# On Homoclinic Solutions for First-Order Superquadratic Hamiltonian Systems with Spectrum Point Zero

Authors: Feng Li; Juntao Sun
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The existence and multiplicity of homoclinic solutions for a class of first-order periodic Hamiltonian systems with spectrum point zero are obtained. The proof is based on two critical point theorems for strongly indefinite functionals. Some recent results are improved and extended.


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## Body

## 1. Introduction

Consider the following first-order Hamiltonian systems:(HS)u˙(t)=𝒥Hu(t,u),t∈ℝ,
where u=(y,z)∈ℝ2N, 𝒥 is the standard symplectic matrix in ℝ2N(1)𝒥=(0-ININ0),
and H∈C1(ℝ×ℝ2N,ℝ) has the form
(2)H(t,u)=12Lu·u+W(t,u)
with L∈C(ℝ,ℝ4N2) being a 2N×2N symmetric matrix-valued function and W∈C1(ℝ×ℝ2N,ℝ). A solution u of (HS) is a homoclinic orbit if u≢0 and u(t)→0 as |t|→∞.To continue the discussion, we need the following notation:(3)A:=-(𝒥ddt+L).Homoclinic orbits of dynamical systems are important in applications for a number of reasons. They may be “organizing centers” for the dynamics in their neighborhood. From their existence one may, under certain conditions, infer the existence of chaos nearby or the bifurcation behavior of periodic orbits. As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics, and nuclear physics.During the last decades, many authors are devoted to the study of homoclinic orbits for Hamiltonian systems via modern variational methods. For example, see [1–4] for the second-order systems and [5–16] for the first-order systems. To be precise, in 1999, Ding and Willem [15] studied the existence of homoclinic solutions for a class of the first-order periodic Hamiltonian systems (HS) with spectrum point zero under the well-known (AR) condition as follows:
(4)∃β>2,0<βW(t,u)≤Wu(t,u)u,u≠0.
Later, under condition (4), Ding and Girardi [11] obtained that (HS) has infinitely many homoclinic orbits provided that W(t,u) is even in u. However, there are many potentials satisfying superquadratic condition, not satisfying the (AR) condition (4).Motivated by the above facts, in this paper, our aim is to study homoclinic solutions for(HS) under the conditions that zero is a continuous spectrum point and W(t,u) satisfies weak superquadratic conditions.Set(5)θ=(0,0,…,0)∈ℝ2N,W~(t,u)=12Wu(t,u)u-W(t,u).
Given a 2N×2N matrix w, we say that w≥0 if and only if
(6)inf⁡s∈ℝ2N,|s|=1ws·s≥0.
Also letting I2N be the identity matrix in ℝ2N and q∈ℝ, we denote the matrix qI2N by q. Moreover, denote by σ(A) and σcont(A) the spectrum and the continuous spectrum of the operator A, respectively.We make the following assumptions:(L1)
L(t) is 1-periodic on t. 0∈σ(A) and there exists β>0 such that (0,β]∩σ(A)=∅.(H1)
W∈C1(ℝ×ℝ2N,[0,∞)) is 1-periodic in t, W(t,0)≡0.(H2)
Wu(t,u)=o(u) uniformly in t as |u|→0.(H3)
W(t,u)/|u|2→∞ as |u|→∞ uniformly in t.(H4)
W~(t,u):=(1/2)Wu(t,u)u-W(t,u)>0 if u≠0, and there exist r>0 and η>1 such that |Wu(t,u)|η≤c1W~(t,u)|u|η if |u|≥r.Remark 1.
The following functions satisfy weak superquadratic conditions(H1)–(H4) but do not verify (4):
(7)Ex1.W(t,u)=d(t)(|u|ϵϵ(|u|p-1)gggggdd+(p-2)|u|p-ϵsin2(|u|ϵϵ)),Ex2.W(t,u)=d(t)(|u|2(|u|2-1)ln⁡(1+|u|)ggggggg+|u|2(2-|u|)),
where d(t)>0 and is 1-periodic in t and p>2, 0<ϵ<p-2.
Now we only check(Ex1.). It is easy to verify that (H1)–(H4) are satisfied. However, following the discussion of Remark  1.2 in [17], let un=(ϵ(nπ+3π/4))1/ϵI, where I=(1,0,…,0). Then for any β>2, one has
(8)Wu(t,un)un-βW(t,un)=d(t)[|u|ϵϵ(p-β)|un|p+(p-2)(p-ϵ-β)2222222kaa×|un|p-ϵsin2(|u|ϵϵ)bbbbbbbbaa+(p-2)|un|psin2(|u|ϵϵ)]=d(t)|un|p[(p-2)(p-ϵ-β)sin2sin2(|u|ϵ/ϵ)|un|ϵ2-plllllllllllllllllllllgg+(p-2)(p-ϵ-β)sin2sin2(|u|ϵ/ϵ)|un|ϵ]⟶-∞asn⟶∞.
