Abstract and Applied Analysis
Volume 2009 (2009), Article ID 128624, 15 pages
doi:10.1155/2009/128624
Research Article

Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

Jian Ding, Junxiang Xu, and Fubao Zhang

Department of Mathematics, Southeast University, Nanjing 210018, China

Received 20 September 2009; Revised 19 November 2009; Accepted 6 December 2009

Academic Editor: Stephen Clark

Copyright © 2009 Jian Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper concerns solutions for the Hamiltonian system: ̇ 𝑧 = 𝒥 𝐻 𝑧 ( 𝑡 , 𝑧 ) . Here 𝐻 ( 𝑡 , 𝑧 ) = ( 1 / 2 ) 𝑧 ⋅ 𝐿 𝑧 + 𝑊 ( 𝑡 , 𝑧 ) , 𝐿 is a 2 𝑁 × 2 𝑁 symmetric matrix, and 𝑊 ∈ 𝐶 1 ( ℝ × ℝ 2 𝑁 , ℝ ) . We consider the case that 0 ∈ 𝜎 𝑐 ( − ( 𝒥 ( 𝑑 / 𝑑 𝑡 ) + 𝐿 ) ) and 𝑊 satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.

1. Introduction and the Main Results

In this paper, we consider the existence of homoclinic orbits for the following Hamiltonian system: ̇ 𝑧 = 𝒥 𝐻 𝑧 ( 𝑡 , 𝑧 ) , ( 1 . 1 ) where 𝐻 ( 𝑡 , 𝑧 ) = ( 1 / 2 ) 𝑧 ⋅ 𝐿 𝑧 + 𝑊 ( 𝑡 , 𝑧 ) , 𝐿 is a 2 𝑁 × 2 𝑁 symmetric matrix-valued function, and 𝑊 ∈ 𝐶 1 ( ℝ × ℝ 2 𝑁 , ℝ ) is superquadratic both around 0 and at infinity in 𝑧 ∈ ℝ 2 𝑁 .

A solution of (1.1) is called to be homoclinic to 0 if 𝑧 ( 𝑡 ) ≢ 0 and 𝑧 ( 𝑡 ) → 0 as | 𝑡 | → ∞ .

In recent years, the existence and multiplicity of homoclinic orbits for Hamiltonian systems have been investigated in many papers via variational methods. See, for example, [1–7] for the second-order systems and [8–12] for the first-order systems. We note that in most of the papers on the first order system (1.1) it was assumed that

( ◊ ) 𝐿 is constant such that s p ( 𝒥 𝐿 ) ∩ 𝑖 ℝ = ∅ , where s p ( 𝒥 𝐿 ) denotes the set of all eigenvalues of 𝒥 𝐿 .

Thus, if we let 𝜎 ( 𝐴 ) denote the spectrum of 𝐴 , ( ◊ ) means that 𝐿 is independent of 𝑡 and there is 𝛼 > 0 such that ( − 𝛼 , 𝛼 ) ∩ 𝜎 ( 𝐴 ) = ∅ . Consequently, the operator 𝐴 ∶ = − ( 𝒥 ( 𝑑 / 𝑑 𝑡 ) + 𝐿 ) ∶ 𝑊 1 , 𝑝 ( ℝ , ℝ 2 𝑁 ) → 𝐿 𝑝 ( ℝ , ℝ 2 𝑁 ) is a homeomorphism for all 𝑝 > 1 . This is important for the variational arguments. Later in [13], Ding considered the case that 𝐿 depends periodically on 𝑡 . He made assumptions on 𝐿 such that 0 lies in a gap of 𝜎 ( 𝐴 ) . If additionally 𝑊 ( 𝑡 , 𝑧 ) is periodic in 𝑡 and satisfies some superquadratic or asymptotically quadratic conditions in 𝑧 at infinity, then infinitely many homoclinic orbits were obtained.

If 0 ∈ 𝜎 𝑐 ( 𝐴 ) , then the problem is quite different in nature since the operator 𝐴 cannot lead the behavior at 0 of the equation. Ding and Willem considered this case in [14]. They assumed that

( 𝐴 0 ) 𝐿 ( 𝑡 ) ∈ 𝐶 ( ℝ , ℝ 4 𝑁 2 ) is 1-periodic. There exists 𝛼 > 0 such that ( 0 , 𝛼 ) ∩ 𝜎 ( 𝐴 ) = ∅ .

Under ( 𝐴 0 ) , 0 may belong to continuous spectrum of 𝐴 . The authors managed to construct an appropriate Banach space, on which some embedding results necessary for variational arguments were obtained. Using a generalized linking theorem developed by Kryszewski and Szulkin in [15], they got one homoclinic orbit of (1.1). Later, Ding and Girardi obtained infinitely many homoclinic orbits in [16] under the conditions of [14] with an additional evenness assumption on 𝑊 . Note that in both papers 𝑊 satisfies a condition of the type of Ambrosetti-Rabinowitz (see [17]), that is,

∃ 𝜇 > 2 s u c h t h a t 0 < 𝜇 𝑊 ( 𝑡 , 𝑧 ) ≤ 𝑊 𝑧 ( 𝑡 , 𝑧 ) 𝑧 , 𝑡 ∈ ℝ , 𝑧 ∈ ℝ 2 𝑁 ⧵ { 0 } . ( A - R ) The (A-R) condition is essential to prove the Palais-Smale condition since the variational functional Φ is strongly indefinite and 0 ∈ 𝜎 𝑐 ( − 𝒥 ( 𝑑 / 𝑑 𝑡 ) + 𝐿 ) . The argument of Palais-Smale condition is rather technical and not standard without the (A-R) condition. In this paper, we consider the existence of solutions of (1.1) under ( 𝐴 0 ) without the (A-R) condition on 𝑊 .

