R E S E A R C H
Open Access
Existence of fast homoclinic orbits for a class
of second-order non-autonomous problems
Qiongfen Zhang
1*, Qi-Ming Zhang
2and Xianhua Tang
3*Correspondence:
1College of Science, Guilin
University of Technology, Guilin, Guangxi 541004, P.R. China Full list of author information is available at the end of the article
Abstract
By applying the mountain pass theorem and the symmetric mountain pass theorem in critical point theory, the existence and multiplicity of fast homoclinic solutions are obtained for the following second-order non-autonomous problem:
¨
u(t) +q(t)u(t) –˙ a(t)|u(t)|p–2u(t) +∇W(t,u(t)) = 0, wherep≥2,t∈R,u∈RN,a∈C(R,R),
W∈C1(R×RN,R) are not periodic intandq:R→Ris a continuous function and
Q(t) =0sq(s)dswith lim|t|→+∞Q(t) = +∞.
MSC: 34C37; 35A15; 37J45; 47J30
Keywords: fast homoclinic solutions; variational methods; critical point
1 Introduction
Consider fast homoclinic solutions of the following problem:
¨
u(t) +q(t)u˙(t) –a(t)u(t)p–u(t) +∇Wt,u(t)= , t∈R, (.)
wherep≥,t∈R,u∈RN,a∈C(R,R),W∈C(R×RN,R) are not periodic int, and
q:R→Ris a continuous function andQ(t) =sq(s)dswith
lim
|t|→+∞Q(t) = +∞. (.)
Whenq(t)≡, problem (.) reduces to the following special second-order Hamiltonian system:
¨
u(t) –a(t)u(t)p–u(t) +∇Wt,u(t)= , t∈R. (.)
When p= , problem (.) reduces to the following second-order damped vibration problem:
¨
u(t) +q(t)u˙(t) –a(t)u(t) +∇Wt,u(t)= , t∈R. (.)
If we takep= andq(t)≡, then problem (.) reduces to the following second-order Hamiltonian system:
¨
u(t) –a(t)u(t) +∇Wt,u(t)= , t∈R. (.)
The existence of homoclinic orbits plays an important role in the study of the behavior of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic. If it has smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. The first work about homoclinic orbits was done by Poincaré [].
Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For exam-ple, see [–] and references therein. In [, , ], the authors considered homoclinic solutions for the special Hamiltonian system (.) in weighted Sobolev space. Later, Shi
et al.[] obtained some results for system (.) with ap-Laplacian, which improved and generalized the results in [, , ]. However, there is little research as regards the exis-tence of homoclinic solutions for damped vibration problems (.) whenq(t)= . In , Wu and Zhou [] obtained some results for damped vibration problems (.) with some boundary value conditions by variational methods. Zhang and Yuan [, ] studied the existence of homoclinic solutions for (.) whenq(t)≡cis a constant. Later, Chenet al.
[] investigated fast homoclinic solutions for (.) and obtained some new results un-der more relaxed assumptions onW(t,x), which resolved some open problems in []. Zhang [] obtained infinitely many solutions for a class of general second-order damped vibration systems by using the variational methods. Zhang [] investigated subharmonic solutions for a class of second-order impulsive systems with damped term by using the mountain pass theorem.
Motivated by [, , –, –], we will establish some new results for (.) in weighted Sobolev space. In order to introduce the concept of fast homoclinic solutions for problem (.), we first state some properties of the weighted Sobolev spaceEon which the certain variational functional associated with (.) is defined and the fast homoclinic solutions are the critical points of the certain functional.
Let
X=
u∈H,R,RN
Re
Q(t)u˙(t)
+u(t) dt< +∞
,
whereQ(t) is defined in (.) and foru,v∈X, let
u,v=
Re
Q(t)u˙(t),v˙(t)+u(t),v(t) dt.
ThenXis a Hilbert space with the norm given by
u=
Re
Q(t)u˙(t)
+u(t) dt /
.