That is, condition (4) is not satisfied for β>2.Observe that, due to the periodicity ofL and W, if u is a homoclinic orbit of (HS), then so is a*u for each a∈ℤ, where (a*u)(t)=u(t+a). Two solutions u1 and u2 are said to be geometrically distinct if a*u1≠u2 for all a∈ℤ.Now we state our main result.Theorem 2.
Assume that(L1) and (H1)–(H4) hold. Then (HS) has at least one homoclinic orbit. In addition, if W(t,u) is even in u, then (HS) has infinitely many geometrically distinct homoclinic orbits.This paper is organized as follows. In Section2, we formulate the variational setting and recall some critical point theorems required. In Section 3, we discuss linking structure and the Cerami condition of the functional. In Section 4, we prove Theorem 2.Notation.
Throughout the paper we will denote byc>0 various positive constants which may vary from line to line and are not essential to the problem.


## 2. Preliminaries

In what follows by|·|q we denote the usual Lq-norm and by (·,·)2 the usual inner product of L2(ℝ,ℝ2N). A standard Floquet reduction argument shows that σ(A)=σcont(A) (see Proposition  2.2 in [15]).Let{F(λ);λ∈ℝ} be the spectral family of A. We have A=U|A|, called the polar decomposition, where U=I-F(0)-F(-0). By (L1), L2 has an orthogonal decomposition
(9)L2=L2-⊕L2+,
where L2±:={u∈L2;Uu=±u}.LetE be the linear space of the completion of 𝒟(|A|1/2) under the norm
(10)∥u∥0:=||A|1/2u|2.
Then E is a Hilbert space under the inner product
(11)(u,v)0:=(|A|1/2u,|A|1/2v)2.E possesses an orthogonal decomposition
(12)E=E-⊕E+,
where E±⊇L2±∩𝒟(|A|1/2), the corresponding projections being denoted by P+,P-.By(L1), it is easy to check
(13)E+=L2+∩𝒟(|A|1/2),∥·∥0~∥·∥H1/2on⁡E+,
where “~" means “equivalence." Therefore, E+ can be embedded continuously into Lμ(ℝ,ℝ2N) for any μ≥2 and compactly into Lloc⁡μ(ℝ,ℝ2N) for any μ∈[2,∞).However, since0 may belong to a spectrum of A, then ∥·∥0 may not be equivalent to H1/2-norm on E-. Therefore, in the following we use the spectrum family of A to separate σ(A)∩(∞,0] into two segments. That is, for any ϵ>0, set
(14)Lϵ2-=F(-ϵ)L2
and Eϵ-=Lϵ2-∩𝒟(|A|1/2)=Lϵ2-∩E-. Let L~ϵ2-:=L2-⊖Lϵ2-=L2-∩(clL2(∪λ<-ϵF(λ)L2))⊥, where clL2(M) denotes the closure of the set M in L2. Similar to E+, since the spectrum of A restricted to Eϵ- is bounded away from 0, thus, one has
(15)∥·∥0~∥·∥H1/2on⁡Eϵ-.
However, L~ϵ2- is not complete with respect to the norm ∥·∥0; thus it is reasonable to introduce a new norm. Define
(16)∥u∥=(||A|1/2u|22+|u|22)1/2,
and let Eϵ,2- be the completion of L~ϵ2- under the norm ∥·∥.Lemma 3 (see [16], Lemma 2.1).
E
ϵ
,
2
-
⊂
H
loc
⁡
1 and is embedded compactly in Lloc⁡∞ and continuously in Lμ for all μ≥2.LetQ- denote the completion of 𝒟(A)∩L2- with respect to the norm ∥·∥. Since H1/2 is continuously embedded in Lμ for all μ≥2, by (15), Eϵ- is a closed subspace of Q-. Note that Eϵ,2-⊂E- and is orthogonal to Eϵ- with respect to (·,·)0. Then Q- has the following decomposition:
(17)Q-=Eϵ-⊕Eϵ,2-.
Let Q be the completion of 𝒟(A) under the norm ∥·∥. We have the following lemma.Lemma 4.
Suppose that (L1) is satisfied. Then Q has the direct sum decomposition
(18)Q=Q-⊕E+,
and Q is embedded continuously in Lμ and compactly in Lloc⁡μ for any μ∈[2,∞).Proof.