We observed that just recently some abstract linking theorems were developed by Bartsch and Ding in [18]. These theorems are impactful to study the existence and multiplicity of solutions for the strongly indefinite problem. Many new results have been obtained by these theorems based on the use of ( 𝐶 ) 𝑐 sequence. See [19–21] for applications of these ideas. Note that in [19–21] 0 either is not a spectral point or is at most an isolated eigenvalue of finite multiplicity. Thus ( 𝐶 ) 𝑐 condition was checked by virtue of some very technical analysis. However, if 0 ∈ 𝜎 𝑐 ( 𝐴 ) , then we can find a sequence { 𝑧 𝑛 } ⊂ 𝐻 1 with | 𝑧 𝑛 | 𝐿 2 = 1 and | 𝐴 𝑧 𝑛 | 𝐿 2 → 0 . Thus the operator 𝐴 cannot lead the behavior at 0 of the equation. Consequently, besides ( 𝐶 ) 𝑐 condition, it seems also hard to check the following condition necessary for the linking theorems in [19–21]:

( Φ 1 ) for any 𝑐 > 0 , there exists 𝜁 > 0 such that ‖ 𝑧 ‖ < 𝜁 ‖ 𝑃 𝑌 𝑧 ‖ for all 𝑧 ∈ Φ 𝑐 .

Our work benefits from [14] and some weak linking theorem recently developed by Schechter and Zou in [22]. This theorem permits us first to study a sequence of approximating problems Φ 𝜆 for 𝜆 ∈ [ 1 , 2 ] (the initial problem corresponds to 𝜆 = 1 ) for which a bounded Palais-Smale sequence of Φ 𝜆 is given for almost each 𝜆 ∈ [ 1 , 2 ] . Then by monotonicity, we find a sequence of { 𝜆 𝑛 } and { 𝑤 𝑛 } such that 𝜆 𝑛 → 1 , Φ 𝜆 𝑛 ( 𝑤 𝑛 ) = 0 , and Φ 𝜆 𝑛 ( 𝑤 𝑛 ) ≤ 𝑑 . Since the sequence { 𝑤 𝑛 } consists of critical points of Φ 𝜆 𝑛 , then its boundedness can be checked. Consequently one solution of (1.1) is obtained. The idea of first studying approximating problems for which the existence of a bounded Palais-Smale sequence is given freely and then proving that the sequence of approximated critical points is bounded was originally introduced in [23]. See also [24].

We make the following assumptions.

( 𝐴 1 ) 𝑊 ( 𝑡 , 𝑧 ) ∈ 𝐶 1 ( ℝ × ℝ 2 𝑁 , ℝ ) is 1-periodic in 𝑡 . 𝑊 ( 𝑡 , 0 ) = 0 for all 𝑡 ∈ ℝ . There exist constants 𝑐 1 > 0 and 𝜇 > 2 such that 𝑊 𝑧 ( 𝑡 , 𝑧 ) 𝑧 ≥ 𝑐 1 | 𝑧 | 𝜇 for ( 𝑡 , 𝑧 ) ∈ ℝ × ℝ 2 𝑁 . ( 𝐴 2 )there exist 𝑐 2 , 𝑟 > 0 such that | 𝑊 𝑧 ( 𝑡 , 𝑧 ) | ≤ 𝑐 2 | 𝑧 | 𝜇 − 1 for 𝑡 ∈ ℝ and | 𝑧 | ≤ 𝑟 . ( 𝐴 3 )there exist 𝑐 3 , 𝑅 ≥ 𝑟 and 𝑝 ≥ 𝜇 such that | 𝑊 𝑧 ( 𝑡 , 𝑧 ) | ≤ 𝑐 3 | 𝑧 | 𝑝 − 1 for 𝑡 ∈ ℝ and | 𝑧 | ≥ 𝑅 . ( 𝐴 4 )there exists 𝑏 0 > 2 such that l i m i n f 𝑧 → 0 ( 𝑊 𝑧 ( 𝑡 , 𝑧 ) 𝑧 / 𝑊 ( 𝑡 , 𝑧 ) ) ≥ 𝑏 0 uniformly for 𝑡 ∈ ℝ ;( 𝐴 5 ) 𝑊 ( 𝑡 , 𝑧 ) ∶ = ( 1 / 2 ) 𝑊 𝑧 ( 𝑡 , 𝑧 ) 𝑧 − 𝑊 ( 𝑡 , 𝑧 ) > 0 for all 𝑡 ∈ ℝ , 𝑧 ∈ ℝ 2 𝑁 ⧵ { 0 } . There exist constants 𝑏 ∞ > 0 and 𝛽 > 𝑝 ( 𝑝 − 2 ) / ( 𝑝 − 1 ) such that l i m i n f | 𝑧 | → ∞ 𝑊 ( 𝑡 , 𝑧 ) / | 𝑧 | 𝛽 ≥ 𝑏 ∞ uniformly for 𝑡 ∈ ℝ .

