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

1. Introduction and Main Results

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

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

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

As usual, we say that H satisfies the subquadratic (or superquadratic) growth condition at infinity if
(6)lim|u|→∞H(t,u)|u|2=0(orlim|u|→∞H(t,u)|u|2=+∞).
If M≠0, that is, the damped vibration system (1), there are only a few authors who have studied homoclinic orbits of the NDVS (1), see [18–23]. Zhu [18] considered the periodic case of (1) (i.e.,L(t)andH(t,u)areT-periodic intwithT>0) and obtained the existence of nontrivial homoclinic solutions of (1). The authors [19–23] considered the nonperiodic case of (1): Zhang and Yuan [19] obtained the existence of at least one homoclinic orbit for (1) whenHsatisfies the subquadratic condition at infinity by using a standard minimizing argument. By a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [20] obtained the existence and multiplicity of homoclinic orbits for (1) whenHsatisfies the local(AR)superquadratic growth condition:
(7)0<μH(t,u)≤(Hu(t,u),u),∀t∈ℝ,∀|u|≥r,
whereμ>2andr>0are two constants. We should notice that the matrixL(t)in (1) is required to satisfy condition (4) in the Previously mentioned two papers [19, 20]. Later, Sun et al. [21] obtained the existence of at least one homoclinic orbit for (1) whenHsatisfies the superquadratic condition at infinity by using the following conditions which are weaker than condition (4).

There exists a constantβ>1such that
(8)meas{t∈ℝ:|t|-βL(t)<bIN}<+∞,∀b>0.

There exists a constantγ≥0such that
(9)l(t):=inf|u|=1(L(t)u,u)≥-γ,∀t∈ℝ.

Recently, by using conditions(L1)and(L2), Chen [22, 23] obtained infinitely many nontrivial homoclinic orbits of (1) whenHsatisfies the subquadratic [22] (or superquadratic [23]) growth condition at infinity. In fact, conditions (L1)and(L2)are first used in [14]. As mentioned in [21], there are some matrix-valued functionsL(t)satisfying(L1)and(L2) but not satisfying (4). For example, L(t):=(t4sin2t+1)IN.That is, conditions(L1)and(L2)are weaker than condition (4).
Remark 1.

To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for the NDVS (1) whenHsatisfies the asymptotically quadratic condition at infinity (see the following condition(H3)).

LetH~(t,u):=H(t,u)-(1/2)(Hu(t,u),u). We assume the following.

There are constantsμ∈(1,2)andc1,c2,c3>0such that
(10)c3|u|μ≤|H(t,u)|≤c1|u|,∀t∈ℝ,|u|≤c2.

We should mention that the coercive-type assumption (see(H4)) of the functionH~was first observed and used by Costa and Magalhães [24].

Now, our main result reads as follows.

Theorem 2.

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

Example 3.

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

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

2. Variational Frameworks and the Proof of Our Main Result

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

In the following, we will use∥·∥pto denote the norm ofLp(ℝ,ℝN)for anyp∈[1,∞]. Let E:=H1(ℝ,ℝN)be a Hilbert space with the inner product and the norm given, respectively, by
(13)〈u,v〉E=∫ℝ[(u˙(t),v˙(t))+(u(t),v(t))]dt,∥u∥E=〈u,u〉E1/2,∀u,v∈E.
It is well known thatEis continuously embedded inLp(ℝ,ℝN)forp∈[2,∞). We define an operator J:E→Eby
(14)(Ju,v):=∫ℝ(Mu(t),v˙(t))dt,∀u,v∈E.
SinceMis an antisymmetricN×Nconstant matrix,Jis self-adjoint onE. Moreover, we denote byχthe self-adjoint extension of the operator-d2/dt2+L(t)+Jwith the domain𝒟(χ)⊂L2(ℝ,ℝN).

LetW:=𝒟(|χ|1/2), the domain of|χ|1/2. We define, respectively, onWthe inner product and the norm
(15)〈u,v〉W≔(|χ|1/2u,|χ|1/2v)2+(u,v)2,∥u∥W=〈u,u〉W1/2,
where(·,·)2denotes the inner product inL2(ℝ,ℝN).

If conditions(L1)and(L2)hold, thenWis compactly embedded into Lp(ℝ,ℝN)for all 1≤p≤+∞.

