Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

Abstract

We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.

1. Introduction and Main Results

In the end of 19th century, Poincaré recognized the importance of homoclinic orbits for dynamical systems. Since then the existence and multiplicity of homoclinic solutions have become one of the most important problems in the research of dynamical systems. In this paper, we consider the following nonperiodic damped vibration system (NDVS): 𝑢̈(𝑡)+𝑀𝑢̇(𝑡)−𝐿(𝑡)𝑢(𝑡)+𝐻_𝑢(𝑡,𝑢(𝑡))=0,𝑡∈ℝ, (1) where 𝑀 is an antisymmetric 𝑁×𝑁 constant matrix,𝐿(𝑡)∈𝐶(ℝ,ℝ^𝑁×𝑁)is a symmetric matrix, 𝐻(𝑡,𝑢)∈𝐶¹(ℝ×ℝ^𝑁,ℝ)and𝐻_𝑢(𝑡,𝑢)denotes its gradient with respect to the𝑢variable. We say that a solution𝑢(𝑡)of (1) is homoclinic (to 0) if𝑢(𝑡)∈𝐶²(ℝ,ℝ^𝑁)such that 𝑢(𝑡)⟶0,𝑢̇(𝑡)⟶0as|𝑡|→∞. (2) If𝑢(𝑡)≢0, then𝑢(𝑡)is called a nontrivial homoclinic solution.

If𝑀=0(zero matrix), then (1) reduces to the following second-order Hamiltonian system: 𝑢̈(𝑡)−𝐿(𝑡)𝑢(𝑡)+𝐻_𝑢(𝑡,𝑢(𝑡))=0,𝑡∈ℝ, (3) which is a classical equation which can describe many mechanical systems, such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (3) have been studied by many authors via variational methods; see [1–17] and the references therein. The periodic assumptions are very important in the study of homoclinic orbits for (3) since periodicity is used to control the lack of compactness due to the fact that (3) is set on allℝ.

Nonperiodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [10] introduced a type of coercivity condition on the matrix𝐿(𝑡): 𝑙(𝑡)∶=inf_|𝑢|=1(𝐿(𝑡)𝑢,𝑢)⟶+∞as|𝑡|⟶∞ (4) and obtained the existence of homoclinic orbit for nonperiodic (3) under the usual Ambrosetti-Rabinowitz (𝐴𝑅) superquadratic condition: 0<𝜇𝐻(𝑡,𝑢)≤𝐻_𝑢(𝑡,𝑢),𝑢,∀𝑡∈ℝ,∀𝑢∈ℝ^𝑁⧵{0}, (5) where𝜇>2is a constant,(⋅,⋅)denotes the standard inner product inℝ^𝑁, and the associated norm is denoted by|⋅|.

As usual, we say that 𝐻 satisfies the subquadratic (or superquadratic) growth condition at infinity if lim_|𝑢|→∞𝐻(𝑡,𝑢)|𝑢|²=0orlim_|𝑢|→∞𝐻(𝑡,𝑢)|𝑢|²=+∞. (6) If 𝑀≠0, that is, the damped vibration system (1), there are only a few authors who have studied homoclinic orbits of the NDVS (1), see [18–23]. Zhu [18] considered the periodic case of (1) (i.e.,𝐿(𝑡)and𝐻(𝑡,𝑢)are𝑇-periodic in𝑡with𝑇>0) and obtained the existence of nontrivial homoclinic solutions of (1). The authors [19–23] considered the nonperiodic case of (1): Zhang and Yuan [19] obtained the existence of at least one homoclinic orbit for (1) when𝐻satisfies the subquadratic condition at infinity by using a standard minimizing argument. By a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [20] obtained the existence and multiplicity of homoclinic orbits for (1) when𝐻satisfies the local(𝐴𝑅)superquadratic growth condition: 0<𝜇𝐻(𝑡,𝑢)≤𝐻_𝑢(𝑡,𝑢),𝑢,∀𝑡∈ℝ,∀|𝑢|≥𝑟, (7) where𝜇>2and𝑟>0are two constants. We should notice that the matrix𝐿(𝑡)in (1) is required to satisfy condition (4) in the Previously mentioned two papers [19, 20]. Later, Sun et al. [21] obtained the existence of at least one homoclinic orbit for (1) when𝐻satisfies the superquadratic condition at infinity by using the following conditions which are weaker than condition (4).(𝐿₁)There exists a constant𝛽>1such that meas𝑡∈ℝ∶|𝑡|^−𝛽𝐿(𝑡)<𝑏𝐼_𝑁<+∞,∀𝑏>0. (8) (𝐿₂)There exists a constant𝛾≥0such that 𝑙(𝑡)∶=inf_|𝑢|=1(𝐿(𝑡)𝑢,𝑢)≥−𝛾,∀𝑡∈ℝ. (9)Recently, by using conditions(𝐿₁)and(𝐿₂), Chen [22, 23] obtained infinitely many nontrivial homoclinic orbits of (1) when𝐻satisfies the subquadratic [22] (or superquadratic [23]) growth condition at infinity. In fact, conditions (𝐿₁)and(𝐿₂)are first used in [14]. As mentioned in [21], there are some matrix-valued functions𝐿(𝑡)satisfying(𝐿₁)and(𝐿₂) but not satisfying (4). For example, 𝐿(𝑡)∶=(𝑡⁴sin²𝑡+1)𝐼_𝑁.That is, conditions(𝐿₁)and(𝐿₂)are weaker than condition (4).