Theorem 1.1. Let ( 𝐴 0 ) , ( 𝐴 1 ) - ( 𝐴 5 ) be satisfied, then (1.1) has at least one homoclinic orbit.

Remark 1.2. We can easily check that the (A-R) condition implies ( 𝐴 4 ) and ( 𝐴 5 ) . But the converse proposition is not true. See the following example: 𝑊 ( 𝑡 , 𝑧 ) = | 𝑧 | 𝜇 + ( 𝜇 − 2 ) | 𝑧 | 𝜇 − 𝜖 s i n 2 | 𝑧 | 𝜖 𝜖 , ( 1 . 2 ) where 2 < 𝜇 < ∞ , 0 < 𝜖 < m i n { 𝜇 − 2 , 𝜇 / ( 𝜇 − 1 ) } (see [25] or [26] for details).

If 𝑊 𝑧 ( 𝑡 , 𝑧 ) = 𝑎 | 𝑧 | 𝜇 − 2 𝑧 + 𝑅 𝑧 ( 𝑡 , 𝑧 ) , 𝑎 > 0 , 𝜇 ∈ ( 2 , ∞ ) with 𝑅 satisfying

( 𝐵 1 ) 𝑅 ∈ 𝐶 1 ( ℝ × ℝ 2 𝑁 , ℝ ) is 1-periodic in 𝑡 and 𝑅 𝑧 ( 𝑡 , 𝑧 ) = 𝑜 | 𝑧 | 𝜇 − 1 𝑅 a s | 𝑧 | ⟶ 0 , 𝑧 ( 𝑡 , 𝑧 ) = 𝑜 | 𝑧 | 𝜇 − 1 a s | 𝑧 | ⟶ ∞ , ( 1 . 3 ) uniformly in 𝑡 ∈ ℝ , then ( 𝐵 2 ) 0 < 𝑅 𝑧 ( 𝑡 , 𝑧 ) 𝑧 ≤ ( 𝑎 ( 𝜇 − 2 ) / 2 ) | 𝑧 | 𝜇 for all 𝑡 ∈ ℝ , 𝑧 ∈ ℝ 2 𝑁 ⧵ { 0 } .

Theorem 1.3. Let ( 𝐴 0 ) , ( 𝐵 1 ) , and ( 𝐵 2 ) be satisfied, then (1.1) has at least one homoclinic orbit.

This paper is organized as follows. In Section 2 we will construct some appropriate variational space and obtain some embedding results necessary for our variational arguments. In Section 3 we will recall a weak linking theorem, by which we will give the proof of Theorems 1.1 and 1.3 in Section 4.

2. Some Embedding Results

In what follows, by | ⋅ | 𝑝 we denote the usual 𝐿 𝑝 -norm and by ( ⋅ ) 2 the usual 𝐿 2 -inner product. A standard Floquet reduction argument shows that 𝜎 ( 𝐴 ) = 𝜎 𝑐 ( 𝐴 ) (see [14]).

Let { 𝐸 ( 𝜆 ) ; 𝜆 ∈ ℝ } be the spectral family of 𝐴 . 𝐴 possesses the polar decomposition 𝐴 = 𝑈 | 𝐴 | with 𝑈 = 𝐼 − 𝐸 ( 0 ) − 𝐸 ( − 0 ) . By ( 𝐴 0 ) , 0 is at most a continuous spectrum of 𝐴 . 𝐿 2 has an orthogonal decomposition

𝐿 2 = 𝐿 2 − ⊕ 𝐿 2 + , ( 2 . 1 ) where 𝐿 2 ± ∶ = { 𝑢 ∈ 𝐿 2 ; 𝑈 𝑢 = ± 𝑢 } .

Let 𝒟 ( | 𝐴 | 1 / 2 ) denote the domain of | 𝐴 | 1 / 2 and let 𝐸 be the space of the completion of 𝒟 ( | 𝐴 | 1 / 2 ) under the norm

‖ 𝑢 ‖ 𝐸 | | | | | 𝐴 | | ∶ = 1 / 2 𝑢 | | | 2 . ( 2 . 2 ) 𝐸 becomes a Hilbert space under the inner product

( 𝑢 , 𝑣 ) 𝐸 | | 𝐴 | | ∶ = 1 / 2 | | 𝐴 | | 𝑢 , 1 / 2 𝑣 2 . ( 2 . 3 ) 𝐸 possesses an orthogonal decomposition

𝐸 = 𝐸 − ⊕ 𝐸 + , ( 2 . 4 ) where 𝐸 ± ⊇ 𝐿 2 ± ∩ 𝒟 ( | 𝐴 | 1 / 2 ) .

