Homoclinic orbits for the second-order Hamiltonian systems

Nonlinear Analysis: Real World Applications 36 (2017) 116–138

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Homoclinic orbits for the second-order Hamiltonian systems✩ Zhisu Liu a , Shangjiang Guo b,∗ , Ziheng Zhang c a

School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, PR China College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China c Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, PR China b

article

info

Article history: Received 28 May 2015 Received in revised form 29 December 2016 Accepted 30 December 2016

abstract We deal with the following second-order Hamiltonian systems

u ¨ − L(t)u + ∇W (t, u) = 0, 2

Keywords: Second-order Hamiltonian system Homoclinic orbit Generalized Nehari manifold Ground state

where L ∈ C(R, RN ) is a symmetric and positive define matrix for all t ∈ R, W ∈ C 1 (R × RN , R) and ∇W (t, u) is the gradient of W with respect to u. Under the superquadratic condition, we obtain the existence of ground state homoclinic orbits by means of the generalized Nehari manifold developed by Szulkin and Weth. Under the subquadratic condition, we employ variational techniques and the concentration-compactness principle to establish new criteria guaranteeing the multiplicity of classical homoclinic orbits. Recent results in literature are generalized and significantly improved. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we are interested in the existence and also the multiplicity of homoclinic orbits for the following second-order Hamiltonian systems u ¨ − L(t)u + ∇W (t, u) = 0,

(HS)

2

where L ∈ C(R, RN ) is a symmetric and positive define matrix for all t ∈ R, W ∈ C 1 (R × RN , R) and ∇W (t, u) is the gradient of W with respect to u. As usual, we say that a solution u(t) of (HS) is homoclinic (to 0) if u(t) → 0 as t → ±∞. If u(t) ̸≡ 0, u(t) is called a nontrivial homoclinic solution. ✩ Zhisu Liu is supported by the NSFC (Grant Nos. 11626127); Shangjiang Guo is supported by the NSFC (Grant Nos. 11671123 and 11271115). ∗ Corresponding author. Fax: +86 731 88822570. E-mail addresses: [email protected] (Z. Liu), [email protected] (S. Guo), [email protected] (Z. Zhang).

http://dx.doi.org/10.1016/j.nonrwa.2016.12.006 1468-1218/© 2017 Elsevier Ltd. All rights reserved.

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Homoclinic orbits of Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics, and nuclear physics. In recent years, the existence and multiplicity of homoclinic orbits for system (HS) have been studied extensively by means of critical point theory, and many interesting results have been obtained. For instance, many authors have studied the existence of homoclinic solutions of (HS) in periodic case (that is, assuming that L(t) and W (t, u) are periodic in t). See, for instance, [1–4] and the references therein. In particular, Yang and Han [1] studied (HS) under the following assumptions: (L0 ) (W0 ) (W1 ) (W2 ) (W3 ) (W4 ) (W5 )

L(t) is T -periodic in t and 0 lies in a spectral gap of the operator −¨ u + Lu; 1 N W (t, u) ∈ C (R × R , R) is T -periodic in t; |∇W (t, u)| = o(|u|) as u → 0 uniformly in t ∈ R, W (t, 0) ≡ 0 and W (t, u) ≥ 0 for all (t, u) ∈ R × RN ; for each fixed (t, u) ∈ (R, RN ), the mapping s → s−1 (∇W (t, su), u) is non-decreasing function in s ∈ (0, 1]; W (t, u)/|u|2 → ∞ uniformly in t as |u| → ∞; there exist c > 0, 2 < p < +∞ such that |∇W (t, u)| ≤ c(1 + |u|p−1 ) for all (t, u) ∈ R × RN ; there exist constants t1 , t2 ∈ [p − 1, 2p − 2], a1 , a2 > 0 such that 1 (∇W (t, u), u) − W (t, u) ≥ a1 |u|t1 , 2 1 (∇W (t, u), u) − W (t, u) ≥ a2 |u|t2 , 2

∀t ∈ R, |u| ≥ 1, ∀t ∈ R, |u| ≤ 1.

By using the variant mountain pass theorem and generalized linking theorem, Yang and Han [1] investigated the existence of ground state homoclinic solutions (i.e., non-trivial solutions with least possible energy). To our best knowledge, there are few papers on the existence of ground state homoclinic orbits for second-order Hamiltonian systems, for example, [1,5,6]. Chen [5] studied system (HS) when W does not satisfy hypotheses (W2 ), (W4 ) and (W5 ) but satisfies the following condition: ′ ˜ (t, u) := 1 (∇W (t, u), u) − W (t, u) > 0, there exist d1 , d2 > 0 and α > 1 such that (W5 ) W 2

|∇W (t, u)|α ˜ (t, u), ≤ d1 W |u|α

if |u| ≥ d2 .

′

Under the hypotheses (L0 ), (W0 )–(W1 ), (W5 ) and (W3 ), Chen [5] also verified the existence of ground state homoclinic solutions of (HS) via a variant generalized weak linking theorem. Moreover, the existence of ground state homoclinic solutions for (HS) was also investigated in [5], when W is asymptotically quadratic at infinity. Szulkin and Weth [6] established the method of generalized Nehari manifold to study a Newtonian system (L(t) = Id in system (HS)) and showed that this system has at least a ground state homoclinic solution. It is very important to notice that, conditions (W1 ) and (W3 ) imply that W (t, u) is superquadratic both at the origin and at infinity, which is different from the well-known Ambrosetti–Rabinowitz (AR) condition for W . That is, there exists µ > 2 such that 0 < µW (t, u) ≤ (∇W (t, u), u)

for all (t, u) ∈ R × RN ,

which has been introduced in the literature [7,3,2,8]. Some other general Hamiltonian systems were also considered in the recent papers [3,9,10], where the existence of homoclinic solutions was investigated by passing to the limit of periodic solutions of the approximating problems.

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For the nonperiodic case (that is, L(t) and W (t, u) are neither autonomous nor periodic in t), the study of existence of homoclinic solutions for (HS) is quite different from that of the periodic systems, because of the lack of compactness of the Sobolev embedding. To verify the Palais–Smale compactness condition for the corresponding functional of (HS), the coercive assumption on L is often supposed. In particular, Rabinowitz [11] proved the existence of homoclinic solutions of (HS) by assuming that L(t) is positive definite for all t and satisfies the coercive condition l(t) := inf |x|=1 (L(t)x, x) → ∞ as |t| → ∞. Under the same coercive condition on L(t) as that in [11], Omana and Willem [12] introduced a compact embedding theorem and proved the existence of homoclinic solutions by using the usual mountain pass lemma. In [7], Ding considered the case where L(t) is unnecessarily global positive definite and employed an improved compact embedding theorem to prove that (HS) possesses homoclinic solutions. In recent years, some other types of coercive conditions on L have been utilized to obtain the corresponding compact embedding theorems, and then system (HS) has been investigated under all kinds of quadratic assumptions of potential W , such as superquadratic, subquadratic, asymptotically quadratic. See, for instance, [13,8,14–21] and the references therein. It is worthy of pointing out that Tang and Lin [22] employed the symmetric mountain pass theorem to investigate system (HS) under the coercive condition on L(t) mentioned above and the following assumptions: ˜ 1 ) W (t, u) = W1 (t, u) − W2 (t, u), W1 , W2 ∈ C 1 (R × RN , R), and there is R > 0 such that (W 1 |∇W (t, u)| = o(|u|) l(t)

as u → 0

uniformly in t ∈ (−∞, −R] ∪ [R, +∞), and W1 satisfies (AR) condition; ˜ 2 ) W2 (t, 0) ≡ 0 and there is a constant ϱ ∈ [2, µ) such that (W W2 (t, x) ≥ 0,

(∇W2 (t, x), x) ≤ ϱW2 (t, x),

∀ (t, x) ∈ (R × RN ).