Remark 1. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for the NDVS (1) when𝐻satisfies the asymptotically quadratic condition at infinity (see the following condition(𝐻₃)).
Let𝐻(𝑡,𝑢)∶=𝐻(𝑡,𝑢)−(1/2)(𝐻_𝑢(𝑡,𝑢),𝑢). We assume the following. (𝐻₁) There are constants𝜇∈(1,2)and𝑐₁,𝑐₂,𝑐₃>0such that 𝑐₃|𝑢|^𝜇≤|𝐻(𝑡,𝑢)|≤𝑐₁|𝑢|,∀𝑡∈ℝ,|𝑢|≤𝑐₂. (10) (𝐻₂)𝐻(𝑡,𝑢)≥(1/2)(𝐻_𝑢(𝑡,𝑢),𝑢)≥0for all(𝑡,𝑢)∈ℝ×ℝ^𝑁. (𝐻₃)lim_|𝑢|→∞((𝐻(𝑡,𝑢))/|𝑢|²)=𝑉(𝑡)uniformly in𝑡, where0<inf_𝑡∈ℝ𝑉(𝑡)≤sup_𝑡∈ℝ𝑉(𝑡)<+∞. (𝐻₄)𝐻(𝑡,𝑢)→+∞as|𝑢|→∞and limsup_|𝑢|→0𝐻_𝑢(𝑡,𝑢)^{𝜇/(𝜇−1)}𝐻()(𝑡)|𝑃(𝑡𝑡,𝑢=𝑃uniformlyin𝑡,)|<∞. (11)
We should mention that the coercive-type assumption (see(𝐻₄)) of the function𝐻was first observed and used by Costa and Magalhães [24].
Now, our main result reads as follows.

Theorem 2. If(𝐿₁),(𝐿₂), (𝐻₁)–(𝐻₄), and 𝐻(𝑡,𝑢)are even in𝑢hold, then (1) possesses infinitely many nontrivial homoclinic orbits.

Example 3. Let 𝐻(𝑡,𝑢)∶=𝑉(𝑡)|𝑢|²+|𝑢|^𝜇,𝜇∈(1,2), (12) where𝑉(𝑡)is defined in(𝐻₃). It is not hard to check that it satisfies conditions(𝐻₁)–(𝐻₄) with 𝑃(𝑡)=(2[2𝑉(𝑡)+𝜇]^{𝜇/(𝜇−1)})/(2−𝜇)in(𝐻₄).

The rest of our paper is organized as follows. In Section 2, we establish the variational framework associated with (1) and give some preliminary lemmas, which are useful in the proof of our result, and then we give the detailed proof of our main result.

2. Variational Frameworks and the Proof of Our Main Result

In this section, we always assume that(𝐿₁), (𝐿₂), (𝐻₁)–(𝐻₄), and𝐻(𝑡,𝑢)are even in𝑢hold.