By (13), (15), and (17) and Lemma 3, Q- and E+ are closed, and using the decomposition of E, it is easy to check that Q-∩E+={0}, and so (18) holds. Using the same facts above and Lemma 3, one can obtain easily the desired conclusion on embedding.It is easy to verify thatQ is a Hilbert space under the inner product (u,v)=(u,v)0+(u,v)2, and ∥·∥ is its induced norm. From now on, we consider the space Q as our working space. Clearly, 𝒟(|A|1/2)⊂Q⊂E and all norms ∥·∥0, ∥·∥H1/2, and ∥·∥ are equivalent to E+. It is not difficult to check that ∥·∥ is uniformly convex, so Q is reflexive. Set
(19)φ(u)=∫ℝ‍W(t,u).
From Lemma 4 and (25) in Section 3, it follows that φ is defined on the Banach space Q and belongs to C1(Q,ℝ), and
(20)φ′(u)·v=∫ℝ‍Wu(t,u)v∀u,v∈Q.
Consider the functional
(21)Φ(u)=12(∥u+∥02-∥u-∥02)-φ(u)
for u=u-+u+∈Q. Then Φ∈C1(Q,ℝ) and a standard argument shows that critical points of Φ are homoclinic orbits of (HS) (see [15]).In order to study the critical points ofΦ, we now recall some abstract critical point theory developed recently in [18].LetE be a Banach space with direct sum decomposition E=X⊕Y and the corresponding projections PX,PY onto X,Y, respectively. For a functional Φ∈C1(E;ℝ) we write Φa={z∈E:Φ(z)≥a}, Φb={z∈E:Φ(z)≤b}, and Φab=Φa∩Φb. Recall that Φ is said to be weakly sequentially lower semicontinuous if for any zn⇀z in E one has Φ(z)≤lim⁡inf⁡n→∞Φ(zn). Φ′ is said to be weakly sequentially continuous if lim⁡n→∞Φ′(zn)u=Φ′(z)u for each u∈E. A sequence {zn}⊂E is said to be a (C)c-sequence if Φ(zn)→c and (1+∥zn∥)Φ′(zn)→0. Φ is said to satisfy the (C)c-condition if any (C)c-sequence has a convergent subsequence. A set 𝒜⊂E is said to be a (C)c-attractor if for any ɛ,δ>0 and any (C)c-sequence {zj} there is j0 such that zj∈Uɛ(𝒜∩Φc-δc+δ) for j≥j0. Given an interval I⊂ℝ, A is said to be a (C)I-attractor if it is a (C)c-attractor for all c∈I.From now on, we assume thatX is separable and reflexive and fix a countable dense subset 𝒮⊂X*. For each s∈𝒮 there exists a seminorm on E defined by
(22)ps:E⟶ℝ,ps(z)=|s(x)|+∥y∥forz=x+y∈X⊕Y.
We denote by 𝒯𝒮 the induced topology. Let u* denote the weak*-topology on E*.Suppose the following.(𝒩0)
For anyc∈ℝ, Φc is 𝒯𝒮-closed, and Φ′:(Φc,𝒯𝒮)→(E*,u*) is continuous.(𝒩1)
For anyc>0, there exists ζ>0 such that ∥z∥<ζ∥PYz∥ for all z∈Φc.(𝒩2)
There existsρ>0 with κ:=inf⁡Φ(SρY)>0, where SρY:={z∈Y:∥z∥=ρ}.(𝒩3)
There exists an increasing sequenceYn⊂Y of finite-dimensional subsequences and a sequence {Rn} of positive numbers such that, letting En=X⊕Yn and Bn=BRn∩En, sup⁡Φ(En)<∞ and sup⁡Φ(En∖Bn)<inf⁡Φ(Bρ∩Y).(𝒩4)
For any intervalI⊂(0,∞), there exists a (C)I-attractor A with PXA bounded and inf⁡{∥PY(z-w)∥:z,w∈A,PY(z-w)≠0}>0.Now we state two critical point theorems which will be used later.Theorem 5.
Let(𝒩0)–(𝒩2) be satisfied and suppose that there are R>ρ>0 and e∈Y with ∥e∥=1 such that sup⁡Φ(∂Q)≤κ, where Q={z=x+te:t≥0,x∈X,∥z∥<R}. Then Φ has a (C)c-sequence with κ≤c≤c¯:=sup⁡Φ(Q).Theorem 6.
AssumeΦ is even with Φ(0)=0 and let (𝒩0)–(𝒩4) be satisfied. Then Φ possesses an unbounded sequence of positive critical values.