Under ( 𝐴 0 ) , it is easy to check

𝐸 + = 𝐿 2 + | | 𝐴 | | ∩ 𝒟 1 / 2 , ‖ ⋅ ‖ 𝐸 ∼ ‖ ⋅ ‖ 𝐻 1 / 2 o n 𝐸 + . ( 2 . 5 ) Therefore, 𝐸 + can be embedded continuously into 𝐿 𝑝 ( ℝ , ℝ 2 𝑁 ) for any 𝑝 ≥ 2 and compactly into 𝐿 𝑝 l o c ( ℝ , ℝ 2 𝑁 ) for any 𝑝 ∈ [ 1 , ∞ ) .

For any 𝜖 > 0 , set 𝐿 𝜖 2 − ∶ = 𝐸 ( − 𝜖 ) 𝐿 2 and 𝐸 − 𝜖 ∶ = 𝐿 𝜖 2 − ∩ 𝒟 ( | 𝐴 | 1 / 2 ) = 𝐿 𝜖 2 − ∩ 𝐸 − . Then on 𝐸 − 𝜖 , we also have ‖ ⋅ ‖ 𝐸 ∼ ‖ ⋅ ‖ 𝐻 1 / 2 and the same embedding conclusion as that of 𝐸 + .

Let 𝐿 𝜖 2 − ∶ = 𝐿 2 − ∩ ( c l 𝐿 2 ( ⋃ 𝜆 < − 𝜖 𝐸 ( 𝜆 ) 𝐿 2 ) ) ⟂ where c l 𝐿 2 ( ⋅ ) stands for the closure of ⋅ in 𝐿 2 .

For 𝜇 > 2 , let 𝐸 − 𝜖 , 𝜇 be the completion of 𝐿 𝜖 2 − under the norm ‖ 𝑢 ‖ 𝜇 | | | | | 𝐴 | | ∶ = 1 / 2 𝑢 | | | 2 2 + | 𝑢 | 2 𝜇 1 / 2 , ( 2 . 6 ) and let 𝐸 − 𝜇 denote the completion of 𝒟 ( 𝐴 ) ∩ 𝐿 2 − with respect to the norm ‖ ⋅ ‖ 𝜇 . Then 𝐸 − 𝜖 is a closed subspace of 𝐸 − 𝜇 , and 𝐸 − 𝜇 possesses the following decomposition:

𝐸 − 𝜇 = 𝐸 − 𝜖 ⊕ 𝐸 − 𝜖 , 𝜇 . ( 2 . 7 ) Moreover, 𝐸 − 𝜖 is orthogonal to 𝐸 − 𝜖 , 𝜇 with respect to ( ⋅ ) 𝐸 .

Let 𝐸 𝜇 be the completion of 𝒟 ( 𝐴 ) under the norm ‖ ⋅ ‖ 𝜇 . The following result holds true.

Lemma 2.1 (see [14]). Under ( 𝐴 0 ) , 𝐸 𝜇 has the direct sum decomposition 𝐸 𝜇 = 𝐸 − 𝜇 ⊕ 𝐸 + , ( 2 . 8 ) and 𝐸 𝜇 is embedded continuously in 𝐿 𝜈 for any 𝜈 ∈ [ 𝜇 , ∞ ) and compactly in 𝐿 𝜈 l o c   for any 𝜈 ∈ [ 2 , ∞ ) .

3. A Weak Linking Theorem

In this section we state some weak linking theorem due to [22] which was first built in a Hilbert space. This theorem is still true in a reflexive Banach space (cf. Willem and Zou [25]).

Let 𝐸 be a reflexive Banach space with norm ‖ ⋅ ‖ and possess a direct sum decomposition 𝐸 = 𝑁 ⊕ 𝑀 , where 𝑁 ⊂ 𝐸 is a closed and separable subspace. Since 𝑁 is separable, we can define a new norm | 𝑧 | 𝑤 satisfying | 𝑧 | 𝑤 ≤ ‖ 𝑧 ‖ , f o r a l l 𝑧 ∈ 𝑁 such that the topology induced by this norm is equivalent to the weak topology of 𝑁 on bounded subsets of 𝑁 . For 𝑧 = 𝑣 + 𝑤 ∈ 𝐸 with 𝑣 ∈ 𝑁 and 𝑤 ∈ 𝑀 , we define | 𝑧 | 2 𝑤 = | 𝑣 | 2 𝑤 + ‖ 𝑤 ‖ 2 , then | 𝑧 | 𝑤 ≤ ‖ 𝑧 ‖ , f o r a l l 𝑧 ∈ 𝐸 . In particular, if 𝑧 𝑛 = 𝑣 𝑛 + 𝑤 𝑛 is | ⋅ | 𝑤 -bounded and 𝑧 𝑛 → 𝑧 under the norm | ⋅ | 𝑤 in 𝐸 , then 𝑣 𝑛 ⇀ 𝑣 weakly in 𝑁 , 𝑤 𝑛 → 𝑤 strongly in 𝑀 , and 𝑧 𝑛 ⇀ 𝑣 + 𝑤 weakly in 𝐸 . Let 𝑄 ⊂ 𝑁 be a ‖ ⋅ ‖ -bounded open convex subset and let 𝑝 0 ∈ 𝑄 be a fixed point. Let 𝐹 be a | ⋅ | 𝑤 -continuous map from 𝐸 onto 𝑁 satisfying the following.