In the following, we will use‖⋅‖_𝑝to denote the norm of𝐿^𝑝(ℝ,ℝ^𝑁)for any𝑝∈[1,∞]. Let 𝐸∶=𝐻¹(ℝ,ℝ^𝑁)be a Hilbert space with the inner product and the norm given, respectively, by ⟨𝑢,𝑣⟩_𝐸=_ℝ𝑢̇(𝑡),𝑣̇(𝑡+(𝑢(𝑡)(𝑡‖𝑢‖[()),𝑣))]𝑑𝑡,_𝐸=⟨𝑢,𝑢⟩_𝐸^1/2,∀𝑢,𝑣∈𝐸. (13) It is well known that𝐸is continuously embedded in𝐿^𝑝(ℝ,ℝ^𝑁)for𝑝∈[2,∞). We define an operator 𝐽∶𝐸→𝐸by (𝐽𝑢,𝑣)∶=_ℝ(𝑀𝑢(𝑡),𝑣̇(𝑡))𝑑𝑡,∀𝑢,𝑣∈𝐸. (14) Since𝑀is an antisymmetric𝑁×𝑁constant matrix,𝐽is self-adjoint on𝐸. Moreover, we denote by𝜒the self-adjoint extension of the operator−𝑑²/𝑑𝑡²+𝐿(𝑡)+𝐽with the domain𝒟(𝜒)⊂𝐿²(ℝ,ℝ^𝑁).

Let𝑊∶=𝒟(|𝜒|^1/2), the domain of|𝜒|^1/2. We define, respectively, on𝑊the inner product and the norm ⟨𝑢,𝑣⟩_𝑊∶=𝜒^1/2𝑢,𝜒^1/2𝑣₂+(𝑢,𝑣)₂,‖𝑢‖_𝑊=⟨𝑢,𝑢⟩_𝑊^1/2, (15) where(⋅,⋅)₂denotes the inner product in𝐿²(ℝ,ℝ^𝑁).

Lemma 4 ([21], Lemma 4). If conditions(𝐿₁)and(𝐿₂)hold, then𝑊is compactly embedded into 𝐿^𝑝(ℝ,ℝ^𝑁)for all 1≤𝑝≤+∞.

By Lemma 4, it is easy to prove that the spectrum𝜎(𝜒)has a sequence of eigenvalues (counted with their multiplicities) 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑘≤⋯⟶∞, (16) and the corresponding system of eigenfunctions {𝑒_𝑘∶𝑘∈ℕ}(𝜒𝑒_𝑘=𝜆_𝑘𝑒_𝑘)forms an orthogonal basis in 𝐿²(ℝ,ℝ^𝑁). Let 𝑘₁∶=♯𝑗∶𝜆_𝑗<0,𝑘₀∶=♯𝑗∶𝜆_𝑗,𝑘=0₂∶=𝑘₀+𝑘₁,𝑊⁻∶=span𝑒₁,…,𝑒_𝑘1,𝑊⁰∶=span𝑒_𝑘1+1,…,𝑒_𝑘2,𝑊⁺∶=cl_𝑊span𝑒_𝑘2+1,…. (17) Then, one has the orthogonal decomposition 𝑊=𝑊⁻⊕𝑊⁰⊕𝑊⁺ (18) with respect to the inner product⟨⋅,⋅⟩_𝑊.

Now, we introduce, respectively, on𝑊the following new inner product and norm: ⟨𝑢,𝑣⟩∶=𝑢⁰,𝑣⁰₂+𝜒^1/2𝑢,𝜒^1/2𝑣₂,‖𝑢‖=⟨𝑢,𝑢⟩^1/2, (19) where𝑢,𝑣∈𝑊=𝑊⁻⊕𝑊⁰⊕𝑊⁺with𝑢=𝑢⁻+𝑢⁰+𝑢⁺, and𝑣=𝑣⁻+𝑣⁰+𝑣⁺. Clearly, the two norms ‖⋅‖ and ‖⋅‖_𝑊 are equivalent (see [3]), and the decomposition𝑊=𝑊⁻⊕𝑊⁰⊕𝑊⁺is also orthogonal with respect to both inner products⟨⋅,⋅⟩and(⋅,⋅)₂.