It is obvious that
X⊂LeQ(t)
with the embedding being continuous. HereLp(eQ(t)) (≤p< +∞) denotes the Banach
spaces of functions onRwith values inRNunder the norm
up=
Re
Q(t)u(t)p
dt /p
Ifσ is a positive, continuous function onRand <s< +∞, let
LsσeQ(t)=
u∈LloceQ(t)
Rσ(t)e
Q(t)u(t)s
dt< +∞
.
Ls
σ equipped with the norm
us,σ =
Rσ(t)e
Q(t)u(t)s
dt /s
is a reflexive Banach space.
SetE=X∩Lpa(eQ(t)), whereais the function given in condition (A). ThenEwith its
standard norm · is a reflexive Banach space. Similar to [, ], we have the following definition of fast homoclinic solutions.
Definition . If (.) holds, a solution of (.) is called a fast homoclinic solution ifu∈E. The functionalϕcorresponding to (.) onEis given by
ϕ(u) =
Re
Q(t)
u˙(t)
+a(t)
p u(t)
p
–Wt,u(t)dt, u∈E. (.)
Clearly, it follows from (W) or (W) thatϕ:E→R. By Theorem . of [], we can deduce that the map
u→a(t)eQ(t)u(t)p–u(t)
is continuous fromLpa(eQ(t)) in the dual spaceLpa–/(p–)(eQ(t)), wherep=pp–. As the
embed-dingsE⊂X⊂Lγ(eQ(t)) for allγ ≥ are continuous, if (A) and (W) or (W)hold, then ϕ∈C(E,R) and one can easily check that
ϕ(u),v=
Re
Q(t)u˙(t),v˙(t)+a(t)u(t)p–u(t),v(t)
–∇Wt,u(t),v(t) dt, u∈E. (.) Furthermore, the critical points ofϕinEare classical solutions of (.) withu(±∞) = .
Now, we state our main results.
Theorem . Suppose that a,q,and W satisfy(.)and the following conditions:
(A) Letp> ,a(t)is a continuous,positive function onRsuch that for allt∈R a(t)≥α|t|β, α> ,β> (p– )/.
(W) W(t,x) =W(t,x) –W(t,x),W,W∈C(R×RN,R),and there exists a constant
R> such that
a(t)∇W(t,x)=o
|x|p– asx→
(W) There is a constantμ>psuch that
<μW(t,x)≤
∇W(t,x),x
, ∀(t,x)∈R×RN\{}. (W) W(t, ) = and there exists a constant∈(p,μ)such that
W(t,x)≥,
∇W(t,x),x
≤W(t,x), ∀(t,x)∈R×RN.
Then problem(.)has at least one nontrivial fast homoclinic solution.
Theorem . Suppose that a,q,and W satisfy(.), (A), (W),and the following condi-tions:
(W) W(t,x) =W(t,x) –W(t,x),W,W∈C(R×RN,R),and
a(t)∇W(t,x)=o
|x|p– asx→
uniformly int∈R.
(W) W(t, ) = and there exists a constant∈(p,μ)such that
∇W(t,x),x
≤W(t,x), ∀(t,x)∈R×RN.
Then problem(.)has at least one nontrivial fast homoclinic solution.
Theorem . Suppose that a,q,and W satisfy(.), (A), (W)-(W),and the following assumption:
(W) W(t, –x) =W(t,x),∀(t,x)∈R×RN.
Then problem(.)has an unbounded sequence of fast homoclinic solutions.
Theorem . Suppose that a,q,and W satisfy(.), (A), (W), (W), (W),and(W).
Then problem(.)has an unbounded sequence of fast homoclinic solutions.
Remark . It is easy to see that our results hold true even ifp= . To the best of our knowledge, similar results for problem (.) cannot be seen in the literature, from this point, our results are new. As pointed out in [], condition (A) can be replaced by more general assumption:a(t)→+∞as|t| →+∞.
The rest of this paper is organized as follows: in Section , some preliminaries are pre-sented. In Section , we give the proofs of our results. In Section , some examples are given to illustrate our results.