(i) 𝐹 | 𝑄 = id; 𝐹 maps bounded sets to bounded sets.(ii)there exists a fixed finite-dimensional subspace 𝐸 0 of 𝐸 such that 𝐹 ( 𝑢 − 𝑣 ) − ( 𝐹 ( 𝑢 ) − 𝐹 ( 𝑣 ) ) ⊂ 𝐸 0 , f o r a l l 𝑢 , 𝑣 ∈ 𝐸 . (iii) 𝐹 maps finite-dimensional subspaces of 𝐸 into finite-dimensional subspaces of 𝐸 .

Set 𝐴 ∶ = 𝜕 𝑄 , 𝐵 ∶ = 𝐹 − 1 ( 𝑝 0 ) , where 𝜕 𝑄 denotes the ‖ ⋅ ‖ -boundary of 𝑄 . For Φ ∈ 𝐶 1 ( 𝐸 , ℝ ) , we introduce the class Γ of mappings ℎ ∶ [ 0 , 1 ] × 𝑄 → 𝐸 with the following properties.

(a) ℎ ∶ [ 0 , 1 ] × 𝑄 → 𝐸 is | ⋅ | 𝑤 -continuous.(b)for any ( 𝑠 0 , 𝑢 0 ) ∈ [ 0 , 1 ] × 𝑄 , there is a | ⋅ | 𝑤 -neighborhood 𝑈 ( 𝑠 0 , 𝑢 0 ) such that { 𝑢 − ℎ ( 𝑡 , 𝑢 ) ∶ ( 𝑡 , 𝑢 ) ∈ 𝑈 ( 𝑠 0 , 𝑢 0 ) ∩ ( [ 0 , 1 ] × 𝑄 ) } ⊂ 𝐸 𝑓 𝑖 𝑛 , where 𝐸 𝑓 𝑖 𝑛 denotes some finite-dimensional subspaces of 𝐸 . (c) ℎ ( 0 , 𝑢 ) = 𝑢 , Φ ( ℎ ( 𝑠 , 𝑢 ) ) ≤ Φ ( 𝑢 ) , f o r a l l 𝑢 ∈ 𝑄 .

The following is a variant weak linking theorem in [22].

Theorem 3.1. Let the family of 𝐶 1 -functionals ( Φ 𝜆 ) have the form Φ 𝜆 [ ] . ( 𝑢 ) ∶ = 𝐼 ( 𝑢 ) − 𝜆 𝐽 ( 𝑢 ) , ∀ 𝜆 ∈ 1 , 2 ( 3 . 1 ) Assume that the following conditions hold. (1) 𝐽 ( 𝑢 ) ≥ 0 , f o r a l l 𝑢 ∈ 𝐸 ; Φ 1 ∶ = Φ .(2) 𝐼 ( 𝑢 ) → ∞ or 𝐽 ( 𝑢 ) → ∞ as ‖ 𝑢 ‖ → ∞ .(3) Φ 𝜆 is | ⋅ | 𝑤 -upper semicontinuous; Φ 𝜆 is weakly sequentially continuous on 𝐸 . Moreover, Φ 𝜆 maps bounded sets into bounded sets.(4) s u p 𝐴 Φ 𝜆 < i n f 𝐵 Φ 𝜆 , f o r a l l 𝜆 ∈ [ 1 , 2 ] .Then for almost all 𝜆 ∈ [ 1 , 2 ] , there exists a sequence { 𝑢 𝑛 } such that s u p 𝑛 ‖ ‖ 𝑢 𝑛 ‖ ‖ < ∞ , Φ 𝜆 𝑢 𝑛 ⟶ 0 , Φ 𝜆 𝑢 𝑛 ⟶ 𝐶 𝜆 , ( 3 . 2 ) where 𝐶 𝜆 ∶ = i n f ℎ ∈ Γ s u p 𝑢 ∈ 𝑄 Φ 𝜆 ( ℎ ( 1 , 𝑢 ) ) ∈ [ i n f 𝐵 Φ 𝜆 , s u p 𝑄 Φ 𝜆 ] .