For problem (1), we consider the following functional: Φ(𝑢)=12_ℝ|𝑢̇(𝑡)|²+((𝑡),𝑢̇(𝑡+(𝐿(𝑡)𝑢(𝑡)(𝑡−𝑀𝑢)),𝑢))𝑑𝑡_ℝ𝐻(𝑡,𝑢)𝑑𝑡,𝑢∈𝑊. (20) Then,Φcan be rewritten as Φ(𝑢)=12𝑢⁺²−12‖𝑢⁻‖²−_ℝ𝐻(𝑡,𝑢)𝑑𝑡,𝑢=𝑢⁻+𝑢⁰+𝑢⁺∈𝑊. (21) Let𝐼(𝑢)∶=∫_ℝ𝐻(𝑡,𝑢)𝑑𝑡. By the assumptions of𝐻, we know that Φ,𝐼∈𝐶¹(𝑊,ℝ)and the derivatives are given by 𝐼^′(𝑢)𝑣=_ℝ𝐻_𝑢()Φ𝑡,𝑢,𝑣𝑑𝑡,_′(𝑢)𝑣=⟨𝑢⁺,𝑣⁺⟩−⟨𝑢⁻,𝑣⁻⟩−𝐼^′(𝑢)𝑣, (22) for any𝑢,𝑣∈𝑊=𝑊⁻⊕𝑊⁰⊕𝑊⁺with𝑢=𝑢⁻+𝑢⁰+𝑢⁺and𝑣=𝑣⁻+𝑣⁰+𝑣⁺. By the discussion of [25], the (weak) solutions of system (1) are the critical points of the𝐶¹functionalΦ∶𝑊→ℝ. Moreover, it is easy to verify that if𝑢≢0is a solution of (1), then 𝑢(𝑡)→0 and 𝑢̇(𝑡)→0 as |𝑡|→∞(see Lemma  3.1 in [26]).

Let𝑊be a Banach space with the norm‖⋅‖ and𝑊∶=⨁_𝑚∈ℕ𝑋_𝑚withdim𝑋_𝑚<∞for any𝑚∈ℕ. Set 𝑌_𝑘∶=^𝑘_𝑚=1𝑋_𝑚,𝑍_𝑘∶=^∞_𝑚=𝑘𝑋_𝑚. (23) Consider the following𝐶¹-functionalΦ_𝜆∶𝑊→ℝdefined by Φ_𝜆(𝑢)=𝐴(𝑢)−𝜆𝐵(𝑢),𝜆∈[1,2]. (24)

To continue the discussion, we give the following variant fountain theorem.

Lemma 5 (see [27]). Assume that the functionalΦ_𝜆defined previously satisfies  (𝑇₁)Φ_𝜆maps bounded sets to bounded sets uniformly for𝜆∈[1,2], and Φ_𝜆(−𝑢)=Φ_𝜆(𝑢)∀(𝜆,𝑢)∈[1,2]×𝑊; (25) (𝑇₂)𝐵(𝑢)≥0for all𝑢∈𝑊and𝐵(𝑢)→+∞as‖𝑢‖→∞on any finite-dimensional subspace of 𝑊; (𝑇₃)there exist𝜌_𝑘>𝑟_𝑘>0such that 𝛼_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}Φ_𝜆(𝑢)≥0>𝛽_𝑘(𝜆)∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}Φ_𝜆(𝑢)[],𝜉,∀𝜆∈1,2_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}Φ_𝜆(𝑢)[].⟶0𝑎𝑠𝑘⟶∞𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑓𝑜𝑟𝜆∈1,2 (26)(27)Then, there exist0<𝜆_𝑗→1and𝑢_𝜆𝑗∈𝑌_𝑗such that Φ^′_𝜆𝑗|_𝑌𝑗𝑢_𝜆𝑗Φ=0,_𝜆𝑗𝑢_𝜆𝑗→𝜂_𝑘∈𝜉_𝑘(2),𝛽_𝑘(1)𝑎𝑠𝑗→∞. (28) Particularly, if{𝑢_𝜆𝑗}has a convergent subsequence for every𝑘, thenΦ₁has infinitely many nontrivial critical points{𝑢_𝑘}⊂𝑊⧵{0}satisfyingΦ₁(𝑢_𝑘)→0⁻as𝑘→∞.