Tang and Lin [22] obtained an unbounded sequence of homoclinic solutions for system (HS), which generalized some results of [11,12]. Our interests mainly concentrate on system (HS) with more general potentials compared with the above mentioned literature on the existence of ground state homoclinic solutions. We first consider the periodic case. We investigate the existence of ground state homoclinic solutions for system (HS) when L(t) is positive definite and W satisfies some monotone condition instead of the (AR) condition (Theorem 1.1). We also demonstrate the existence of ground-state homoclinic solutions of system (HS) when L(t) and W (t, u) are non-periodic in t and the potential W satisfies a growth restriction (Theorem 1.2). Furthermore, if W (t, u) is symmetric at u, infinitely many nontrivial homoclinic orbits can also be obtained via an abstract critical point theorem. The arguments are based on the method of the generalized Nehari manifold developed by Szulkin and Weth [6]. Moreover, based on the mountain-pass lemma developed by Ambrosetti and Rabinowitz [23] together with the concentration-compactness principle, we also prove the existence of infinitely many homoclinic solutions for a class of subquadratic Hamiltonian system. In the present paper we assume that L(t) is bounded in the sense that: 2

(L1 ) L ∈ C(R, RN ) is a symmetric and positive definite matrix for all t ∈ R and there are constants 0 < τ1 < τ2 such that τ1 |u|2 ≤ (L(t)u, u) ≤ τ2 |u|2

for all (t, u) ∈ R × RN .

Very few papers (see, for example, [24,14,25]) deal with the case where L is bounded from below and maybe not satisfy the coercive condition introduced before, or some more demanding coercive condition.

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We are now ready to formulate our main results. The first result concerns the existence of ground state homoclinic solutions to (HS) with the periodic potential. Theorem 1.1. In addition to assumptions (L1 ), (W0 )– (W1 ), and (W3 ), suppose that ′

(W2 ) s → s−1 (∇W (t, su), u) is strictly increasing of s > 0 for all u ̸= 0 and t ∈ R. If L(t) is T -periodic in t, then (HS) has at least one ground state homoclinic solution. ′

Remark 1.1. As we known, some strict monotone conditions like (W2 ) have been often used to prove the existence of ground state solutions for elliptic partial differential equation. See, for instance, [26]. Indeed, it turns ′ out that such a strict monotone condition as (W2 ) is an essential hypothesis in applying Nehari manifold method to find ground state solutions. However, as far as we know, the study of the existence of homoclinic ′ solutions of (HS) under assumption (W2 ) is much rare; see, for instance, [11,6]. Rabinowitz and Tanaka [11] firstly introduced this monotone condition to prove that a class of non-periodic second-order Hamiltonian systems possess at least one homoclinic orbit. In the present paper, in order to apply generalized Nehari ′ manifold method to investigate the ground state homoclinic orbits, it is natural to require hypothesis (W2 ). Remark 1.2. Under the same conditions of Theorem 1.1, Szulkin and Weth [6] employed the method of the generalized Nehari manifold to investigate a class of Newtonian system, i.e., (HS) with L(t) ≡ Id. In [6], however, an additional growth restriction on the potential W was used to guarantee the compactness of Palais–Smale sequence on Nehari manifold with the help of concentration-compactness principle. In our paper, arguments on Theorem 1.1 are very distinct from those in [6] and we do not need such a growth restriction. Yang [1] and Chen [5] considered the existence of ground state homoclinic solutions under ′ some growth restrictions on W (i.e., (W4 )–(W5 )) and a strong restriction assumption (W5 ), respectively. The following Examples 1.1 and 1.2 can be well covered by Theorem 1.1 but not by the results in [1,5]. Therefore, the results in [1,5] are extended and complemented. Example 1.1. Consider (HS) with 3 3 τ1 + τ2 Id and W (t, u) = e2|u| − e|u| . 2 It is easy to check that all conditions of Theorem 1.1 are satisfied. In addition, we have 3 3 3 3 1 1 (∇W (t, u), u) − W (t, u) = (6e2|u| |u|3 − 3e|u| |u|3 ) − (e2|u| − e|u| ) 2 2   3 3 |u|3 |u|3 3 |u|3 =e 3e |u| − |u| + 1 − e 2 > 0,

L(t) =

due to f (t) := 3tet − 32 t + 1 − et satisfying that f (0) = 0 and f (t) is strictly increasing for t > 0. Thus, Lemma 3.2 holds for W . However,   3 3 3 3   |∇W (t, u)| = 6e2|u| |u|u − 3e|u| |u|u = 3|u|2 (2e2|u| − e|u| ), which does not suffer from such a growth restriction as (W4 ). Therefore, W does not satisfy Theorem 1.2 in [1]. Example 1.2. Consider (HS) with L(t) =

τ1 + τ2 Id 2

3

and W (t, u) = g(t)(e|u| − 1),

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where g(t) > 1 is T -periodic in t. It is not hard to check that all conditions of Theorem 1.1 are satisfied. However,    1 3 3 |u|3 ˜ W (t, u) = (∇W (t, u), u) − W (t, u) = g(t) |u| − 1 e + 1 > 0 for all u ̸= 0, 2 2 and for any given d1 > 0 and α > 1, we have     |∇W (t, u)|α α α α|u|3 3 |u|3 ˜ − d W (t, u) ≥ lim g(t) 3 |u| e − d (3|u| − 1)e − d lim 1 1 1 |u|α |u|→∞ |u|→∞   3 3 3 = lim g(t)e|u| 3α |u|α e(α−1)|u| − d1 (3|u|3 − 1) − d1 e−|u| |u|→∞

= +∞, ′

which does not satisfy assumption (W5 ). Thus, function W contradicts with the conditions of Theorem 1.1 in [5]. Remark 1.3. There are also many periodic functions W (t, u) satisfying the conditions of Theorem 1.1, but not satisfying the (AR) condition; for example, W (t, u) = W (u) = u2 ln(|u|2 + 1) or |u|2 ln(1 + |u|). For non-periodic case, we assume that W satisfies ′

(W4 ) There exists 2 < p < +∞ such that |∇W (t, u)| = b(t)|u|p−1 , where b(t) > 0, b ∈ Lr (R, R) ∩ L∞ (R, R), 1r + ps = 1 with 2 < s < ∞. Moreover, W (t, u) ≥ 0 and W (t, 0) = 0. Next, let us state the following results about non-periodic second order Hamiltonian system. ′

Theorem 1.2. Under the assumptions (L1 ), and (W4 ), (HS) has at least one ground state homoclinic solution. Furthermore, if W (t, u) = W (t, −u), then (HS) has infinitely many homoclinic solutions. Remark 1.4. To our best knowledge, it seems that Theorem 1.2 is the first result on existence of ground state homoclinic solutions for the non-periodic second order Hamiltonian system and is also the first result on the application of generalized Nehari manifold developed by Szulkin and Weth to such a second-order non-periodic case. Let us state the following example to illustrate our Theorem 1.2. Example 1.3. Consider (HS) with L(t) = K1 (t)Id

and W (t, u) =

1 b(t)|u|p , p ∈ (2, +∞), p

where K1 ∈ C 1 (R, R+ ), b ∈ Lr (R, R+ ) are non-periodic and τ1 ≤ K1 (t) ≤ τ2 . It is not hard to check that L and W satisfy all conditions of Theorem 1.2. Here, L(t) is in contrast to the coercive condition of matrix L in Theorems 1.3 and 1.4 of [22]. Therefore, Theorem 1.2 generalizes the relevant results of [22]. At last, let us state a new sufficient criterion for the subquadratic case. Theorem 1.3. Suppose that W (t, u) =

1 p+1 ,0 p+1 a(t)|u|

< p < 1, L satisfies (L1 ) and

(L2 ) There exists a positive defined symmetric matrix A such that |L(t) − A| → 0 in terms of matrix norm as |t| → ∞,

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and that a(t) satisfies the following assumption: (A0 ) a ∈ L∞ (R, R), lim|t|→+∞ a(t) = a∞ < 0 and there exist y0 ∈ R, R0 > 0 such that a(t) > 0,

for all t ∈ (y0 − R0 , y0 + R0 ).