Remark 3.2. Consider 𝐸 𝜇 = 𝐸 − 𝜇 ⊕ 𝐸 + defined as in Section 2. Obviously, 𝐸 𝜇 is reflexive. For 𝑧 0 ∈ 𝐸 + with ‖ 𝑧 0 ‖ 𝜇 = 1 , 𝐸 + = ℝ 𝑧 0 ⊕ 𝐸 + 1 . Let 𝑁 = 𝐸 − 𝜇 ⊕ ℝ 𝑧 0 and 𝑀 = 𝐸 + 1 , then 𝐸 𝜇 = 𝑁 ⊕ 𝑀 . It is easy to see that 𝑁 is a closed and separable subspace of 𝐸 𝜇 . For any 𝑢 ∈ 𝐸 𝜇 , 𝑢 can be written as 𝑢 = 𝑢 − + 𝑠 𝑧 0 + 𝑤 + with 𝑢 − ∈ 𝐸 − 𝜇 and 𝑤 + ∈ 𝐸 + 1 . For 𝑅 > 0 , let 𝑄 ∶ = { 𝑢 ∶ = 𝑢 − + 𝑠 𝑧 0 , 𝑠 > 0 , 𝑢 − ∈ 𝐸 − 𝜇 , ‖ 𝑢 ‖ 𝜇 < 𝑅 } , then 𝑝 0 ∶ = 𝑠 0 𝑧 0 ∈ 𝑄 for 0 < 𝑠 0 < 𝑅 . Define 𝐹 ∶ 𝐸 𝜇 → 𝑁 as 𝐹 𝑢 ∶ = 𝑢 − + ‖ 𝑠 𝑧 0 + 𝑤 + ‖ 𝜇 𝑧 0 . Then it is easy to check that 𝐹 , 𝑄 , a n d 𝑝 0 satisfy (i), (ii), and (iii). If we let 𝐴 ∶ = 𝜕 𝑄 and 𝐵 ∶ = 𝐹 − 1 ( 𝑠 0 𝑧 0 ) = { 𝑢 ∶ = 𝑠 𝑧 0 + 𝑤 + , 𝑠 ≥ 0 , 𝑤 + ∈ 𝐸 + 1 , ‖ 𝑠 𝑧 0 + 𝑤 + ‖ 𝜇 = 𝑠 0 } , then 𝐴 links 𝐵 (see Lemmas 4.2 and 4.3 in Section 4).

4. The Proof of the Main Results

Consider the functional

1 Φ ( 𝑧 ) ∶ = 2 ‖ ‖ 𝑧 + ‖ ‖ 2 𝐸 − 1 2 ‖ 𝑧 − ‖ 2 𝐸 − ℝ 𝑊 ( 𝑡 , 𝑧 ) , ( 4 . 1 ) for 𝑧 = 𝑧 + + 𝑧 − ∈ 𝐸 𝜇 . Then by assumptions ( 𝐴 1 ) - ( 𝐴 3 ) and Lemma 2.1, Φ ∈ 𝐶 1 ( 𝐸 𝜇 , ℝ ) . A standard argument shows that any critical point of Φ is a homoclinic orbit of (1.1) (cf. [14]).

Set

Φ 𝜆 1 ( 𝑧 ) ∶ = 2 ‖ ‖ 𝑧 + ‖ ‖ 2 𝐸 1 − 𝜆 2 ‖ 𝑧 − ‖ 2 𝐸 + ℝ [ ] 𝑊 ( 𝑡 , 𝑧 ) = 𝐼 ( 𝑧 ) − 𝜆 𝐽 ( 𝑧 ) , 𝜆 ∈ 1 , 2 . ( 4 . 2 ) Then Φ 1 = Φ and 𝐽 ( 𝑧 ) ≥ 0 . By ( 𝐴 2 ) and ( 𝐴 3 ) ,

| | 𝑊 𝑧 | | ( 𝑡 , 𝑧 ) ≤ 𝐶 | 𝑧 | 𝜇 − 1 + | 𝑧 | 𝑝 − 1 , ( 4 . 3 ) where, as below, 𝐶 stands for some generic positive constant.

Together with ( 𝐴 1 ) , one has

𝑐 1 𝜇 | 𝑧 | 𝜇 ≤ 𝑊 ( 𝑡 , 𝑧 ) ≤ 𝐶 ( | 𝑧 | 𝜇 + | 𝑧 | 𝑝 ) . ( 4 . 4 ) Thus 𝐼 ( 𝑧 ) → ∞ or 𝐽 ( 𝑧 ) → ∞ if ‖ 𝑧 ‖ 2 𝜇 = ‖ 𝑧 + ‖ 2 𝐸 + ‖ 𝑧 − ‖ 2 𝐸 + | 𝑧 | 2 𝜇 → ∞ .

Lemma 4.1. Φ 𝜆 is | ⋅ | 𝑤 -upper semicontinuous and Φ 𝜆 is weakly sequentially continuous.