For𝑚∈ℕ, let𝑋_𝑚∶=ℝ𝑒_𝑚(the sequence{𝑒_𝑚}is defined in Section 2 just below Lemma 4); then𝑍_𝑘and𝑌_𝑘can be defined as before. In order to apply the previously mentioned variant fountain theorem to prove our main result, we define the functionals 𝐴, 𝐵, andΦ_𝜆on𝑊by 𝐴(𝑢)1∶=2𝑢⁺²(𝑢)1,𝐵∶=2‖𝑢⁻‖²+_ℝ𝐻()Φ𝑡,𝑢𝑑𝑡,_𝜆(𝑢)(𝑢)(𝑢)=1∶=𝐴−𝜆𝐵2𝑢⁺²1−𝜆2‖𝑢⁻‖²+_ℝ𝐻(𝑡,𝑢)𝑑𝑡 (29) for all 𝑢=𝑢⁰+𝑢⁻+𝑢⁺∈𝑊=𝑊⁰⊕𝑊⁻⊕𝑊⁺ and 𝜆∈[1,2]. Obviously, Φ_𝜆∈𝐶¹(𝑊,ℝ) for all 𝜆∈[1,2].

Next, we will prove that conditions(𝑇₂)and(𝑇₃)of Lemma 5 hold, that is, the following two lemmas.

Lemma 6. 𝐵(𝑢)≥0for all𝑢∈𝑊and𝐵(𝑢)→∞as‖𝑢‖→∞on any finite-dimensional subspace of 𝑊.

Proof. Obviously, condition(𝐻₂)and the definition of𝐵imply that𝐵(𝑢)≥0for all𝑢∈𝑊. We claim that for any finite-dimensional subspace𝑋⊂𝑊, there exists a constant𝜖>0such that 𝑚({𝑡∈ℝ∶|𝑢|≥𝜖‖𝑢‖})≥𝜖,∀𝑢∈𝑋⧵{0}, (30) where𝑚(⋅)denotes the Lebesgue measure inℝ. In fact, the detailed proof of (30) has been given by Chen (Lemma 2.3 in [22]).
For the𝜖given in (30), let Λ_𝑢∶={𝑡∈ℝ∶|𝑢|≥𝜖‖𝑢‖},∀𝑢∈𝑋⧵{0}. (31) Then, by (30), 𝑚Λ_𝑢≥𝜖,∀𝑢∈𝑋⧵{0}. (32) By(𝐻₃), there exist constants𝑅₁,𝑅₂>0such that 𝐻(𝑡,𝑢)≥𝑅₁|𝑢|²,∀(𝑡,𝑢)∈ℝ×ℝ^𝑁with|𝑢|≥𝑅₂. (33) The definition ofΛ_𝑢implies that for any𝑢∈𝑋with ‖𝑢‖≥𝑅₂/𝜖there holds |𝑢|≥𝑅₂,∀𝑡∈Λ_𝑢. (34) Combining(𝐻₂), (32)–(34), and the definition ofΛ_𝑢, for any𝑢∈𝑋with‖𝑢‖≥𝑅₂/𝜖, we have 𝐵(𝑢)=12‖𝑢⁻‖²+_ℝ𝐻()≥𝑡,𝑢𝑑𝑡_Λ𝑢𝐻()≥𝑡,𝑢𝑑𝑡_Λ𝑢𝑅₁|𝑢|²𝑑𝑡≥𝑅₁𝜖²‖𝑢‖²⋅𝑚Λ_𝑢≥𝑅₁𝜖³‖𝑢‖². (35) It implies that 𝐵(𝑢)→∞as‖𝑢‖→∞on any finite-dimensional subspace𝑋⊂𝑊. The proof is finished.

Lemma 7. There exist a positive integer𝑙₀and two sequences0<𝑟_𝑘<𝜌_𝑘→0as𝑘→∞such that 𝛼_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}Φ_𝜆(𝑢)>0,∀𝑘≥𝑙₀,𝜉_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}Φ_𝜆(𝑢)[],𝛽⟶0𝑎𝑠𝑘→∞𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑓𝑜𝑟𝜆∈1,2_𝑘(𝜆)∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}Φ_𝜆(𝑢)<0,∀𝑘∈ℕ, (36)(37)(38) where𝑌_𝑘=⨁^𝑘_𝑚=1𝑋_𝑚and𝑍_𝑘=⨁^∞_𝑚=𝑘𝑋_𝑚for all𝑘∈ℕ.