## 3. Linking Structure and the(C)c-Sequence

We now study the linking structure and the(C)c-sequence of Φ. Observe that if (H4) holds, then |Wu(t,u)|η≤c1|Wu(t,u)||u|η+1; hence
(23)|Wu(t,u)|≤c2|u|p-1,if|u|≥r
for p≥τ:=2η/(η-1)>2. Remark that (23) and (H1)-(H2) imply that, given p≥τ, for any ɛ>0, there is Cɛ>0 such that
(24)|Wu(t,u)|≤ɛ|u|+Cɛ|u|p-1,(25)|W(t,u)|≤ɛ|u|2+Cɛ|u|p
for all (t,u).Lemma 7.
Assume that(H1)-(H2) and (H4) hold. Then there exists ρ>0 such that κ:=inf⁡Φ(Sρ+)>0, where Sρ+=∂Bρ∩E+.Proof.
Choosep≥τ such that (25) holds for any ɛ>0. This yields
(26)φ(u)≤ɛ|u|22+Cɛ|u|pp≤c(ɛ∥u∥2+Cɛ∥u∥p)
for all u∈Q; together with the equivalence of ∥·∥0 and ∥·∥ on E+, the lemma follows from the form of Φ.Setλ_=inf⁡[σ(A)∩(0,∞)]. (L1) implies that λ_≥β>0. Choose ω=2λ_. Then we can take a number λ¯ such that
(27)λ_<λ¯<ω.
Since σ(A)=σcont(A), the subspace Y0:=(Fλ¯-F0)L2 is infinite dimensional, where {Fλ:λ∈ℝ} denotes the spectral family of A. Note that Y0⊂E+ and
(28)λ_|u|22≤∥u∥≤λ¯|u|22foranyu∈Y0.
Let {αn} be a sequence with
(29)λ_=α0<α1<α2<⋯≤λ¯.
For each n∈ℕ, take an element en∈(Fαn-Fαn-1)L2 with ∥en∥=1 and define Yn:=span⁡{e1,…,en}. Then Yn⊂Y0 is an increasing sequence of finite dimensional subsequence of E+. Set Qn=Q-⊕Yn.Lemma 8.
Suppose that(H1) and (H3)-(H4) are satisfied and ρ>0 is given by Lemma 7. Then sup⁡Φ(Qn)<∞, and there exists a sequence {Rn>0}n∈N such that sup⁡Φ(Qn∖Bn)<inf⁡Φ(Bρ∩E+), where Bn:={z∈Qn:∥z∥≤Rn}.Proof.
It suffices to show thatΦ(u)→-∞ as ∥u∥→∞ in Qn. If not, we assume that, for some sequence {uj}⊂Qn with ∥uj∥→∞, there exists M0>0 such that Φ(uj)≥-M0 for all j. Define vj=uj/∥uj∥; we have ∥vj∥=1. Passing to subsequence vj⇀v, vj-⇀v-, vj→v in Lloc⁡μ for μ∈[2,∞) and vj+→v+ with respect to ∥·∥ and ∥·∥0. Then, by (H1) we have
(30)12(∥vj+∥02-∥vj-∥02)≥12(∥vj+∥02-∥vj-∥02)-∫ℝ‍W(t,uj)∥uj∥2=Φ(uj)∥uj∥2≥-M0∥uj∥2.
We claim that v+≠0. Indeed, if not, it follows from (30) that ∥vj-∥0→0. Also ∫ℝ(W(t,uj)/∥uj∥2)‍→0.
By(H3), there exists r′>0 such that
(31)W(t,u)≥ω|u|2if|u|>r′.
For 0≤l<m, let
(32)Ωj(l,m)={t∈ℝ:l≤|uj(t)|<m}.
Thus, by (31) and (32), one has
(33)∫ℝ‍W(t,uj)∥uj∥2≥∫Ωj(r′,∞)‍W(t,uj)∥uj∥2≥ω∫Ωj(r′,∞)‍|vj|2.
From (33) and ∫ℝ(W(t,uj)/∥uj∥2)‍→0 and together with ∫Ωj(0,r′)‍|vj|2→0 it follows that |vj|22→0. Therefore, ∥vj∥→0, which contradicts ∥vj∥=1. So v+≠0.
By (27)-(28) and the fact that |v|22=|v+|22+|v-|22, one has(34)∥v+∥02-∥v-∥02-ω∫ℝ‍v2≤∥v+∥2-∥v-∥02-ω|v|22≤-(ω-λ¯)|v+|22-∥v-∥02<0.
Then there exists a finite interval I⊂ℝ such that
(35)∥v+∥02-∥v-∥02-ω∫I‍v2<0.