Proof. For any 𝑐 ∈ ℝ , assume that 𝑧 𝑛 ∈ { 𝑧 ∈ 𝐸 𝜇 , Φ 𝜆 ( 𝑧 ) ≥ 𝑐 } with 𝑧 𝑛 ⇀ 𝑧 . Let 𝑧 𝑛 = 𝑧 + 𝑛 + 𝑧 − 𝑛 with 𝑧 + 𝑛 ∈ 𝐸 + and 𝑧 − 𝑛 ∈ 𝐸 − 𝜇 . Then 𝑧 + 𝑛 → 𝑧 + in 𝐸 𝜇 and hence s u p ‖ 𝑧 + 𝑛 ‖ < ∞ . Since Φ 𝜆 ( 𝑧 𝑛 ) ≥ 𝑐 and 𝑊 ( 𝑡 , 𝑧 𝑛 ) ≥ 0 , we have s u p ‖ 𝑧 − 𝑛 ‖ < ∞ .
By (4.4),
𝐶 | | 𝑧 𝑛 | | 𝜇 𝜇 ≤ ℝ 𝑊 𝑡 , 𝑧 𝑛 1 d 𝑡 ≤ 𝜆 1 2 ‖ ‖ 𝑧 + 𝑛 ‖ ‖ 2 𝐸 − 𝜆 2 ‖ ‖ 𝑧 − 𝑛 ‖ ‖ 2 𝐸 − 𝑐 < ∞ . ( 4 . 5 ) Then s u p ‖ 𝑧 𝑛 ‖ 𝜇 < ∞ . By Lemma 2.1, 𝑧 𝑛 ⇀ 𝑧 in 𝐸 𝜇 , 𝑧 𝑛 → 𝑧 in 𝐿 𝜇 l o c , and 𝑧 𝑛 ( 𝑡 ) → 𝑧 ( 𝑡 ) a.e. for 𝑡 ∈ ℝ . By Fatou's Lemma, Φ 𝜆 ( 𝑧 ) ≥ 𝑐 . Therefore, Φ 𝜆 is | ⋅ | 𝑤 -upper semicontinuous.
Let 𝑧 𝑛 ⇀ 𝑧 in 𝐸 𝜇 , then 𝑧 𝑛 → 𝑧 in 𝐿 𝑝 l o c , 2 ≤ 𝑝 < ∞ . By (4.3), 𝑊 𝑧 ( 𝑡 , 𝑧 𝑛 ) → 𝑊 𝑧 ( 𝑡 , 𝑧 ) in 𝐿 𝜇 / ( 𝜇 − 1 ) l o c and ∫ ℝ 𝑊 𝑧 ( 𝑡 , 𝑧 𝑛 ∫ ) 𝑣 → ℝ 𝑊 𝑧 ( 𝑡 , 𝑧 ) 𝑣 for any 𝑣 ∈ 𝐸 𝜇 . Therefore, Φ 𝜆 ( 𝑧 𝑛 ) → Φ 𝜆 ( 𝑧 ) .

Lemma 4.2. There exist 𝑏 > 0 , 𝑟 > 0 such that Φ 𝜆 ( 𝑧 ) ≥ 𝑏 > 0 , f o r a l l 𝑧 ∈ 𝐸 + with ‖ 𝑧 ‖ 𝜇 = 𝑟 , f o r a l l 𝜆 ∈ [ 1 , 2 ] .

Proof. By (4.4) and Lemma 2.1, ℝ | 𝑊 ( 𝑡 , 𝑧 ) ≤ 𝐶 𝑧 | 𝜇 𝜇 + | 𝑧 | 𝑝 𝑝 ≤ 𝐶 ‖ 𝑧 ‖ 𝜇 𝜇 + ‖ 𝑧 ‖ 𝑝 𝜇 . ( 4 . 6 ) The conclusion is obvious.

Lemma 4.3. There exists 𝑅 > 𝑟 > 0 such that Φ 𝜆 | 𝜕 𝑀 = 0 and s u p 𝑀 Φ 𝜆 < 𝑑 < ∞ for all 𝜆 ∈ [ 1 , 2 ] , where 𝑀 ∶ = { 𝑧 = 𝑥 + 𝑠 𝑧 0 , 𝑥 ∈ 𝐸 − 𝜇 , ‖ 𝑧 ‖ 𝜇 ≤ 𝑅 , 𝑠 > 0 } and 𝑧 0 ∈ 𝐸 + , ‖ 𝑧 0 ‖ 𝜇 = 1 .

Proof. For 𝑧 = 𝑥 + 𝑠 𝑧 0 , by (4.4), Φ 𝜆 𝑠 ( 𝑧 ) ≤ 2 2 ‖ ‖ 𝑧 0 ‖ ‖ 2 𝐸 − 1 2 ‖ 𝑥 ‖ 2 𝐸 − 𝐶 ℝ | | 𝑥 + 𝑠 𝑧 0 | | 𝜇 . ( 4 . 7 )
Since 𝐸 𝜇 is continuously embedded in 𝐿 𝑡 for 𝜇 ≤ 𝑡 < ∞ , there exists a continuous projection from 𝐸 − 𝜇 ⊕ ℝ 𝑧 0 in 𝐿 𝜇 to ℝ 𝑧 0 . Thus, | 𝑠 𝑧 0 | 𝜇 ≤ 𝐶 | 𝑥 + 𝑠 𝑧 0 | 𝜇 for some 𝐶 > 0 and then
Φ 𝜆 ( 𝑧 ) ≤ 𝐶 𝑠 2 − 𝐶 ‖ 𝑥 ‖ 2 𝐸 − 𝐶 𝑠 𝜇 , ( 4 . 8 ) and thus the lemma follows easily.

Combining Lemmas 4.1–4.3 and Theorem 3.1, we get the following.