Proof. (a) First, we show that (36) holds. Note that𝑍_𝑘⊂𝑊⁺for all𝑘≥𝑘₂+1, where𝑘₂is the integer defined in (17) just below Lemma 4. By Lemma 4, there is a constant𝜀₀>0such that ‖𝑢‖_∞≤𝜀₀‖𝑢‖for any𝑢∈𝑊. It follows that for any𝑢∈𝑊with ‖𝑢‖≤𝑐₂/𝜀₀there holds |𝑢|≤‖𝑢‖_∞≤𝑐₂, (39) where𝑐₂is the constant in(𝐻₁). It follows from(𝐻₁)and the definition ofΦ_𝜆that for any𝑘≥𝑘₂+1and𝑢∈𝑍_𝑘with ‖𝑢‖≤𝑐₂/𝜀₀there holds Φ_𝜆(𝑢)≥12‖𝑢‖²−2_ℝ𝐻()≥1𝑡,𝑢𝑑𝑡2‖𝑢‖²−2𝑐₁‖𝑢‖₁,∀𝜆∈[1,2]. (40) Let 𝑙_𝑘∶=sup_{𝑢∈𝑍𝑘⧵{0}}‖𝑢‖₁‖𝑢‖,∀𝑘∈ℕ. (41) Then 𝑙_𝑘⟶0as𝑘⟶∞ (42) by Lemma 4 and the Rellich embedding theorem (see [28]). Consequently, (40) and (41) imply that Φ_𝜆(𝑢)≥12‖𝑢‖²−2𝑐₁𝑙_𝑘‖𝑢‖ (43) for any𝑘≥𝑘₂+1and𝑢∈𝑍_𝑘with ‖𝑢‖≤𝑐₂/𝜀₀. For any𝑘∈ℕ, let 𝜌_𝑘∶=8𝑐₁𝑙_𝑘. (44) Then, by (42), we have 0<𝜌_𝑘⟶0as𝑘⟶∞. (45) Evidently, (45) implies that there exists a positive integer𝑙₀>𝑘₂+1such that 𝜌_𝑘≤𝑐₂𝜀₀,∀𝑘≥𝑙₀. (46) (43) together with (44) and (46) implies that 𝛼_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝜌𝑘}Φ_𝜆(𝑢)≥𝜌²_𝑘2−𝜌²_𝑘4=𝜌²_𝑘4>0,∀𝑘≥𝑙₀. (47) That is, (36) holds.
(b) Second, we show that (37) holds. By (43), for any𝑘≥𝑙₀and𝑢∈𝑍_𝑘with ‖𝑢‖≤𝜌_𝑘, we have Φ_𝜆(𝑢)≥−2𝑐₁𝑙_𝑘𝜌_𝑘. (48) Observing thatΦ_𝜆(0)=0by(𝐻₁), thus 0≥inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}Φ_𝜆(𝑢)≥−2𝑐₁𝑙_𝑘𝜌_𝑘,∀𝑘≥𝑙₀, (49) which together with (42) and (45) implies that 𝜉_𝑘(𝜆)∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖≤𝜌𝑘}Φ_𝜆(𝑢)[].⟶0as𝑘⟶∞uniformlyfor𝜆∈1,2 (50) That is, (37) holds.
(c) Last, we show that (38) holds. For any𝑘∈ℕand𝑢∈𝑌_𝑘with ‖𝑢‖≤𝑐₂/𝜀₀(𝜀₀is the constant above (39)), similar to (39), we have |𝑢|≤𝑐₂. (51) Therefore, by (51) and(𝐻₁), for any𝑘∈ℕand𝑢∈𝑌_𝑘with ‖𝑢‖≤𝑐₂/𝜀₀, we have Φ_𝜆(𝑢)≤12𝑢⁺²−_ℝ𝐻()≤1𝑡,𝑢𝑑𝑡2‖𝑢‖²−𝑐₃‖𝑢‖^𝜇_𝜇≤12‖𝑢‖²−𝐶_𝑘‖𝑢‖^𝜇,∀𝜆∈[1,2], (52) where the last inequality follows by the equivalence of the two norms‖⋅‖_𝜇and‖⋅‖on finite dimensional space𝑌_𝑘, and𝐶_𝑘>0is a constant depending on𝑌_𝑘. For any𝑘∈ℕ, if we choose 0<𝑟_𝑘<min𝜌_𝑘,𝐶_𝑘^1/(2−𝜇),𝑐₂𝜀₀, (53) Then, by (52), direct computation shows that 𝛽_𝑘(𝜆)∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝑟𝑘}Φ_𝜆(𝑢)𝑟≤−^2𝑘2<0,∀𝑘∈ℕ. (54) That is, (38) holds.
Therefore, the proof is finished by (a), (b), and (c).