Then (HS) possesses a sequence of nontrivial homoclinic solutions converging to zero. Compared with the literature available for W (t, u) being superquadratic as |u| → +∞, the study of the existence of homoclinic solutions of (HS) under the assumption that W (t, u) is subquadratic at infinity is much more recent and the number of such references is considerably small, for instance, [7,15,27,24,28]. Zhang and Yuan [28] studied the existence of homoclinic solutions for (HS) with W (t, u) = a(t)|u|γ and 2 a ∈ L2 (R, R+ ) ∩ L 2−γ (R, R+ ) (integrable condition), 1 < γ < 2, and obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument. Subsequently, Sun, Chen and Nieto [27] employed the variant fountain theorem to improve the above results and obtained infinitely many ho2 moclinic solutions for (HS) with W (t, u) = a(t)|u|γ and a ∈ L 2−γ (R, R+ ). Notice that the coercive condition on L(t) is introduced in [28] and [27]. Moreover, in Tang and Lin [24], W satisfies the following conditions ¯ 1 ) W ∈ C 1 (R × RN , R) and there exist two constants 1 < γ1 < γ2 < 2 and two functions a1 , a2 ∈ (W 2 L 2−γ1 (R, R+ ) such that |W (t, u)| ≤ a1 (t)|u|γ1 ,

∀(t, u) ∈ R × RN , |t| ≤ 1,

|W (t, u)| ≤ a2 (t)|u|γ2 ,

∀(t, u) ∈ R × RN , |t| ≥ 1;

¯ 2 ) there exist two functions b ∈ L2/(2−γ1 ) (R, R+ ) and ϕ ∈ C(R+ , R+ ) such that (W |∇W (t, u)| ≤ b(t)ϕ(|u|),

∀(t, u) ∈ R × RN ,

where ϕ(s) = O(sγ1 −1 ) as s → 0+ ; ¯ (W3 ) there exist an open set Ω ∈ R and two constants γ3 ∈ (1, 2) and η > 0 such that W (t, u) ≥ η|u|γ3 ,

∀(t, u) ∈ Ω × RN , |t| ≤ 1

and W (t, u) is an even function at u. Tang and Lin [24] showed that (HS) possesses infinitely many homoclinic solutions. It easy to check that our Theorem 1.3 is novel based on the above mentioned literature. Indeed, comparing the results in [28,27] with our Theorem 1.3, we do not need to require that the positive define matrix function L(t) is coercive. Besides, in [28,27], a(t) is positive and satisfies integrable condition 2 L 2−γ (R, R+ ). However, a(t) is a sign-changing function and a(t) ∈ L∞ (R, R) in Theorem 1.3. Thus, our ¯ 3 ) is an result improves and generalizes those relevant results of [28,27]. Moreover, it turns out that (W essential condition in applying the genus properties to find infinitely many homoclinic solutions of (HS) ¯ 3 ). Of course, in [24]. But, in present paper, (A0 ), as a necessary condition, is totally different from (W ¯ ¯ there also exist a lot of functions satisfying (A0 ) instead of (W1 )–(W2 ). Thus, Theorem 1.3 generalizes and improves the above mentioned results on subquadratic case. Example 1.4. Consider (HS) with L(t) =

τ1 + τ2 Id = A, 2

R0 =

1 , 2

y0 = 0

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and a(t) =

   1,   −1,

  1 1 , t∈ − ,  2 2    1 1 t ∈ −∞, − ∪ ,∞ . 2 2 2

It is easy to verify all conditions of Theorem 1.1. However, a(t) ̸∈ L 2−γ (R, R+ ) and L(t) is not coercive at infinity. Therefore, a(t) does not satisfy the associated conditions of [27,28]. Moreover, it is easy to ¯ 1 ) and (W ¯ 2 ). Note that, in [25], the authors considered the see that a(t) contradicts with assumptions (W ϑ−1 subquadratic case where |∇W (t, u)| ≤ b(t)|u| and b(t) ∈ Lε (R, R+ ), ε ∈ [1, 2], which contradicts with the definition of a(t). Thus, in some sense, Theorem 1.3 generalizes results in [24,27,28,25]. Throughout this paper, C > 0 denotes a universal positive constant. The remainder of this paper is organized as follows. We will give some notations and preliminaries in Section 2. We only consider the superquadratic cases and the proof of Theorems 1.1 and 1.2 are presented in Section 3, and Section 4 is devoted to the result (Theorem 1.3) about subquadratic case. 2. Preliminaries Hereafter, let us fix some notations and preliminaries which are useful later. Take    1 N 2 E := u ∈ H (R, R ) : [|u| ˙ + (L(t)u(t), u(t))]dt < +∞ . R

Then E is a Hilbert space with the norm given by   12 1 2 2 ∥u∥ = ⟨u, u⟩ = [|u| ˙ + (L(t)u(t), u(t))]dt . R q

N

Note that E ⊂ L (R, R ) for all q ∈ [2, +∞] with the embedding being continuous, then there exists a positive constant Cq such that ∥u∥q ≤ Cq ∥u∥,

∀u ∈ E,

(2.1)

where Lq (R, RN ) denotes the Banach space of functions on R with values in RN under the norm   q1 q ∥u∥q := |u(t)| dt . R

In order to properly obtain our main results, we define the energy functional associated with (HS):   1 2 I : E → R, I(u) := (|u| ˙ + (L(t)u(t), u(t)))dt − W (t, u)dt. 2 R R We can easily check that I ∈ C 1 (E, R) and  ⟨I ′ (u), v⟩ = [(u(t), ˙ v(t)) ˙ + (L(t)u(t), v(t)) − (∇W (t, u), v(t))]dt

(2.2)

(2.3)

R

for any u, v ∈ E. 3. Superquadratic case In this section, we only consider the superquadratic cases. Using similar arguments to [1], we have the following mountain pass geometry of functional I.

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Lemma 3.1. Assume that the conditions of Theorems 1.1 and 1.2 hold, then (1) there exist positive constants β, α such that I(u) ≥ β for all u satisfying ∥u∥ = α; (2) there exists e ∈ E with ∥e∥ > α such that I(e) < 0. From Lemma 3.1 and the Ambrosetti–Rabinowitz Mountain Pass Theorem without (PS)c condition (see [29,30]), it follows that there exists a (PS)c sequence {un } ⊂ E such that I(un ) → c = inf max I(γ(t)) > β γ∈Γ 0≤t≤1

and I ′ (un ) → 0,

where Γ = {γ ∈ C(E, RN ) : I(γ(0)) = 0, I(γ(1)) < 0}. ′ With the help of hypotheses (W1 ) and (W2 ), we state the following lemma. Lemma 3.2. H(t, su) := (∇W (t, su), su) − 2W (t, su) is strictly increasing in s > 0 for all t ∈ R and u ̸= 0. Moreover, H(t, u) > 0. Proof. Let 0 < s1 < s2 , then we have   1 H(t, s2 u) − H(t, s1 u) = 2 ((∇W (t, s2 u), s2 u) − (∇W (t, s1 u), s1 u)) − (W (t, s2 u) − W (t, s1 u)) 2  s2   s1  s2 (∇W (t, s2 u), u) (∇W (t, s1 u), u) (∇W (t, ξu), u) =2 ξdξ ξdξ − ξdξ − s2 s1 ξ 0 0 s1   s2  (∇W (t, s2 u), u) (∇W (t, ξu), u) =2 ξdξ − s2 ξ s1   s1  (∇W (t, s2 u), u) (∇W (t, s1 u), u) +2 − ξdξ > 0, s2 s1 0 which, together with the fact that H(t, 0) = 0, yields the conclusion.



3.1. Behaviors of Nehari manifold Since we shall use the Nehari manifold methods to find critical points of I, we define Nehari manifold corresponding to I as N = {u ∈ E \ {0} : ⟨I ′ (u), u⟩ = 0}. Thus for u ∈ N , one sees that ∥u∥2 =

 (∇W (t, u), u)dt. R

Lemma 3.3. Under the assumptions of Theorem 1.1, we have (i) for each u ∈ S, there exists a unique su > 0 such that m(u) := su u ∈ N . Moreover, m(u) is the unique maximum point of I on E, where S = {u ∈ E : ∥u∥ = 1}; (ii) N is closed in E and bounded away from 0, I is bounded below away from 0 on N ; (iii) there exists δ > 0 such that su ≥ δ for all u ∈ S, and for each compact subset W ⊂ S there exists a constant CW such that su ≤ CW for all w ∈ W; (iv) if un → u in S, then we have m(un ) → m(u) as n → ∞; (v) N is a regular manifold homeomorphism to the sphere of E.

