𝑡<0,(136) when 𝑟 and 𝑀 are both large enough.
By Lemma 10, 𝐼 possesses a critical value 𝑐≥𝛼>0 given by 𝑐=inf_𝑓∈Γmax_[]𝑡∈0,1𝐼𝑓(𝑡),(137) where Γ=𝑓∈𝐶([0,1],𝐵)∶𝑓(0)=0,𝑓(1)=𝑒.(138) Hence there is a 𝑢∈𝐸 such that 𝐼(𝑢)=𝑐,𝐼^′(𝑢)=0.(139) Therefore 𝑢 is a nontrivial homoclinic orbit of problem (1). Theorem 3 is proved now.

Acknowledgements

The authors would like to thank the referee for valuable suggestions. This paper is supported by National Natural Science Foundation of China (no. 11071198) and Fundamental Funds for the Central Universities (XDJK2009C075).