By Lemma 4, it is easy to prove that the spectrumσ(χ)has a sequence of eigenvalues (counted with their multiplicities)
(16)λ1≤λ2≤⋯≤λk≤⋯⟶∞,
and the corresponding system of eigenfunctions {ek:k∈ℕ}(χek=λkek)forms an orthogonal basis in L2(ℝ,ℝN). Let
(17)k1:=♯{j:λj<0},k0:=♯{j:λj=0},k2:=k0+k1,W-:=span{e1,…,ek1},W0:=span{ek1+1,…,ek2},W+:=clW(span{ek2+1,…}).
Then, one has the orthogonal decomposition
(18)W=W-⊕W0⊕W+
with respect to the inner product〈·,·〉W.

Now, we introduce, respectively, onWthe following new inner product and norm:
(19)〈u,v〉≔(u0,v0)2+(|χ|1/2u,|χ|1/2v)2,∥u∥=〈u,u〉1/2,
whereu,v∈W=W-⊕W0⊕W+withu=u-+u0+u+, andv=v-+v0+v+. Clearly, the two norms ∥·∥ and ∥·∥W are equivalent (see [3]), and the decompositionW=W-⊕W0⊕W+is also orthogonal with respect to both inner products〈·,·〉and(·,·)2.

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

LetWbe a Banach space with the norm∥·∥ andW:=⨁m∈ℕXm¯withdimXm<∞for anym∈ℕ. Set
(23)Yk:=⨁m=1kXm,Zk:=⨁m=k∞Xm¯.
Consider the followingC1-functionalΦλ:W→ℝdefined by
(24)Φλ(u)=A(u)-λB(u),λ∈[1,2].

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

Lemma 5 (see [<xref ref-type="bibr" rid="B27">27</xref>]).

Assume that the functionalΦλdefined previously satisfies

Φλmaps bounded sets to bounded sets uniformly forλ∈[1,2], and
(25)Φλ(-u)=Φλ(u)∀(λ,u)∈[1,2]×W;

B(u)≥0for allu∈WandB(u)→+∞as∥u∥→∞on any finite-dimensional subspace of W;

there existρk>rk>0such that
(26)αk(λ)≔infu∈Zk,∥u∥=ρkΦλ(u)≥0>βk(λ):=maxu∈Yk,∥u∥=rkΦλ(u),∀λ∈[1,2],(27)ξk(λ)≔infu∈Zk,∥u∥≤ρkΦλ(u)⟶0ask⟶∞≔infu∈Zk,∥u∥≤ρkuniformlyforλ∈[1,2].

Then, there exist0<λj→1anduλj∈Yjsuch that
(28)Φλj′|Yj(uλj)=0,Φλj(uλj)→ηk∈[ξk(2),βk(1)]asj→∞.
Particularly, if{uλj}has a convergent subsequence for everyk, thenΦ1has infinitely many nontrivial critical points{uk}⊂W∖{0}satisfyingΦ1(uk)→0-ask→∞.

Form∈ℕ, letXm:=ℝem(the sequence{em}is defined in Section 2 just below Lemma 4); thenZkandYkcan be defined as before. In order to apply the previously mentioned variant fountain theorem to prove our main result, we define the functionals A, B, andΦλonWby
(29)A(u):=12∥u+∥2,B(u):=12∥u-∥2+∫ℝH(t,u)dt,Φλ(u)≔A(u)-λB(u)Φλ(u)=12∥u+∥2-λ(12∥u-∥2+∫ℝH(t,u)dt)
for all u=u0+u-+u+∈W=W0⊕W-⊕W+ and λ∈[1,2]. Obviously, Φλ∈C1(W,ℝ) for all λ∈[1,2].

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

Lemma 6.

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

Proof.

Obviously, condition(H2)and the definition ofBimply thatB(u)≥0for allu∈W. We claim that for any finite-dimensional subspaceX⊂W, there exists a constantϵ>0such that
(30)m({t∈ℝ:|u|≥ϵ∥u∥})≥ϵ,∀u∈X∖{0},
wherem(·)denotes the Lebesgue measure inℝ. In fact, the detailed proof of (30) has been given by Chen (Lemma 2.3 in [22]).