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Proof. (i) For each u ∈ S and s > 0, define h(s) = I(su). It is easy to see that h(0) = 0 and h(s) < 0 for s > 0 large. For otherwise, we may assume that there exists sequence {sn } ⊂ R+ , sn → +∞ such that I(sn u) > 0 for all n. Then, we have  I(sn u) 1 W (t, sn u) 2 = − u dt > 0. (3.1) s2n 2 |sn u|2 R Since ∥u∥ = 1 and u ∈ E, we have |sn u(t)| → +∞ for t ∈ supp u, which contains a positive measure. By (W3 ) and the Fatou’s lemma, we have  W (t, sn u) 2 u dt → +∞ as n → ∞, |sn u|2 R which contradicts with (3.1). Moreover, we claim that h(s) > 0 for s > 0 small. Indeed, from conditions (W1 ), we deduce that for each ϵ > 0, there exists Cϵ > 0 such that |W (t, u)| ≤ ϵ|u|2 for all |u| ≤ Cϵ . Take s ∈ (0, CC∞ϵ ), then by the Sobolev inequality, we have |su| ≤ ∥su∥∞ ≤ C∞ ∥su∥ ≤ Cϵ . So, for sufficiently small ϵ,  1 I(u) = ∥su∥2 − W (t, su)dt 2 R   1 = s2 − ϵC22 > 0. (3.2) 2 Therefore, the maximum of h(s) on (0, +∞) is attained at s = su > 0 and hence h′ (su ) = 0, su u ∈ N . Assume that there exist s′u > su > 0 such that s′u u, su u ∈ N , then we have  2 2 (su ) ∥u∥ − (∇W (t, su u), su u)dt = 0 R

and (s′u )2 ∥u∥2

 −

(∇W (t, s′u u), s′u u)dt = 0.

R

It follows that   R

 ∇W (t, s′u u) ∇W (t, su u) − , u dt = 0, s′u su

′

which contradicts with (W2 ) and s′u > su > 0. So the conclusion (i) holds. (ii) First we show that N is bounded away from 0. Assume by contradiction that for {un } ⊂ N , we have un → 0 in E. Set vn = ∥uunn ∥ , then there exists sn = ∥un ∥ > 0 such that sn vn ∈ N . It is easy to see that sn → 0 and from the definition of N , we have    ∇W (t, sn vn ) 1= , vn dt. (3.3) sn R Moreover, (2.1) implies that ∥vn ∥∞ ≤ C∞ . Therefore, ∥sn vn ∥∞ → 0 as n → ∞. This, together with assumption (W1 ) and (2.1), yields a contradiction with (3.3). So, there exists some C > 0, such that ∥u∥ ≥ C for all u ∈ N . Now, we prove the set N is closed in E. Let {un } ⊂ N be such that un → u in E. In the following, we show that u ∈ N . In view of the definition of I and the following fact ⟨I ′ (un ), un ⟩ − ⟨I ′ (u), u⟩ = ⟨I ′ (un ), un − u⟩ + ⟨I ′ (un ) − I ′ (u), u⟩ = o(1), we have ⟨I ′ (u), u⟩ = 0. Furthermore, it is easy to see that ∥u∥ = limn→∞ ∥un ∥ ≥ c > 0. So u ∈ N . It remains to prove that there exists C > 0 such that I(u) ≥ C for all u ∈ N . For each u ∈ N , it follows from (2.2)

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and Lemma 3.2 that 1 1 I(u) = I(u) − ⟨I ′ (u), u⟩ = 2 2

 ((∇W (t, u), u) − 2W (t, u))dt > 0.

(3.4)

R

Define c∗ = inf u∈N I(u). Then it follows from conclusion (i) and the above facts that c∗ = inf I(u) = u∈N

inf

max I(su) = inf max I(su).

u∈E\{0} s>0

u∈S s>0

We claim that c∗ = c = inf max I(γ(t)),

(3.5)

γ∈Γ 0≤t≤1

where c has been defined. Our arguments are similar to those in [26]. Notice that for any u ∈ S, there exists some s0 > 0 large, so that I(s0 u) < 0. Define a path γ : [0, 1] → E by γ(t) = s0 tu. Clearly, γ ∈ Γ and consequently, c ≤ inf u∈S maxs>0 I(su) = c∗ . In what follows we prove c∗ ≤ c. From the conclusion (i), we know that the manifold N separates E into two components. We see that I(u) ≥ 0 for all u in one component, because for all 0 ≤ s ≤ su , by (W3 ) we have      ∇W (t, su) ′ ′ 2 , u dt h (s) = ⟨I (su), u⟩ = s ∥u∥ − s R s ≥ 2 ⟨I ′ (su u), su u⟩ = 0. su Furthermore, by (3.2), this component contains the origin and also contains a small ball around the origin. Thus, every path γ ∈ Γ has to cross N and c∗ ≤ c. Therefore, our claim is true. Therefore, c∗ ≥ β, where β has been given in Lemma 3.1. (iii) Assume on the contrary that there exists {un } ⊂ S such that sun un ∈ N due to conclusion (i), where sun → 0. From the conclusion (ii) we deduce that ∥sun un ∥ ≥ C > 0 for some positive constant C, which is impossible. Therefore, there exists δ > 0 such that su ≥ δ for all u ∈ S. In order to prove su ≤ CW for all u ∈ W ⊂ S, we argue by contradiction. Suppose that there exists {un } ⊂ W ⊂ S such that sn = sun → ∞ as n → ∞. Since W is compact, there exists u ∈ W such that un → u in E and un (t) → u(t) a.e. on R after passing to a subsequence. Therefore, it follows from (2.2) that I(sn un ) = I(sn u) = o(1). Using the same arguments as the proof of conclusion (i), we have that I(sn un ) → −∞ as n → ∞. However, from the conclusion (ii) we deduce that I(sn un ) > 0. This is a contradiction. (iv) Assume that {un } ⊂ S and un → u in E, there exists a unique sn > 0 such that m(un ) = sn un due to the conclusion (i). Using similar arguments as the proof of the conclusion (iii), we can prove that I(sn un ) → −∞ as sn → ∞. However, I(sn un ) > 0. Therefore, {sn } is bounded. Assume that sn → s∗ . Then by the fact that N is closed, we have m(un ) → s∗ u, s∗ u ∈ N . Moreover, by conclusion (i) we know that m(s∗ u) = m(u) = su u and then s∗ = su . So m(un ) = m(u) + o(1). (v) By the conclusions (i)–(iii), it follows from Proposition 3.1 in [6] that the mapping m is a homeomorphism between S and N , and the inverse of m is given by u m(u) ˇ = m−1 (u) = for u ∈ N . ∥u∥ Thus N is a regular manifold homeomorphic to the sphere of E.



Remark 3.1. Similar to the above arguments, it is easy to obtain the same conclusions as Lemma 3.3 under the assumptions of Theorem 1.2. As in [6], we have the following lemma.

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Lemma 3.4. Assume that the conditions of Theorems 1.1 and 1.2 hold. Then we have (1) φ ∈ C 1 (S, R) and ⟨φ′ (w), z⟩ = ∥m(w)∥⟨I ′ (m(w)), z⟩ for all z ∈ Tw (S) = {u ∈ E : ⟨w, u⟩ = 0}; (2) if {wn } is a Palais–Smale sequence for φ, then {m(wn )} is a Palais–Smale sequence for I. If {wn } ⊂ N is a bounded Palais–Smale sequence for I, then {m−1 (wn )} is a Palais–Smale sequence for φ; (3) inf S φ = inf N I. Moreover, w is a critical point of φ if and only if m(w) is a nontrivial critical point of I and the corresponding critical values coincide. 3.2. Proof of Theorem 1.1 In this subsection, we mainly give the proofs of Theorem 1.1 on superquadratic case. The Proof of Theorem 1.1. From the conditions of Theorem 1.1, we have proved that Lemmas 3.3 and 3.4  hold. However, Φ ′ is not compact, where Φ : E → R is defined as Φ(u) = R W (t, u)dt. Moreover, I does not satisfy the Palais–Smale condition on N . Indeed, if I ′ (u) = 0 and un := u(t − yn T ), where yn ∈ Z, then {un } is a Palais–Smale sequence which converges weakly but not strongly to 0 if |yn | → ∞. In view of Lemmas 3.3 and 3.4, we see that φ is bounded below on S. Let {wn } ⊂ S be a minimizing sequence for φ. By the Ekeland’s variational principle [31] we may assume that φ′ (wn ) → 0. Hence, it follows from Lemma 3.4 that I ′ (un ) → 0 and I(un ) → c∗ as n → ∞, where un := m(wn ). We shall show that {un } is bounded. Suppose that ∥un ∥ → ∞, then vn := ∥uunn ∥ ⇀ v in E and vn → v a.e. in R after passing to a subsequence. For any n ∈ N, there exists kn ∈ Z such that ∥vn (· + kn T )∥∞ = maxt∈R |vn (t)| occurs in [0, T ]. Let v¯n := vn (· + kn T ). Since {¯ vn } is also bounded in E, passing to a subsequence, we may assume that v¯n → v¯ in E, v¯n ⇀ v¯ a.e. in R and v¯n → v¯ in L∞ loc (R). It is easy to see that I is invariant under translation of the form v → v(· − yT ), y ∈ Z. Therefore, I(vn ) = I(¯ vn ) and ∥vn ∥ = ∥¯ vn ∥ = 1. Case 1. v¯ ≡ 0. Note that ∥vn ∥∞ = ∥¯ vn ∥∞ = ∥¯ vn ∥L∞ [0,T ] → 0 as n → ∞. Then by (W1 ), for any given ϵ > 0, we have, for n large enough, |W (t, v¯n )| ≤ ϵ|¯ vn |2 for all t ∈ R. Therefore, for n large enough, we have  |W (t, v¯n )|dt ≤ ϵ∥¯ vn ∥22 . R