Note that
(36)Φ(uj)∥uj∥2≤12(∥vj+∥02-∥vj-∥02)-∫I‍W(t,uj)∥uj∥2=12(∥vj+∥02-∥vj-∥02-ω∫I‍|vj|2)-∫I‍W(t,uj)-(ω/2)|uj|2∥uj∥2≤12(∥vj+∥02-∥vj-∥02-ω∫I‍|vj|2)+ωr2|I|2∥uj∥2,
where |I| denotes the Lebesgue measure of I. Combining this with (30) and (35), one has
(37)0≤lim⁡j→∞(12∥vj+∥02-12∥vj-∥02-∫Ω‍W(t,uj)∥uj∥2)≤12(∥v+∥02-∥v-∥02-ω∫I‍v2)<0,
which is a contradiction.As a special case we have the following.Lemma 9.
Suppose that(H1) and (H3)-(H4) are satisfied and ρ>0 is given by Lemma 7. Then, letting e∈Y0 with ∥e∥=1, there exists r1>0 such that sup⁡Φ(∂Q1)≤κ, where Q1:={u=u-+se:u-∈Q-,s≥0,∥u∥≤r1}.Lemma 10.
Suppose that(H1)–(H4) are satisfied. Then any (C)c-sequence is bounded in Q.Proof.
Let{uj}⊂Q be such that
(38)Φ(uj)⟶c,(1+∥uj∥)Φ′(uj)⟶0.
Then, for C0>0,
(39)C0≥Φ(uj)-12Φ′(uj)uj=∫ℝ‍W~(t,uj).
Suppose to the contrary that ∥uj∥ is unbounded. Setting vj=uj/∥uj∥, then ∥vj∥=1 and |vj|μ≤c∥vj∥=c for all μ∈[2,∞). Passing to subsequence, vj⇀v in Q, vj→v in Lloc⁡μ for μ∈[2,∞), and vj(t)→v(t) for a.e., t∈ℝ.
Note that(40)o(1)=Φ′(uj)(uj+-uj-)=∥uj∥02-∫ℝ‍Wu(t,uj)(uj+-uj-)=∥uj∥02-∥uj∥2∫ℝ‍Wu(t,uj)∥uj∥(vj+-vj-)=∥uj∥2(∥vj∥02-∫ℝ‍Wu(t,uj)∥uj∥(vj+-vj-))=∥uj∥2(1-|vj|22-∫ℝ‍Wu(t,uj)∥uj∥(vj+-vj-)).
From (40), we obtain
(41)∫ℝ‍(|vj|2+Wu(t,uj)(vj+-vj-)∥uj∥)⟶1.
Set, for x≥0,
(42)h(x):=inf⁡{W~(t,u):t∈ℝ,u∈ℝ2Nwith|u|≥x}.
By (H3) and (H4), h(x)>0 for all x>0 and h(x)→∞ as x→∞.
Let(43)Clm=inf⁡{W~(t,u)|u|2:t∈ℝwithl≤|u(t)|<m}.
Since W(t,u) depends periodically on t and W~(t,u)>0 if u≠0, one has Clm>0 and
(44)W~(t,uj)≥Clm|uj|2∀t∈Ωj(l,m),
where Ωj(l,m) is defined in (32). It follows from (32) and (39) that
(45)C0≥∫Ωj(0,l)‍W~(t,uj)+∫Ωj(l,m)‍W~(t,uj)+∫Ωj(m,∞)‍W~(t,uj)≥∫Ωj(0,l)‍W~(t,uj)+Clm∫Ωj(l,m)‍|uj|2+h(m)|Ωj(m,∞)|.
Using (45) we obtain
(46)|Ωj(m,∞)|≤C0h(m)⟶0,
as m→∞ uniformly in j, and for any fixed 0<l<m,
(47)∫Ωj(l,m)‍|vj|2=1∥uj∥2∫Ωj(l,m)‍|uj|2≤C0Clm∥uj∥2⟶0,
as j→∞. It follows from (46) that, for any s∈[2,+∞),
(48)∫Ωj(m,∞)‍|vj|s≤(∫Ωj(m,∞)‍|vj|2s)1/2·|Ωj(m,∞)|1/2≤c|Ωj(m,∞)|1/2⟶0,
as m→∞ uniformly in j.
Let0<ϵ<1/5. By (H2) there is lϵ>0 such that
(49)|Wu(t,u)|<ϵc|u|
for all |u|≤lϵ. Consequently,
(50)∫Ωj(0,lϵ)‍Wu(t,uj)(vj+-vj-)|vj||uj|≤∫Ωj(0,lϵ)‍ϵc|vj+-vj-||vj|≤ϵc|vj|22<ϵ
for all j.