Lemma 4.4. Under ( 𝐴 0 ) and ( 𝐴 1 ) - ( 𝐴 3 ) , for almost every 𝜆 ∈ [ 1 , 2 ] , there exist { 𝑧 𝑛 } ⊆ 𝐸 𝜇 such that ‖ ‖ 𝑧 s u p 𝑛 ‖ ‖ 𝜇 < ∞ , Φ 𝜆 𝑧 𝑛 ⟶ 0 , Φ 𝜆 𝑧 𝑛 ⟶ 𝐶 𝜆 ∈ [ ] . 𝑏 , 𝑑 ( 4 . 9 )

We need the following lemma which is a special case of a more general result due to Lions [27, 28].

Lemma 4.5. Let 𝑎 > 0 and 2 ≤ 𝑝 < ∞ . If { 𝑧 𝑛 } ⊂ 𝐻 1 is bounded and if s u p 𝑠 ∈ ℝ 𝐵 ( 𝑠 , 𝑎 ) | | 𝑢 𝑛 | | 𝑝 ⟶ 0 , 𝑛 ⟶ ∞ , ( 4 . 1 0 ) where 𝐵 ( 𝑠 , 𝑎 ) ∶ = ( 𝑠 − 𝑎 , 𝑠 + 𝑎 ) , then 𝑢 𝑛 → 0 in 𝐿 𝑡 for 2 < 𝑡 < ∞ .

Lemma 4.6. Under ( 𝐴 0 ) - ( 𝐴 3 ) , let 𝜆 ∈ [ 1 , 2 ] be fixed. For the sequence { 𝑧 𝑛 } in Lemma 4.4, there exist { 𝑘 𝑛 } ⊂ ℤ such that, up to a subsequence, 𝑢 𝑛 ( 𝑡 ) ∶ = 𝑧 𝑛 ( 𝑡 + 𝑘 𝑛 ) satisfies 𝑢 𝑛 ⇀ 𝑢 𝜆 ≠ 0 , Φ 𝜆 ( 𝑢 𝜆 ) = 0 and Φ 𝜆 ( 𝑢 𝜆 ) ≤ 𝑑 .

Proof. Write 𝑧 𝑛 = 𝑧 + 𝑛 + 𝑧 − 𝑛 with 𝑧 + 𝑛 ∈ 𝐸 + and 𝑧 − 𝑛 ∈ 𝐸 − 𝜇 . Since s u p ‖ 𝑧 𝑛 ‖ 𝜇 < ∞ , s u p ‖ 𝑧 + 𝑛 ‖ 𝐸 < ∞ , let 𝐴 + denote the part of 𝐴 in 𝒟 ( 𝐴 ) ∩ 𝐿 2 + = 𝐻 1 ∩ 𝐿 2 + ∶ = 𝐻 1 + . Then by ( 𝐴 0 ) , 𝐴 + = ∞ 𝛼 𝜆 d 𝐸 ( 𝜆 ) . ( 4 . 1 1 ) Obviously, 𝐴 + ∶ 𝐻 1 + ⊂ 𝐿 2 + → 𝐿 2 + has a bounded inverse 𝐴 + − 1 . Since | | 𝐴 + 𝑧 | | 2 2 = ∞ 𝛼 𝜆 | | | | d 𝐸 ( 𝜆 ) 𝑧 2 2 ≥ 𝛼 | 𝑧 | 2 2 , | ̇ 𝑧 | 2 = | | | | 𝐴 𝑧 + 𝐿 𝑧 2 ≤ | | 𝐴 + 𝑧 | | 2 + | | | | 𝐿 𝑧 2 f o r 𝑧 ∈ 𝐻 1 + , ( 4 . 1 2 ) then we have ‖ 𝑧 ‖ 𝐻 1 | | 𝐴 ≤ 𝐶 + 𝑧 | | 2 f o r 𝑧 ∈ 𝐻 1 + . ( 4 . 1 3 ) Set 𝑣 𝑛 = 𝐴 + − 1 𝑧 + 𝑛 ∈ 𝐻 1 + , then ‖ ‖ 𝑣 𝑛 ‖ ‖ 𝐻 1 | | 𝑧 ≤ 𝐶 + 𝑛 | | 2 ‖ ‖ 𝑧 ≤ 𝐶 + 𝑛 ‖ ‖ 𝐸 < ∞ . ( 4 . 1 4 ) We claim that 𝑣 𝑛 is nonvanishing, that is, there exist 𝑀 > 0 , 𝑎 > 0 , and 𝑦 𝑛 ∈ ℝ such that l i m i n f 𝑛 → ∞ 𝐵 ( 𝑦 𝑛 , 𝑎 ) | | 𝑣 𝑛 | | 2 d 𝑡 ≥ 𝑀 . ( 4 . 1 5 ) Indeed, if not, by (4.14), { 𝑣 𝑛 } is bounded in 𝐻 1 . Lemma 4.5 shows that 𝑣 𝑛 → 0 in 𝐿 𝑡 for 2 < 𝑡 < ∞ . By (4.3), | | | | ℝ 𝑊 𝑧 𝑡 , 𝑧 𝑛 𝑣 𝑛 | | | | ≤ 𝐶 ℝ | | 𝑧 𝑛 | | 𝜇 − 1 + | | 𝑧 𝑛 | | 𝑝 − 1 | | 𝑣 𝑛 | | | | 𝑧 ≤ 𝐶 𝑛 |