Since ϵ is arbitrary and v¯n is bounded in E, by (2.1) we have  |W (t, v¯n )|dt → 0 as n → ∞. R

It follows from Lemma 3.3 that, for any given K > 0, c∗ + o(1) = I(un ) ≥ I(Kvn ) = I(K v¯n ) =

K2 − 2

 W (t, K v¯n )dt → R

K2 , 2

which is a contradiction with K large. Case 2. v¯ ̸≡ 0. Without loss of generality, we assume that v¯ is larger than some positive constant C on a nonempty bounded positive measure set Π . Therefore, from v¯n → v¯ in L∞ loc (R) we deduce that |un (t + kn T )| → +∞ as n → ∞ for all t ∈ Π .

(3.6)

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Observe that un (· + kn T ) = v¯n ∥un ∥. By the invariance of ZT , we have   W (t, un )dt = W (t + kn T, un (t + kn T ))dt. R

R

It follows from Fatou’s lemma, (2.2), (3.6) and (W3 ) that  1 I(un ) W (t, un (t)) 2 = − vn dt 0≤ ∥un ∥2 2 u2n (t) R 1 W (t + kn T, un (t + kn T )) = − vn (t + kn T )2 dt 2 u2n (t + kn T ) R  1 W (t + kn T, un (t + kn T )) 2 = − v¯n dt → −∞. 2 u2n (t + kn T ) Π This is a contradiction. Therefore, {un } is a bounded sequence in E. Let u ¯n = un (· + kn T ), where kn ∈ Z is such that ∥un (·+kn T )∥∞ = maxt∈R |¯ un (t)| occurs in [0, T ]. Then, by the invariance of I we have ∥un ∥ = ∥¯ un ∥ and I(un ) = I(¯ un ) = c∗ + o(1),

I ′ (un ) = I ′ (¯ un ) = o(1).

(3.7)

Assume that u ¯n ⇀ u ¯ in E and u ¯n ⇀ u ¯ a.e. in R after passing to a subsequence. We show that u ¯ ̸≡ 0. Otherwise, we have ∥¯ un ∥∞ = ∥¯ un ∥L∞ [0,T ] → 0 as n → ∞. Therefore, by (W1 ), for any ϵ > 0, we have for n large enough |W (t, u ¯n )| ≤ ϵC|¯ un |2 , |(∇W (t, u ¯n ), u ¯n )| ≤ ϵ|¯ un |2 for all t ∈ R. Therefore, by (2.2) and (2.3) one has    1 1 ′ ∗ un )¯ un = (∇W (t, u ¯n ), u ¯n ) − W (t, u ¯n ) dt ≤ ϵC∥¯ un ∥22 , c + o(1) = I(¯ un ) − I (¯ 2 R 2 which contradicts with c∗ > 0 since ϵ is arbitrary and {¯ un } is bounded in E. Therefore, u ¯ ̸≡ 0. It follows form (3.7) that u ¯ is a nontrivial critical point of I and in particular, u ¯ ∈ N . We still need to show that I(¯ u) = c∗ = inf N I. Since we may assume, passing to a subsequence, that u ¯n → u ¯ a.e. in R. Fatou’s lemma implies that 1 c∗ = I(¯ un ) − ⟨I ′ (¯ un ), u ¯n ⟩ + o(1) 2    1 = (∇W (t, u ¯n ), u ¯n ) − W (t, u ¯n ) dt + o(1) R 2    1 ≥ (∇W (t, u ¯), u ¯) − W (t, u ¯) dt + o(1) R 2 1 = I(¯ u) − ⟨I ′ (¯ u), u ¯⟩ = I(¯ u). 2 Hence I(¯ u) = c∗ due to the fact that u ¯ is a nontrivial critical point of I. The proof is complete.  3.3. Proof of Theorem 1.2 Now we state the following two classical critical point theorems which can be found in [29,30] and play an important role in our arguments. Theorem 3.1. Let E be a Banach space such that the unit sphere S in E is a submanifold of (at least) C 1 class and let φ ∈ C 1 (S; R). If φ is bounded below and satisfies the Palais–Smale condition, then c∗ := inf S φ is attained and is a critical value of φ.

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Theorem 3.2. If E is infinite-dimensional, φ ∈ C 1 (S; R) is even and bounded below and satisfies the Palais–Smale condition, then φ has infinitely many pairs of critical points. The following lemma is used to ensure that the Palais–Smale condition holds for the functional I. Lemma 3.5. The following hold under the assumptions of Theorem 1.2: (a) If {un } ⊂ N is a sequence such that supn∈N I(un ) < ∞, then, passing to a subsequence, we have un ⇀ u ̸= 0 as n → ∞, and there is su > 0 such that su u ∈ N and I(su u) ≤ lim inf n→∞ I(un ); (b) I satisfies the Palais–Smale condition on N . Proof. (a) Let {un } ⊂ N be a sequence such that I(un ) ≤ C for all n ∈ N and some positive constant C. Note that 1 p−2 I(un ) − I ′ (un )un = ∥un ∥2 . p 2p Thus {un } is a bounded sequence in E. Here we assume that the operator Φ ′ is compact and Φ is weakly continuous. Let un ⇀ u in E after passing to a subsequence. If u = 0, then 0 = I ′ (un )un = ∥un ∥2 − ⟨Φ ′ (un ), un ⟩ = ∥un ∥2 + o(1), which contradicts with the conclusion (ii) in Lemma 3.3. Hence, u ̸= 0. It is easy to check that there exists s′u > 0 such that s′u u ∈ N . Moreover, as Φ(s′u un ) → Φ(s′u u) and un ∈ N , we have I(s′u u) ≤ lim inf I(s′u un ) ≤ lim inf I(un ). n→∞

n→∞

(b) Let {un } ⊂ N be a Palais–Smale sequence. By conclusion (a), un ⇀ u in E after passing to a subsequence. Moreover, I ′ (u) = 0

and

Φ ′ (un ) → Φ ′ (u)

as n → ∞.

Furthermore, by the definition of I we have o(1) = ⟨(I ′ (un ) − I ′ (u0 )), (un − u0 )⟩ = ∥un − u0 ∥2 − ⟨Φ ′ (un ) − Φ ′ (u0 ), un − u0 ⟩ = ∥un − u0 ∥2 . So I satisfies the Palais–Smale condition on N . Now we give the following fact. Claim. The operator Φ ′ is compact and Φ is weakly continuous. Indeed, assume that un is bounded in E, ′ then after extracting a subsequence we have un ⇀ u in E. For any choice of ϵ > 0, by (W4 ) we know that there exists R > 0 such that   r1 |b(t)|r dt

≤ ϵ.

(3.8)

|t|≥R

Therefore, it follows from (3.8) and the H¨ older inequality that   1 |W (t, un ) − W (t, u)|dt ≤ b(t)(|un |p + |u|p )dt p |t|≥R |t|≥R    r1   r−1   r−1 r r   rp rp 1 ≤ b(t)r dt |un | r−1 dt + |u| r−1 dt  |t|≥R  p |t|≥R |t|≥R    ps   ps   ϵ s s ≤ |un | dt + |u| dt  p  |t|≥R |t|≥R ≤ ϵC.