2 Preliminaries
LetEand · be given in Section , by a similar argument in [], we have the following important lemma.
Lemma . For any u∈E,
u∞≤√
e
u=√ e
Re
Q(s)u˙(s)
+u(s) ds /
u(t)≤
+∞
t
e–Q(s)eQ(s)u˙(s)+u(s) ds /
≤ √
e
+∞
t
eQ(s)u˙(s)+u(s) ds /
(.)
and
u(t)≤
t
–∞e
–Q(s)eQ(s)u˙(s)
+u(s) ds /
≤ √
e
t
–∞
eQ(s)u˙(s)+u(s) ds /
, (.)
whereu∞=ess supt∈R|u(t)|,e=emin{Q(t):t∈R}.
The following lemma is an improvement result of [].
Lemma . If a satisfies assumption(A),then
the embedding LpaeQ(t)⊂LeQ(t)is continuous. (.)
Moreover,there exists a Hilbert space Z such that
the embeddings LpaeQ(t)⊂Z⊂LeQ(t)are continuous; (.)
the embedding X∩Z⊂LeQ(t)is compact. (.)
Proof Letθ=p/(p– ),θ=p/, we have
u =
Re
Q(t)u(t)dt
=
Ra
–/θ
a/θeQ(t)/θeQ(t)/θu(t)dt
≤
Ra
–θ/θ eQ(t)dt
/θ
Rae
Q(t)u(t)θ dt
/θ
=a
Rae
Q(t)u(t)p
dt /p
=aup,a,
where from (A) and (.),a= (
Ra–/(p–)eQ(t)dt)(p–)/p< +∞. Then (.) holds.
By (A), there exists a continuous positive functionρsuch thatρ(t)→+∞as|t| →+∞ and
a=
R
ρθa–θ/θeQ(t)dt /θ
< +∞.
Since
u,ρ=
Rρe
Q(t)u(t)dt
=
R
≤
Rρ θ
a–θ/θeQ(t)dt /θ
Rae
Q(t)u(t)pdt
/θ
=aup,a,
(.) holds by takingZ=Lρ(eQ(t)).
Finally, asX∩Zis the weighted Sobolev space,(R,ρ, ), it follows from [] that (.)
holds.
The following two lemmas are the mountain pass theorem and the symmetric mountain pass theorem, which are useful in the proofs of our theorems.
Lemma .[] Let E be a real Banach space and I∈C(E,R)satisfying(PS)-condition.
Suppose I() = and:
(i) There exist constantsρ,α> such thatI∂Bρ()≥α. (ii) There exists ane∈E\ ¯Bρ()such thatI(e)≤.
Then I possesses a critical value c≥αwhich can be characterized as
c=inf h∈smax∈[,]I
h(s),
where={h∈C([, ],E)|h() = ,h() =e}and Bρ()is an open ball in E of radiusρ
centered at.
Lemma .[] Let E be a real Banach space and I∈C(E,R)with I even.Assume that
I() = and I satisfies(PS)-condition,assumption(i)of Lemma.and the following con-dition:
(iii) For each finite dimensional subspaceE⊂E,there isr=r(E) > such thatI(u)≤
foru∈E\Br(),Br()is an open ball inEof radiusrcentered at.
Then I possesses an unbounded sequence of critical values.
Lemma . Assume that(W)and(W)or(W)hold.Then for every(t,x)∈R×RN, (i) s–μW
(t,sx)is nondecreasing on(, +∞); (ii) s–W
(t,sx)is nonincreasing on(, +∞).
The proof of Lemma . is routine and we omit it.
3 Proofs of theorems
Proof of Theorem. Step . The functionalϕ satisfies the (PS)-condition. Let{un} ⊂E
satisfyingϕ(un) is bounded andϕ(un)→ asn→ ∞. Then there exists a constantC>
such that
ϕ(un)≤C, ϕ(un)E∗≤μC. (.)