Proof of Theorem 2. By the assumptions of𝐻and the definition ofΦ_𝜆, we easily get thatΦ_𝜆maps bounded sets to bounded sets uniformly for𝜆∈[1,2]. Note that𝐻(𝑡,−𝑢)=𝐻(𝑡,𝑢), so we have Φ_𝜆(−𝑢)=Φ_𝜆(𝑢)for all(𝜆,𝑢)∈[1,2]×𝑊. Thus, the condition(𝑇₁)of Lemma 5 holds. Lemma 6 shows that the condition(𝑇₂)of Lemma 5 holds. Lemma 7 implies that the condition(𝑇₃)of Lemma 5 holds for all𝑘≥𝑙₀, where𝑙₀is given in Lemma 7. Therefore, by Lemma 5, for each𝑘≥𝑙₀, there exist 0<𝜆_𝑗→1, 𝑢_𝜆𝑗∈𝑌_𝑗such that Φ^′_𝜆𝑗|_𝑌𝑗𝑢_𝜆𝑗Φ=0,_𝜆𝑗𝑢_𝜆𝑗⟶𝜂_𝑘∈𝜉_𝑘(2),𝛽_𝑘(1)as𝑗⟶∞. (55)
Next, we only need to prove the following two claims to complete the proof of Theorem 2.

Claim 1. {𝑢_𝜆𝑗}is bounded in𝑊.

Proof of Claim 1. By (55), we have (1/2)Φ^′_𝜆𝑗∣_𝑌𝑗𝑢_𝜆𝑗𝑢_𝜆𝑗−Φ_𝜆𝑗𝑢_𝜆𝑗𝜆_𝑗≤𝐶₁ (56) for some constant 𝐶₁>0. It follows from the definitions ofΦ_𝜆𝑗and𝐻that _ℝ𝐻𝑡,𝑢_𝜆𝑗()Φ𝑑𝑡=1/2^′_𝜆𝑗∣_𝑌𝑗𝑢_𝜆𝑗𝑢_𝜆𝑗−Φ_𝜆𝑗𝑢_𝜆𝑗𝜆_𝑗≤𝐶₁. (57) Since(𝐻₄)implies∫_ℝ𝐻(𝑡,𝑢)𝑑𝑡→+∞as|𝑢|→+∞, it follows from (57) that 𝑢_𝜆𝑗≤𝐶₂,∀𝑗∈ℕ (58) for some constant𝐶₂>0. Note that𝐻(𝑡,𝑢)∈𝐶¹(ℝ×ℝ^𝑁,ℝ); it follows from(𝐻₄)that there is a constant𝐶₃>0such that 𝐻_𝑢𝑡,𝑢_𝜆𝑗^{𝜇/(𝜇−1)}≤𝐶₃𝐻𝑡,𝑢_𝜆𝑗,𝑡∈ℝ,𝑢_𝜆𝑗≤𝐶₂. (59) Thus, by (57)–(59),Φ^′_𝜆𝑗∣_𝑌𝑗(𝑢_𝜆𝑗)𝑢⁺_𝜆𝑗=0, Hölder’s inequality, and Lemma 4, 𝑢⁺_𝜆𝑗²=𝜆_𝑗ℝ𝐻_𝑢𝑡,𝑢_𝜆𝑗,𝑢⁺_𝜆𝑗𝑑𝑡≤𝜆_𝑗ℝ𝐻_𝑢𝑡,𝑢_𝜆𝑗^{𝜇/(𝜇−1)}𝑑𝑡^{(𝜇−1)/𝜇}_ℝ𝑢⁺_𝜆𝑗^𝜇𝑑𝑡^1/𝜇≤𝐶_4ℝ𝐶₃𝐻𝑡,𝑢_𝜆𝑗𝑑𝑡^{(𝜇−1)/𝜇}𝑢⁺_𝜆𝑗≤𝐶₅𝑢⁺_𝜆𝑗 (60) for some positive constant𝐶₄and𝐶₅. It implies that‖𝑢⁺_𝜆𝑗‖≤𝐶₅. On the other hand, (𝐻₂) and Φ^′_𝜆𝑗|_𝑌𝑗(𝑢_𝜆𝑗)𝑢_𝜆𝑗=0imply that 𝑢⁺_𝜆𝑗²−𝜆_𝑗𝑢⁻_𝜆𝑗²=𝜆_𝑗ℝ𝐻_𝑢𝑡,𝑢_𝜆𝑗,𝑢_𝜆𝑗𝑑𝑡≥0; (61) that is, 𝜆_𝑗𝑢⁻_𝜆𝑗²≤𝑢⁺_𝜆𝑗². (62) It follows from‖𝑢⁺_𝜆𝑗‖≤𝐶₅that{𝑢_𝜆𝑗}is bounded in𝑊. Therefore, Claim 1 is true.