(3.9)

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′

On the other hand, using the mean value theorem, the H¨older inequality and (W4 ), it is not hard to check that  |W (t, un ) − W (t, u)|dt |t|≤R  = |(∇W (t, u + λ(t)(un − u)), un − u)|dt |t|≤R  ≤C b(t)(|un |p−1 + |u|p−1 )|un − u|dt |t|≤R

≤C

≤C

    |t|≤R    |t|≤R



|un |

r(p−1) r−1

 r−1 r |un − u|

r r−1

dt

 |u|

+

r(p−1) r−1

|un − u|

  r−1 r  dt 

r r−1

|t|≤R

 p−1  s s |un | dt

 1s |un − u|s dt

|t|≤R

 + |t|≤R

 p−1  s s |u| dt

 1s   |un − u|s dt  |t|≤R

≤ ϵC

(3.10)

for large n, where λ(t) ∈ (0, 1) and ϵ has been given in (3.8). Combining (3.9) and (3.10) we obtain Φ is weakly continuous. Now we are going to show the operator Φ ′ is compact. In fact, by the definition of Φ we have     ′ ′  (3.11) ∥Φ (un ) − Φ (u)∥E ∗ = sup  (∇W (t, un ) − ∇W (t, u), v)dt , ∥v∥=1

R

∗

where E is the dual space of E. For any fixed ϵ > 0, there exists R > 0 such that (3.8) holds. Therefore, it follows from the H¨ older inequality that       (∇W (t, un ) − ∇W (t, u), v)dt   |t|≥R   ≤ b(t)(|un |p−1 + |u|p−1 )|v|dt |t|≥R

  r1   r−1   r−1 r r   r(p−1) r(p−1) r r ≤ b(t)r dt |un | r−1 |v| r−1 dt + |u| r−1 |v| r−1 dt  |t|≥R  |t|≥R |t|≥R   p−1   1s   p−1   1s  s s    ≤ϵ |un |s dt |v|s dt + |u|s dt |v|s dt  |t|≥R  |t|≥R |t|≥R |t|≥R 

≤ ϵC. On the other hand, we can use the Lebesgue’s dominated convergence theorem to get       (∇W (t, un ) − ∇W (t, u), v)dx ≤ ϵ   |t|≤R 

(3.12)

(3.13)

for large n due to un → u in Lkloc (R, RN ) for k ∈ [1, ∞). Combining (3.11), (3.12) with (3.13), we have ∥Φ ′ (un ) − Φ ′ (u)∥E ∗ → 0 as n → ∞. The proof is complete.  The proof of Theorem 1.2. From Lemma 3.5 we know that I satisfies the Palais–Smale condition on N . Let {wn } ⊂ S be a Palais–Smale sequence for φ and take un = m(wn ) ∈ N . It follows from Lemma 3.4 that {un } is a Palais–Smale sequence for I. Hence we have un → u after passing to a subsequence and wn → m−1 (u). It follows that φ satisfies the Palais–Smale condition. Let {wn } be a minimizing sequence for φ. By the

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130

Ekeland’s variational principle [31] we assume that φ′ (wn ) → 0, and by the Palais–Smale condition, wn → w after passing to a subsequence. By Theorem 3.1 and Lemma 3.5, we obtain that w is a minimizer for φ and u is a ground state solution for (HS) satisfying I(u) = c∗ . Moreover, if W (t, −u) = W (t, u) uniformly for t ∈ R, then functional I is even. And so is φ. Since inf S φ > 0, φ is bounded from below. Hence, it follows from Theorem 3.2 that (HS) has infinitely many homoclinic solutions.  4. Subquadratic case In this section, we consider a class of subquadratic Hamiltonian system (HS) with W (t, u) = < p < 1, where a(t) is bounded in R. Let X = H 1 (R, RN ) ∩ Lp+1 (R, RN ) be the reflexive Banach space endowed with the norm 1 p+1 ,0 p+1 a(t)|u|

∥u∥X = ∥u∥ + ∥u∥p+1 . It is natural to check that I are well defined and of class C 1 on X. Unlike Lemma 3.1 in the superquadratic case, it is difficult to ensure the geometric conditions required by the mountain pass theorem [23]. We use a geometrical construction of subset to overcome this difficulty. Let us give a basic definition and recall the mountain pass theorem of Ambrosetti and Rabinowitz [23]. Definition 4.1. Let E be a Banach space and a subset A of E is said to be symmetric if u ∈ A implies −u ∈ A. For a closed symmetric set A which does not contain the origin, we define the genus γ(A) of A by the smallest integer k such that there exists an odd continuous mapping from A to Rk \ {0}. If there does not exist such a k, we define γ(A) = +∞. We set γ(∅) = 0. Let Γk denote the family of closed symmetric subsets A of E such that 0 ̸∈ A and γ(A) ≥ k. Theorem 4.1 (See [23]). Let I be an even C 1 functional on E and satisfy the Palais–Smale condition and (1) I(0) = 0; (2) For each k ∈ N, there exists Ak ∈ Σk such that supu∈Ak I(u) < 0. Define bk by bk = inf sup I(u). A∈Σk u∈A

Then each bk is a critical value of I, bk ≤ bk+1 < 0 for k ∈ N and {bk } converges to zero. Moreover, if bk = bk+1 = · · · = bk+q = b; then γ(Kb ) ≥ q + 1, where q ∈ N. The critical set Kb is defined by Kb = {u ∈ E; I ′ (u) = 0, I(u) = b}. The following lemma plays an essential role in demonstrating that the corresponding functional satisfies the Palais–Smale condition in our latter consideration. It is a special case of a result due to Lions [32]. Lemma 4.1. Let {un } be a bounded sequence in E. If there is r > 0 such that  y+r lim sup |un |2 dt = 0, n→∞ y∈R

s

y−r

N

then un → 0 in L (R, R ) for all s ∈ (2, +∞). Lemma 4.2. Under the conditions of Theorem 1.3, each Palais–Smale sequence of I is bounded in X.

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Proof. Let {un } ⊂ X be a Palais–Smale sequence of I. There exists C > 0 such that I(un ) ≤ C. Applying the H¨ older inequality and condition (A0 ), we have  1 1 2 a(t)|u|p+1 dt C ≥ I(un ) = ∥u∥ − 2 p+1 R   1 1 1 ≥ ∥u∥2 − a+ (t)|u|p+1 dt + a− (t)|u|p+1 dt 2 p+1 R p+1 R  1 1 ≥ ∥u∥2 − a+ (t)|u|p+1 dt, (4.1) 2 p+1 R where a+ (t) = max{0, a(t)} and a− (t) = a+ (t) − a(t). It follows from (A0 ) that there exists R > 0 such that a∞ <0 2 Combining (4.1) and (4.2), we have

for all |t| ≥ R and a+ ∈ Ls (R, RN ), 1 ≤ s ≤ +∞.

− ∥a∥∞ ≤ a(t) ≤

1 C ≥ I(un ) ≥ 2

 R

(4.2)

1+p 2 ∥un ∥ (|u| ˙ 2 + τ1 |u|2 )dt − ∥a+ ∥ 1−p , 2

(4.3)

which yields that ∥un ∥2 ≤ C for some positive number C due to 1 + p < 2. Therefore, it follows that there exists some positive constant C such that ∥un ∥ ≤ C

∀n ∈ N.

(4.4)

On the other hand, there exist constants d, d1 > 0 such that 1 d + d1 ∥un ∥X ≥ I(un ) − ⟨I ′ (un ), un ⟩ 2   1 1 − a(t)|un |p+1 dt = 2 p+1 R     1 1 1 1 = − a− (t)|un |p+1 dt − − a+ (t)|un |p+1 dt p+1 2 p+1 2 R R     1 1 1 1 − p+1 = − [a (t) + χ[−R,R] (t)]|un | dt − − [a+ (t) + χ[−R,R] (t)]|un |p+1 dt p+1 2 p+1 2 R R      −a  1 1 1 1 ∞ p+1 p+1 2 ∥un ∥ |un | dt − − min ,1 − ∥a+ + χ[−R,R] ∥ 1−p , ≥ 2 p+1 2 2 p+1 2 R

where R has been given in (4.2) and χ[−R,R] (t) = 1 if t ∈ [−R, R]; otherwise, χ[−R,R] (t) = 0. Therefore, there exists a constant C > 0 such that p+1 + 2 ∥un ∥ ∥un ∥p+1 + d). p+1 ≤ C(∥un ∥ + ∥un ∥p+1 + ∥a + χ[−R,R] ∥ 1−p 2

Relation (4.4) yields that ∥un ∥p+1 p+1 ≤ C(1 + ∥un ∥p+1 )

for all n ∈ N.

From (4.4) and (4.5) we infer that {un } is a bounded sequence in X. The proof is complete.

(4.5) 

We need the following lemma (see [33]) to prove that on X the functional I satisfies the Palais–Smale condition. Lemma 4.3. There exists a constant C > 0 such that for all real numbers x, y, ||x + y|p+1 − |x|p+1 − |y|p+1 | ≤ C|x|p |y|. Lemma 4.4. Under the conditions of Theorem 1.3, I satisfies the Palais–Smale condition on X.