For theϵgiven in (30), let
(31)Λu:={t∈ℝ:|u|≥ϵ∥u∥},∀u∈X∖{0}.
Then, by (30),
(32)m(Λu)≥ϵ,∀u∈X∖{0}.
By(H3), there exist constantsR1,R2>0such that
(33)H(t,u)≥R1|u|2,∀(t,u)∈ℝ×ℝNwith|u|≥R2.
The definition ofΛuimplies that for anyu∈Xwith ∥u∥≥R2/ϵthere holds
(34)|u|≥R2,∀t∈Λu.
Combining(H2), (32)–(34), and the definition ofΛu, for anyu∈Xwith∥u∥≥R2/ϵ, we have
(35)B(u)=12∥u-∥2+∫ℝH(t,u)dt≥∫ΛuH(t,u)dt≥∫ΛuR1|u|2dt≥R1ϵ2∥u∥2·m(Λu)≥R1ϵ3∥u∥2.
It implies that B(u)→∞as∥u∥→∞on any finite-dimensional subspaceX⊂W. The proof is finished.

Lemma 7.

There exist a positive integerl0and two sequences0<rk<ρk→0ask→∞such that
(36)αk(λ):=infu∈Zk,∥u∥=ρkΦλ(u)>0,∀k≥l0,(37)ξk(λ)≔infu∈Zk,∥u∥≤ρkΦλ(u)⟶0ask→∞uniformlyforλ∈[1,2],(38)βk(λ):=maxu∈Yk,∥u∥=rkΦλ(u)<0,∀k∈ℕ,
whereYk=⨁m=1kXmandZk=⨁m=k∞Xm¯for allk∈ℕ.

Proof.

(a) First, we show that (36) holds. Note thatZk⊂W+for allk≥k2+1, wherek2is the integer defined in (17) just below Lemma 4. By Lemma 4, there is a constantɛ0>0such that ∥u∥∞≤ɛ0∥u∥for anyu∈W. It follows that for anyu∈Wwith ∥u∥≤c2/ɛ0there holds
(39)|u|≤∥u∥∞≤c2,
wherec2is the constant in(H1). It follows from(H1)and the definition ofΦλthat for anyk≥k2+1andu∈Zkwith ∥u∥≤c2/ɛ0there holds
(40)Φλ(u)≥12∥u∥2-2∫ℝH(t,u)dt≥12∥u∥2-2c1∥u∥1,∀λ∈[1,2].
Let
(41)lk:=supu∈Zk∖{0}∥u∥1∥u∥,∀k∈ℕ.
Then
(42)lk⟶0ask⟶∞
by Lemma 4 and the Rellich embedding theorem (see [28]). Consequently, (40) and (41) imply that
(43)Φλ(u)≥12∥u∥2-2c1lk∥u∥
for anyk≥k2+1andu∈Zkwith ∥u∥≤c2/ɛ0. For anyk∈ℕ, let
(44)ρk:=8c1lk.
Then, by (42), we have
(45)0<ρk⟶0ask⟶∞.
Evidently, (45) implies that there exists a positive integerl0>k2+1such that
(46)ρk≤c2ɛ0,∀k≥l0.
(43) together with (44) and (46) implies that
(47)αk(λ)≔infu∈Zk,∥u∥=ρkΦλ(u)≥ρk22-ρk24=ρk24>0,∀k≥l0.
That is, (36) holds.

(b) Second, we show that (37) holds. By (43), for anyk≥l0andu∈Zkwith ∥u∥≤ρk, we have
(48)Φλ(u)≥-2c1lkρk.
Observing thatΦλ(0)=0by(H1), thus
(49)0≥infu∈Zk,∥u∥≤ρkΦλ(u)≥-2c1lkρk,∀k≥l0,
which together with (42) and (45) implies that
(50)ξk(λ)≔infu∈Zk,∥u∥≤ρkΦλ(u)⟶0ask⟶∞uniformlyforλ∈[1,2].
That is, (37) holds.

(c) Last, we show that (38) holds. For anyk∈ℕandu∈Ykwith ∥u∥≤c2/ɛ0(ɛ0is the constant above (39)), similar to (39), we have
(51)|u|≤c2.
Therefore, by (51) and(H1), for anyk∈ℕandu∈Ykwith ∥u∥≤c2/ɛ0, we have
(52)Φλ(u)≤12∥u+∥2-∫ℝH(t,u)dt≤12∥u∥2-c3∥u∥μμ≤12∥u∥2-Ck∥u∥μ,∀λ∈[1,2],
where the last inequality follows by the equivalence of the two norms∥·∥μand∥·∥on finite dimensional spaceYk, andCk>0is a constant depending onYk. For anyk∈ℕ, if we choose
(53)0<rk<min{ρk,Ck1/(2-μ),c2ɛ0},
Then, by (52), direct computation shows that
(54)βk(λ):=maxu∈Yk,∥u∥=rkΦλ(u)≤-rk22<0,∀k∈ℕ.
That is, (38) holds.