Setσ′:=τ/2. By (H4), (48), and Hölder inequality, we can take mϵ≥r large enough such that
(51)∫Ωj(mϵ,∞)‍Wu(t,uj)(vj+-vj-)|vj||uj|≤(∫Ωj(mϵ,∞)‍|Wu(t,uj)|σ|uj|σ)1/σ×(∫Ωj(mϵ,∞)‍(|vj+-vj-||vj|)σ′)1/σ′≤(∫Ωj(mϵ,∞)‍c1W~(t,uj))1/σ(∫ℝN‍(|vj+-vj-|)τ)1/τ×(∫Ωj(mϵ,∞)‍|vj|τ)1/τ≤ϵ
for all j. Note that there is C=C(ϵ)>0 independent of j such that |Wu(t,uj)|≤C|uj| for t∈Ωj(lϵ,mϵ). By (47) there is j0 such that
(52)∫Ωj(lϵ,mϵ)‍Wu(t,uj)(vj+-vj-)|vj||uj|≤C∫Ωj(lϵ,mϵ)‍|vj+-vj-||vj|≤C|vj|2(∫Ωj(lϵ,mϵ)‍|vj|2)1/2≤ϵ
for all j≥j0. By (50)–(52), one has
(53)lim⁡sup⁡j→∞∫ℝ‍Wu(t,uj)(vj+-vj-)∥uj∥≤3ϵ.
On the other hand, by (48), take mϵ>0 large enough such that
(54)∫Ωj(mϵ,∞)‍|vj|2≤ϵ.
Similar to (47), one has
(55)∫Ωj(0,mϵ)‍|vj|2=1∥uj∥2∫Ωj(0,mϵ)‍|uj|2≤mϵ2|Ωj(0,mϵ)|∥uj∥2≤ϵ.
Thus (54) and (55) imply that
(56)lim⁡sup⁡j→∞∫ℝ‍|vj|2≤2ϵ.
By (53) and (56), one has
(57)lim⁡sup⁡j→∞∫ℝ‍(|vj|2+Wu(t,uj)(vj+-vj-)∥uj∥)≤5ϵ<1,
which contradicts (41). The proof is complete.Let{uj}⊂Q be a (C)c-sequence of Φ; by Lemma 10, it is bounded, up to a subsequence; we may assume uj⇀u in Q, uj→u in Lloc⁡μ for μ∈[2,∞), and uj(t)→u(t) a.e. on ℝ. Plainly, u is a critical point of Φ. Set uj1=uj-u.Lemma 11.
Under the assumptions of Theorem2, one has, as j→∞,(1)
Φ(uj1)→c-Φ(u);(2)
Φ′(uj1)→0.Proof.
The proof of this lemma is similar to the one of Lemma  5.7 in [9] (or see [10]), so we omit it here.Let𝒦:={u∈Q:Φ′(u)=0,u≠0} be the set of nontrivial critical points of Φ.Lemma 12.
Under the assumptions of Theorem2, the following two conclusions hold:(1)
ϕ:=inf⁡{∥u∥:u∈𝒦}>0;(2)
ψ:=inf⁡{Φ(u):u∈𝒦}>0.Proof.
(1) For anyu∈𝒦, there holds
(58)0=Φ′(u)u+=∥u+∥02-∫ℝ‍Wu(t,u)u+,
together with (25), which implies that
(59)∥u+∥02≤ϵ|u+|22+Cϵ|u+|pp≤cϵ∥u+∥02+cCϵ∥u+∥0p,
where p≥τ. Choose ϵ small enough; one has
(60)0<(1-cϵcCϵ)2/(p-2)≤∥u+∥02
for each u∈𝒦. Therefore,
(61)∥u∥2>∥u∥02=∥u+∥02+∥u-∥02>(1-cϵcCϵ)2/(p-2)>0.
(2) Suppose to the contrary that there exists a sequence {uj}⊂𝒦 such that Φ(uj)→0.
Then(62)0=∥uj∥02-∫ℝ‍Wu(t,uj)(uj+-uj-),(63)o(1)=Φ(uj)=Φ(uj)-12Φ′(uj)uj=∫ℝ‍W~(t,uj).
By (1), ∥uj∥≥ϕ. Clearly {uj} is a (C)0-sequence of Φ and hence is bounded by Lemma 10. Using (63) and the discussion in the proof of Lemma 10, we see that, for any 0<l<m and p∈[τ,∞), ∫Ωj(l,m)‍|uj|2→0 and ∫Ωj(m,∞)‍|uj|p→0 as j→∞. Therefore, it follows from (24) and (62) that for any ϵ′>0(64)lim⁡sup⁡j→∞∥uj∥02≤ϵ′,
which contradicts (61). The proof is complete.In the following lemma we discuss further the(C)c-sequence. Let [k] denote the integer part of k∈ℝ. The following lemma is standard by using Lemmas 11 and 12 (see [1, 19]).Lemma 13.