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Proof. Let {un } ⊂ X be a Palais–Smale sequence of I. In view of Lemma 4.2, {un } is bounded in X. Then there exists a subsequence un ⇀ u in X, un → u in Lsloc (R, RN ) for all s ∈ [1, +∞] and un → u a.e. in R. For any given ϕ ∈ C0∞ (R, RN ), it follows from the weak convergence of {un } to u that   [(u˙n , ϕ) ˙ + ⟨(L(t)un , ϕ)⟩]dt → [(u, ˙ ϕ) ˙ + (L(t)u, ϕ)]dt. (4.6) R

R

By the compactness Sobolev embedding, un → u in Lp+1 (supp(ϕ), RN ), there exists a function h ∈ Lp+1 (supp(ϕ), RN ) such that a(t)|un |p−1 un ϕ → a(t)|u|p−1 uϕ a.e. in R and |a(t)||un |p ϕ ≤ ∥a∥∞ |h|p |ϕ| in R. Using the Lebesgue dominated convergence theorem, we have   p−1 a(t)|un | un ϕdt → a(t)|u|p−1 uϕdt. R

(4.7)

R

By combining relations (4.6) and (4.7), we obtain 0 = lim ⟨I ′ (un ), ϕ⟩ = ⟨I ′ (u), ϕ⟩, n→∞

∀ϕ ∈ C0∞ (R, RN ).

Hence I ′ (u) = 0. Since un ⇀ u in X, we have ∥u∥X ≤ lim inf n→∞ ∥un ∥X = limn→∞ ∥un ∥X after extracting a subsequence. We distinguish two cases: (i) ∥u∥X = limn→∞ ∥un ∥X ; (ii) ∥u∥X < lim inf n→∞ ∥un ∥X . If Case (i) occurs, then lim sup ∥un ∥X = lim sup(∥un ∥ + ∥un ∥p+1 ) = ∥u∥ + ∥u∥p+1 , n→∞

n→∞

which implies lim sup ∥un ∥p+1 ≤ ∥u∥ + ∥u∥p+1 − lim inf ∥un ∥. n→∞

n→∞

(4.8)

Note that ∥u∥ ≤ lim inf ∥un ∥, n→∞

∥u∥p+1 ≤ lim inf ∥un ∥p+1 . n→∞

It follows from (4.8) that ∥u∥p+1 ≤ lim inf ∥un ∥p+1 ≤ lim sup ∥un ∥p+1 ≤ ∥u∥p+1 . n→∞

n→∞

Therefore, un → u a.e. in R and ∥un ∥p+1 → ∥u∥p+1 . Set vn = un − u, then by Lemma 4.3 we have ∥un |p+1 − |u|p+1 − |vn |p+1 | = ∥vn + u|p+1 − |u|p+1 − |vn |p+1 | ≤ C|u|p vn . From the above relations we infer that    p+1 p+1 p+1 p+1 |vn | dt ≤ ∥un | − |u| − |vn | |dt + ∥un |p+1 − |u|p+1 |dt R R R  ≤C |u|p vn dt + o(1) → 0 as n → ∞, R

(4.9)

Z. Liu et al. / Nonlinear Analysis: Real World Applications 36 (2017) 116–138

133

which means that ∥un − u∥p+1 → 0.

(4.10)

Note that ∥un ∥ → ∥u∥ as n → ∞. On the other hand,   2 [|u˙ n − u| ˙ + ⟨L(t)(un − u), (un − u)⟩dt] = [|u˙ n |2 + |u| ˙ 2 + ⟨L(t)un , un ⟩]dt R

R



 ⟨L(t)u, u⟩dt − 2

+

[⟨u˙ n , u⟩ ˙ + ⟨L(t)un , u⟩]dt. (4.11)

R

R

It follows from un ⇀ u in X that   [⟨u˙ n , u⟩ ˙ + ⟨L(t)un , u⟩]dt → [⟨u, ˙ u⟩ ˙ + ⟨L(t)u, u⟩]dt R

(4.12)

R

as n → ∞. Combining relations (4.11) and (4.12), we deduce that ∥un − u∥ → 0. Hence, it follows from (4.10) that un → u in X and the Palais–Smale condition for I is satisfied. In what follows, we prove Case (ii) cannot take place. The arguments will now be carried out by dividing into three steps. Step (a) Set vn = un − u, then there exist {tn } ⊂ R and constants R, δ > 0 such that  tn +R lim |vn |2 dt ≥ δ > 0. n→∞

(4.13)

tn −R

Otherwise, for every R > 0 

y+R

|vn |2 dt = 0.

lim sup

n→∞ y∈R

y−R

It follows from Lemma 4.1 that vn → 0

in Ls (R, RN ), 2 < s < +∞.

(4.14)

On the other hand, ′





[|u˙ n | + ⟨L(t)un , un ⟩]dt − a(x)|un |p+1 dt  R 2 = [|v˙ n | + ⟨L(t)vn , vn ⟩]dt + [|u| ˙ 2 + ⟨L(t)u, u⟩]dt 2

⟨I (un ), un ⟩ =

R

R

R



 [⟨v˙ n , u⟩ ˙ + ⟨L(t)vn , u⟩]dt −

+2 R

a(t)|u|p+1 dt −

R



a(t)(|un |p+1 − |u|p+1 )dt. (4.15)

R

By (A0 ), (4.14) and (4.9), we obtain   p+1 p+1 lim a(t)(|un | − |u| )dt = lim a(t)|vn |p+1 dt. n→∞

n→∞

R

(4.16)

R

Using the H¨ older inequality, (4.2) and (4.14), we have  p+1 3 ∥vn ∥ →0 (a+ (t) + χ[−R,R] (t))|vn |p+1 dt ≤ ∥a+ + χ[−R,R] ∥ 2−p 3

(4.17)

R

as n → ∞. From ⟨I ′ (u), u⟩ = 0 and (4.14)–(4.17), we deduce that   0 = lim ⟨I ′ (un ), un ⟩ = ⟨I ′ (u), u⟩ + lim |v˙ n |2 + ⟨L(t)vn , vn ⟩ − a(t)|vn |p+1 dt n→∞ n→∞ R  ′ = ⟨I (u), u⟩ + lim |v˙ n |2 + ⟨L(t)vn , vn ⟩ + (a− (t) + χ[−R,R] (t))|vn |p+1 dt n→∞

R

134

Z. Liu et al. / Nonlinear Analysis: Real World Applications 36 (2017) 116–138



p+1

+

 dt

− lim (a (t) + χ[−R,R] (t))|vn | n→∞  R  2 − p+1 = lim |v˙ n | + ⟨L(t)vn , vn ⟩ + (a (t) + χ[−R,R] (t))|vn | dt n→∞ R    −a∞ ≥ lim |v˙ n |2 + ⟨L(t)vn , vn ⟩ + min , 1 |vn |p+1 dt . n→∞ 2 R Therefore, it follows from the above relation and (4.14) that vn → 0 in X. This is a contradiction. Hence, from (4.13) we can easily see that there is v ∈ X such that vn (· + tn ) ⇀ v ̸= 0 in X. Step (b). We show that sequence {tn } is unbounded. Indeed, suppose on the contrary that {tn } is bounded. Then there exists R′ > 0 such that [tn − R, tn + R] ⊂ [−R′ , R′ ] for all n ∈ N, where R has been given in (4.13). Therefore, it follows from (4.13) that  R′  tn +R 2 lim |vn | dt ≥ lim |vn |2 dt ≥ δ > 0, n→∞

n→∞

−R′

which contradicts with the fact that vn → 0 in

tn −R

L2loc (R, RN )

as n → ∞.

Step (c). We claim that v is a solution of the following system: −u ¨ + Au = a∞ |u|p−1 u,

u ∈ E.

(S∞ )

Since matrix A is positive definite and a∞ < 0, it is natural to check that system (S∞ ) admits only the trivial solution. Thus, this contradicts with the claim that v solves (S∞ ). Since {tn } is not bounded, then un (· + tn ) ⇀ v in X. Indeed, {u(· + tn )} is bounded in X. Suppose that u(· + tn ) ⇀ w ∈ X, then we have   0 = lim u(t + tn )ϕdt = w(t)ϕdt, ∀ϕ ∈ C0∞ (R, RN ). n→∞

R

R

It follows that w = 0 a.e. in R. Therefore, from un ⇀ u in X, we have un (· + tn ) ⇀ v

in X.