Claim 2. {𝑢_𝜆𝑗}has a strongly convergent subsequence in𝑊.

Proof of Claim 2. Note thatdim(𝑊⁰⊕𝑊⁻)<∞. By Claim 1, without loss of generality, we may assume that 𝑢⁻_𝜆𝑗⟶𝑢⁻,𝑢⁰_𝜆𝑗⟶𝑢⁰,𝑢⁺_𝜆𝑗⇀𝑢⁺,𝑢_𝜆𝑗⇀𝑢as𝑗⟶∞ (63) for some𝑢=𝑢⁰+𝑢⁻+𝑢⁺∈𝑊=𝑊⁰⊕𝑊⁻⊕𝑊⁺. By virtue of the Riesz Representation Theorem, Φ^′_𝜆𝑗|_𝑌𝑗∶𝑌_𝑗→𝑌^∗_𝑗and𝐼^′∶𝑊→𝑊^∗can be viewed asΦ^′_𝜆𝑗|_𝑌𝑗∶𝑌_𝑗→𝑌_𝑗and𝐼^′∶𝑊→𝑊, respectively, where𝑌^∗_𝑗and𝑊^∗are the dual spaces of𝑌_𝑗and𝑊, respectively. Note that 0=Φ^′_𝜆𝑗|_𝑌𝑗𝑢_𝜆𝑗=𝑢⁺_𝜆𝑗−𝜆_𝑗𝑢⁻_𝜆𝑗+𝜒_𝑗𝐼^′𝑢_𝜆𝑗,∀𝑗∈ℕ, (64) where𝜒_𝑗∶𝑊→𝑌_𝑗is the orthogonal projection for all𝑗∈ℕ; that is, 𝑢⁺_𝜆𝑗=𝜆_𝑗𝑢⁻_𝜆𝑗+𝜒_𝑗𝐼^′𝑢_𝜆𝑗,∀𝑗∈ℕ. (65) By the assumptions of𝐻and the standard argument (see [29, 30]), we know𝐼^′∶𝑊→𝑊^∗is compact. Therefore,𝐼^′∶𝑊→𝑊is also compact. Due to the compactness of𝐼^′and (63), the right-hand side of (65) converges strongly in𝑊and hence𝑢⁺_𝜆𝑗→𝑢⁺in𝑊. Combining this with (63), we have 𝑢_𝜆𝑗⟶𝑢in𝑊,𝑗⟶∞. (66) Therefore, Claim 2 is true.
Now, from the last assertion of Lemma 5, we know thatΦ=Φ₁has infinitely many nontrivial critical points. Therefore, (1) possesses infinitely many nontrivial homoclinic orbits.

Acknowledgment

This Research was supported by the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).