Under the assumptions of Theorem2, let {uj}⊂Q be a (C)c-sequence of Φ. Then either(i)
uj→0 (and hence c=0), or(ii)
c≥ψ and there exist a positive integer k≤[c/ψ], y1,…,yk∈𝒦, and sequence {aji}⊂ℤ,  i=1,2,…,k, such that, after extraction of a subsequence of {uj},
(65)∥uj-∑i=1kaji*yi∥⟶0,∑i=1kΦ(yi)=cand fori≠q,
(66)|aji-ajq|⟶∞asj→∞.


## 4. Proof of Theorem2

In order to apply the abstract Theorems5 and 6 to Φ, we choose X=Q- and Y=E+. X is separable and reflexive and let 𝒮 be a countable dense subset of X*. First we have the following.Lemma 14.
Φ satisfies (𝒩0).Proof.
We first show thatΦa is 𝒯S-closed for every a∈ℝ. Consider a sequence {uj} in Φa which 𝒯S-converges to u∈Q and write uj=uj-+uj+ and u=u-+u+. Observe that {uj+} converges to u+ in norm topology. Since W(t,u)≥0, there exists c>0 such that
(67)12∥uj-∥02=12∥uj+∥02-Φ(uj)-∫ℝ‍W(t,uj)<c.
Therefore, ∥uj-∥0 is bounded. So ∥uj∥0 is bounded. Now we claim that |uj-|2 is bounded. Assume by contradiction that |uj-|2 is unbounded. Clearly, ∥uj∥ is also unbounded. Setting vj=uj/∥uj∥, then ∥vj∥=1. Passing to subsequence, vj⇀v in Q, vj→v in Lloc⁡μ for μ∈[2,∞). Note that
(68)∫ℝ‍W(t,uj)<-a+12∥uj+∥02-12∥uj-∥02.
Therefore, ∫ℝ‍W(t,uj) is bounded and ∫ℝ(W(t,uj)/∥uj∥2)‍→0. Similar to Lemma 8, we can obtain (33). From (33) and ∫ℝ‍(W(t,uj)/∥uj∥2)→0 and together with ∫Ωj(0,r)‍|vj|2→0 it follows that |vj|2→0. Since 1=∥vj∥2=∥vj∥02+|vj|22, we obtain ∥vj∥0→1, which contradicts the fact that ∥vj∥0=∥uj∥0/∥uj∥→0. So |uj-|2 is bounded. Therefore, {uj-} is bounded in Q, and we have uj-⇀u-. Therefore, uj⇀u. It is easy to show that Ψ is weakly sequentially lower semicontinuous. Thus, from the form of Φ it follows that Φ(u)≥lim⁡inf⁡Φ(uj)≥a. So u∈Φa and Φa is 𝒯S-closed.
Next we show thatΦ′:(Φc,𝒯𝒮)→(Q*,u*) is continuous. It is sufficient to show that φ′ has the same property. Let uj⇀u in Q. Then uj→u in Lloc⁡μ for μ∈[2,∞). It is obvious that
(69)φ′(uj)v=∫ℝ‍W(t,uj)v⟶∫ℝ‍W(t,u)v=φ′(u)v,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiivvxx∀v∈C0∞(ℝ)
as j→∞. Now using the density of C0∞(ℝ) in Q one can obtain the desired conclusion.Lemma 15.
Φ satisfies (𝒩1).Proof.
We assume by contradiction that, for somec1>0, there exists a sequence {un} with Φ(un)≥c1 and ∥un∥2≥n∥un+∥2. The form of Φ implies that
(70)∥un-∥2≥(n-1)∥un+∥2≥c(n-1)∥un+∥02≥c(n-1)(2c1+∥un-∥02+2∫ℝ‍W(t,un)).
Since c,c1>0 and W(t,un)≥0, it follows that ∥un-∥→∞ as n→∞. Hence ∥un∥→∞. Set vn=un/∥un∥. Then ∥vn∥=1 and ∥vn+∥2=∥un+∥2/∥un∥2≤1/n→0. Therefore, vn⇀v=v- in Q. By (70) we have
(71)1≥∥vn-∥2≥c(n-1)(2c1∥un∥2+∥vn-∥02+2∫ℝ‍W(t,un)∥un∥2).
Thus ∥vn-∥02≤1/c(n-1)→0. By (31)-(32) and (71), one has
(72)12c(n-1)≥∫ℝ‍W(t,un)∥un∥2≥∫Ωn(r,∞)‍W(t,un)∥un∥2≥ω∫Ωn(r′,∞)‍|vn|2,
and together with ∫Ωn(0,r′)‍|vn|2→0, it follows that |vn|22→0. Therefore, ∥vn∥→0, which is in contradiction with ∥vn∥=1. The proof is complete.Now we give the proof of Theorem2.Proof.