(4.18)

For every fixed ϕ ∈ C0∞ (R, RN ), we have   ⟨I ′ (un ), ϕ(· − tn )⟩ = [u˙ n ϕ(t ˙ − tn ) + ⟨L(t)un , ϕ(t − tn )⟩]dt − a(t)|un |p−1 un ϕ(t − tn )dt R R = [u˙ n (t + tn )ϕ(t) ˙ + ⟨L(t + tn )un (t + tn ), ϕ(t)⟩]dt R  − a(t + tn )|un (t + tn )|p−1 un (t + tn )ϕ(t)dt. R

By (4.18), we have 

 u˙ n (t + tn )ϕ(t)dt ˙ →

R

v(t) ˙ ϕ(t)dt. ˙

(4.19)

R

Since un (· + tn ) → v in Lsloc (R, RN ) for each s ∈ [1, ∞] after extracting a subsequence, un (· + tn ) → v a.e. in R and there exists U ∈ Ls (R) such that |un (· + tn )| ≤ |U | in R, 1 ≤ s ≤ ∞. Then, by (L1 ) and (L2 ), we obtain ⟨L(t + tn )un (t + tn ), ϕ⟩ → ⟨Av, ϕ⟩ a.e. in R and |⟨L(t + tn )un (t + tn ), ϕ⟩| ≤ τ2 |U ||ϕ| ∈ L1 (R, RN ).

Z. Liu et al. / Nonlinear Analysis: Real World Applications 36 (2017) 116–138

Using the Lebesgue’s dominated convergence theorem, we deduce that   ⟨L(t + tn )un (t + tn ), ϕ(t)⟩dt → ⟨Av, ϕ(t)⟩dt. R

135

(4.20)

R

On the other hand, by (A0 ), we have a(t + tn )|un (t + tn )|p−1 |un (t + tn )|ϕ → a∞ |v|p−1 vϕ a.e. in R and |a(t + tn )||un (t + tn )|p−1 |ϕ| ≤ ∥a∥∞ |U |p−1 |ϕ| ∈ L1 (R). Applying the Lebesgue’s dominated convergence theorem again, we have   lim a(t + tn )|un (t + tn )|p−1 |un (t + tn )|ϕdt → a∞ |v|p−1 vϕdt. n→∞

R

(4.21)

R

Combining (4.19), (4.20) with (4.21), we deduce that for all ϕ ∈ C0∞ (R, RN ), 0 = lim ⟨I ′ (un ), ϕ(· − tn )⟩ n→∞    = v(t) ˙ ϕ(t)dt ˙ + ⟨Av, ϕ(t)⟩dt + a∞ |v|p−1 vϕdt. R

R

R

So v is a weak solution of system (S∞ ). Therefore, v = 0, which yields a contradiction. It follows from Steps (a), (b), and (c), that case (ii) does not occur. The proof is complete.  Lemma 4.5. Under the conditions of Theorem 1.3, for each k ∈ N, there exists Ak ∈ Σk such that supu∈Ak I(u) < 0. Proof. Here we use some ideas developed in [34]. Let R0 and y0 be fixed by assumption (A0 ) and consider the set D(R0 ) = {t ∈ R : |t − y0 | < R0 }. Fix k ∈ N arbitrary. We divide D(R0 ) into k small equal intervals, denoted by Di (1 ≤ i ≤ k), where the 0 length of Di is 2R k . We construct new intervals Ei in Di such that Ei has the same center as that of Di . Ei has the length of Rk0 . Thus, we can construct a function ψi : R → RN , 1 ≤ i ≤ k, such that supp(|ψi |) ⊂ Di ,

supp(|ψi |) ∩ supp(|ψj |) = ∅, (i ̸= j), for t ∈ Ei , 0 ≤ |ψi (t)| ≤ 1, |ψ˙ i (t)| ≤ C ∀t ∈ R,

|ψi (t)| = 1 where C is independent of i. Let

S k−1 = {(t1 , . . . , tk ) ∈ Rk : max |ti | = 1} 1≤i≤k

and Xk =

 k 

 ti ψi (t) : (t1 , . . . , tk ) ∈ S

k−1

⊂ X.

i=1

k k−1 Since the mapping (t1 , . . . , tk ) → into Xk is odd and homeomorphic, we have i=1 ti ψi (t) from S k−1 γ(Xk ) = γ(S ) = k. It is easy to see that Xk is compact in X. Thus, there is a constant ek > 0 such that ∥u∥2X ≤ ek

for all u ∈ Xk .

Z. Liu et al. / Nonlinear Analysis: Real World Applications 36 (2017) 116–138

136

Let λ > 0 and u =

k

i=1 ti ψi (t)

∈ Xk , then we have k

1  λ2 ek − I(λu) ≤ 2 p + 1 i=1



a(t)|λti ψi |p+1 dt.

(4.22)

Di

In view of the definition of S k−1 , there exists j ∈ {1, . . . , k} such that |tj | = 1 and |ti | ≤ 1 for i ̸= j. Then, we have   k    a(t)|λti ψi |p+1 dt = a(t)|λtj ψj |p+1 dt + a(t)|λtj ψj |p+1 dt + a(t)|λti ψi |p+1 dt. i=1

Di

Dj \Ej

Ej

i̸=j

Di

(4.23) Noticing that |ψj (t)| = 1 for all t ∈ Ej and |tj | = 1, we have   a(t)|λtj ψj |p+1 dt = |λ|p+1

a(t)dt.

(4.24)

a(t)|λti ψi |p+1 dt ≥ 0.

(4.25)

Ej

Ej

On the other hand, by (A0 ) one has   a(t)|λti ψi |p+1 dt + Dj \Ej

i̸=j

Di

It follows from relations (4.22)–(4.25) that I(λu) ≤ λ

2 ek

2

− |λ|

p+1

 a(t)dt < 0

inf

1≤i≤k

(4.26)

Ei

for λ small enough. Therefore, we fix λ so small such that sup{I(u), u ∈ Ak } < 0, where Ak = λXk ∈ Γk . This concludes the proof. Lemma 4.6. Under the conditions of Theorem 1.3, I is bounded from below. Proof. It follows from (4.2) that a+ ∈ Ls (R) for all s ∈ [1, +∞]. Then, for all u ∈ X we have   1 1 I(u) = [|u| ˙ 2 + ⟨L(t)u, u⟩]dt − a(t)|u|p+1 dt 2 R p+1 R   1 1 ≥ (|u| ˙ 2 + τ1 |u|2 )dt − a+ (t)|u|p+1 dt 2 R p+1 R τ1 1+p 2 ∥u∥ . ≥ ∥u∥22 − ∥a+ ∥ 1−p 2 2 The conclusion is immediate.  The proof of Theorem 1.3. We have known that I(0) = 0 and I is even. From Lemmas 4.4–4.6, we deduce that all conditions of Theorem 4.1 are satisfied. Thus, there exists a sequence {un } ⊂ X such that I(un ) < 0, I ′ (un ) = 0 and I(un ) → 0 for all n ≥ 0. Hence un is a nontrivial homoclinic solution of system (HS) and    1 1 1 ′ 2 o(1) = ⟨I (un ), un ⟩ − I(un ) = − (|u˙n | + ⟨L(t)un , un ⟩)dt ≥ τ1 |un |2 dt. (4.27) p+1 p+1 2 R R On the other hand, by the H¨ older inequality, (4.2) and (4.27), we have   1 1 1 o(1) = I(un ) − ⟨I ′ (un ), un ⟩ = − a(t)|un |p+1 dt 2 2 p+1 R      1 1 + = − (a (t) + χ[−R,R] (t))|un |p+1 dt − (a− (t) + χ[−R,R] (t))|un |p+1 dt 2 p+1 R R

Z. Liu et al. / Nonlinear Analysis: Real World Applications 36 (2017) 116–138

   1 1 p+1 − p+1 + 2 ≥ − ∥(a + χ[−R,R] )∥ 1−p ∥un ∥2 − (a (t) + χ[−R,R] (t))|un | dt 2 p+1 R   1 1 − p+1 = − (a (t) + χ[−R,R] (t))|un | dt + o(1) p+1 2 R     1 1 −a∞ p+1 , ≥ − min , 1 ∥un ∥p+1 p+1 2 2

137



(4.28)

which implies that  lim

n→+∞

|un |p+1 dt = 0.

R

It follows from (4.27) and (4.28) that un → 0 in X. This concludes the proof.



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