Therefore, the proof is finished by (a), (b), and (c).

Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>.

By the assumptions ofHand the definition ofΦλ, we easily get thatΦλmaps bounded sets to bounded sets uniformly forλ∈[1,2]. Note thatH(t,-u)=H(t,u), so we have Φλ(-u)=Φλ(u)for all(λ,u)∈[1,2]×W. Thus, the condition(T1)of Lemma 5 holds. Lemma 6 shows that the condition(T2)of Lemma 5 holds. Lemma 7 implies that the condition(T3)of Lemma 5 holds for allk≥l0, wherel0is given in Lemma 7. Therefore, by Lemma 5, for eachk≥l0, there exist 0<λj→1, uλj∈Yjsuch that
(55)Φλj′|Yj(uλj)=0,Φλj(uλj)⟶ηk∈[ξk(2),βk(1)]asj⟶∞.

Next, we only need to prove the following two claims to complete the proof of Theorem 2.

Claim 1.

{uλj}is bounded inW.

Proof of Claim <xref ref-type="statement" rid="claim1">1</xref>.

By (55), we have
(56)(1/2)Φ′λj∣Yj(uλj)uλj-Φλj(uλj)λj≤C1
for some constant C1>0. It follows from the definitions ofΦλjandH~that
(57)∫ℝH~(t,uλj)dt=(1/2)Φλj′∣Yj(uλj)uλj-Φλj(uλj)λj≤C1.
Since(H4)implies∫ℝH~(t,u)dt→+∞as|u|→+∞, it follows from (57) that
(58)|uλj|≤C2,∀j∈ℕ
for some constantC2>0. Note thatH(t,u)∈C1(ℝ×ℝN,ℝ); it follows from(H4)that there is a constantC3>0such that
(59)|Hu(t,uλj)|μ/(μ-1)≤C3H~(t,uλj),t∈ℝ,|uλj|≤C2.
Thus, by (57)–(59),Φλj′∣Yj(uλj)uλj+=0, Hölder’s inequality, and Lemma 4,
(60)∥uλj+∥2=λj∫ℝ(Hu(t,uλj),uλj+)dt≤λj(∫ℝ|Hu(t,uλj)|μ/(μ-1)dt)(μ-1)/μ(∫ℝ|uλj+|μdt)1/μ≤C4(∫ℝC3H~(t,uλj)dt)(μ-1)/μ∥uλj+∥≤C5∥uλj+∥
for some positive constantC4andC5. It implies that∥uλj+∥≤C5. On the other hand, (H2) and Φλj′|Yj(uλj)uλj=0imply that
(61)∥uλj+∥2-λj∥uλj-∥2=λj∫ℝ(Hu(t,uλj),uλj)dt≥0;
that is,
(62)λj∥uλj-∥2≤∥uλj+∥2.
It follows from∥uλj+∥≤C5that{uλj}is bounded inW. Therefore, Claim 1 is true.

Claim 2.

{uλj}has a strongly convergent subsequence inW.

Proof of Claim <xref ref-type="statement" rid="claim2">2</xref>.

Note thatdim(W0⊕W-)<∞. By Claim 1, without loss of generality, we may assume that
(63)uλj-⟶u-,uλj0⟶u0,uλj+⇀u+,uλj⇀uasj⟶∞
for someu=u0+u-+u+∈W=W0⊕W-⊕W+. By virtue of the Riesz Representation Theorem, Φλj′|Yj:Yj→Yj*andI′:W→W*can be viewed asΦλj′|Yj:Yj→YjandI′:W→W, respectively, whereYj*andW*are the dual spaces ofYjandW, respectively. Note that
(64)0=Φλj′|Yj(uλj)=uλj+-λj[uλj-+χjI′(uλj)],∀j∈ℕ,
whereχj:W→Yjis the orthogonal projection for allj∈ℕ; that is,
(65)uλj+=λj[uλj-+χjI′(uλj)],∀j∈ℕ.
By the assumptions ofHand the standard argument (see [29, 30]), we knowI′:W→W*is compact. Therefore,I′:W→Wis also compact. Due to the compactness ofI′and (63), the right-hand side of (65) converges strongly inWand henceuλj+→u+inW. Combining this with (63), we have
(66)uλj⟶uinW,j⟶∞.
Therefore, Claim 2 is true.

Now, from the last assertion of Lemma 5, we know thatΦ=Φ1has infinitely many nontrivial critical points. Therefore, (1) possesses infinitely many nontrivial homoclinic orbits.

Acknowledgment

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

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