(1) Existence of a Homoclinic Orbit. With X=Q- and Y=E+ the condition (𝒩0) holds by Lemma 14 and (𝒩1) holds by Lemma 15. Lemma 7 implies that (𝒩2) holds. Lemma 9 shows that Φ possesses the linking structure of Theorem 5. Therefore, there exists a sequence {uj}⊂Q such that
(73)Φ(uj)⟶c≥κ,(1+∥uj∥)Φ′(uj)⟶0.
By Lemma 10  {uj} is bounded. Consequently, Φ′(uj)→0. A standard argument shows that {uj} is a nonvanishing sequence; that is, there exist ξ1>0, 0<R<∞, and {aj}⊂ℝ such that
(74)lim⁡inf⁡j→∞∫aj-Raj+R‍|uj|2≥ξ1.
Setting u~j(t)=uj(t+aj), by the invariance under translation of Φ, {u~j} is a (C)c-sequence of Φ and u~j⇀u~. From
(75)lim⁡inf⁡j→∞∫-RR‍|u~j|2≥ξ1>0,
we see that u~≠0, and hence u~ is a nontrivial critical point of Φ. Following the idea of [15], we prove that u~ is a homoclinic orbit of (HS). By (23) and Lemma 4, Wu(t,u)∈Lμ for all μ≥2. Hence, a standard regularity theory of solutions of ordinary differential equations shows that u∈Hloc⁡1∩Lμ for all μ≥2. By (HS) and (23),
(76)|u˙|2≤c(|u|2+|u|2(p-1)),
where p≥τ. So for all ϱ∈ℝ, by Hölder inequality, one has
(77)∫ϱ-1/2ϱ+1/2‍|u˙|2≤c{(∫ϱ-1/2ϱ+1/2|u|2(p-1))1/(p-1)+∫ϱ-1/2ϱ+1/2|u|2(p-1)}.
Integrating from ϱ-1/2 to ϱ+1/2 the following equality,
(78)u(ϱ)=u(t)+∫tϱ‍u˙(s)ds,
yields, using Hölder inequality
(79)|u(ϱ)|≤c(∫ϱ-1/2ϱ+1/2‍|u|2(p-1))1/(p-1)+(∫ϱ-1/2ϱ+1/2‍|u˙|2)1/2.
Since u∈L2(p-1), (77) and (79) show that
(80)|u(ϱ)|⟶0as|ϱ|⟶∞;
that is, u is a homoclinic orbit of (HS). The proof is complete.
(2) Multiplicity. Following the idea of [10], the proof will be completed in an indirect way; namely, we show that if
(81)𝒦ℤisafiniteset,
then Φ has an unbounded sequence of critical values, a contradiction. We do this by checking that if (81) is true then Φ satisfies all the assumptions of Theorem 6.
By the assumptionW(t,u) is even in u, hence so is Φ. Φ(0)=0 is deduced from (H1). Lemma 8 shows that Φ satisfies (𝒩3). Let 𝒯 be a set consisting of arbitrarily chosen representatives of the ℤ-orbits of 𝒦. Then 𝒯 is a finite set by (77), and since Φ is odd we may assume that 𝒯=-𝒯. If z∈𝒦, then Φ(z)≥ψ by (2) of Lemma 12. Hence there exists ψ≤χ such that
(82)ψ≤min⁡𝒯Φ=min⁡𝒦Φ≤max⁡𝒦Φ≤max⁡𝒯Φ≤χ.
For k∈ℕ and a finite set ℬ⊂Q we define
(83)(ℬ,k):={∑i=1jai*uj:1≤j≤k,ai∈ℤ,uj∈ℬ}.
As in Coti Zelati and Rabinowitz [1, 19],
(84)inf⁡{∥u-u′∥:u,u′∈(ℬ,k)}>0.
Now we check (𝒩4). Given a compact interval I⊂(0,∞) with s:=max⁡I we set k=[d/ψ] and 𝒞:=(𝒯,k). We have P+(𝒯,k)=(P+𝒯,k). Thus from (84)
(85)inf⁡{∥u1+-u2+∥:u1,u2∈𝒞,u1+≠u2}>0.
In addition, 𝒞 is a (C)I-attractor by Lemma 13 and 𝒞 is bounded because ∥u∥≤kmax⁡{∥u¯∥:u¯∈𝒯} for all u∈𝒞. Therefore, by Theorem 6, Φ has an unbounded sequence of critical values which contradicts the assumption (81), and hence Φ has infinitely many geometrically distinct homoclinic orbits.



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*Source: 313690-2014-04-24.xml*