From (.), (.), (.), (W), and (W), we have
C+ Cun ≥ϕ(un) –
μ
ϕ(un),un
= μ–
μ ˙un +
Re
Q(t)
W
t,un(t)
–
μ
∇W
t,un(t)
,un(t)
–
Re
Q(t)
W
t,un(t)
–
μ
∇W
t,un(t)
,un(t)
dt
+
p–
μ Ra(t)e
Q(t)u n(t)
p
dt
≥μ–
μ ˙un +
p–
μ
unpp,a.
It follows from Lemma .,μ>p> , and the above inequalities that there exists a constant
C> such that
un ≤C, n∈N. (.)
Now we prove thatun→uinE. Passing to a subsequence if necessary, it can be assumed
thatunuinE. SinceQ(t)→ ∞as|t| → ∞, we can chooseT>Rsuch that
Q(t)≥ln
C
ξ
for|t| ≥T. (.)
It follows from (.), (.), and (.) that
un(t)
≤
+∞
t
e–Q(s)eQ(s)u˙n(s)
+un(s)
ds
≤ ξ
C
un≤ξ fort≥Tandn∈N. (.)
Similarly, by (.), (.), and (.), we have
un(t)
≤ξ fort≤–Tandn∈N. (.)
SinceunuinE, it is easy to verify thatun(t) converges tou(t) pointwise for allt∈R.
Hence, it follows from (.) and (.) that
u(t)≤ξ fort∈(–∞, –T]∪[T, +∞). (.)
SinceeQ(t)≥e
> on [–T,T] =J, the operator defined by S:E→X(J) :u→u|J is a
linear continuous map. Soun→uinX(J). The Sobolev theorem implies thatun→u
uniformly onJ, so there isn∈Nsuch that
T
–T
eQ(t)∇Wt,un(t)
–∇Wt,u(t)un(t) –u(t)dt<ε forn≥n. (.)
For any given numberε> , by (W), we can chooseξ> such that
∇W(t,x)≤εa(t)|x|p– for|t| ≥Rand|x| ≤ξ. (.)
From (.), we have
eQ(t)∇Wt,un(t)
–∇Wt,u(t)
≤eQ(t)εa(t)un(t) p–
≤eQ(t)εp–a(t)un(t) –u(t) p–
+ε + p–a(t)u(t) p–
≤pεa(t)eQ(t)u
n(t) –u(t) (p–)
+ (ε) + p–
a(t)eQ(t)u (t)
(p–)
:=gn(t). (.)
Moreover, sincea(t) is a positive continuous function onRandun(t) converges tou(t)
pointwise for allt∈R, it follows from (.) that
lim
n→∞gn(t) = (ε)
+ p–a(t)eQ(t)u (t)
(p–)
:=g(t) for a.e.t∈R
and
lim n→∞
R\(–T,T)
gn(t)dt= lim n→∞
R\(–T,T)
eQ(t)pεa(t)un(t) –u(t) (p–)
+ (ε) + p–a(t)u(t) (p–)
dt
= p(ε) lim n→∞
R\(–T,T)
a(t)eQ(t)un(t) –u(t) (p–)
dt
+ (ε) + p–
R\(–T,T)
a(t)eQ(t)u(t)(p–)dt
= (ε) + p–
R\(–T,T)
a(t)eQ(t)u(t) (p–)
dt
=
Rg(t)dt< +∞.
From Lebesgue’s dominated convergence theorem, (.), (.), (.), (.), and the above inequalities, we have
lim n→∞
R\(–T,T)
eQ(t)∇Wt,un(t)
–∇Wt,u(t)dt= . (.)
From Lemma ., we haveun→uinL(eQ(t)). Hence, by (.),
R\(–T,T)
eQ(t)∇Wt,un(t)
–∇Wt,u(t)un(t) –u(t)dt
≤
R\(–T,T)
eQ(t)∇Wt,un(t)
–∇Wt,u(t)
dt /
×
R\(–T,T)
eQ(t)un(t) –u(t)
dt /
tends to asn→+∞, which together with (.) shows that
Re
Q(t)∇Wt,u n(t)
–∇Wt,u(t)un(t) –u(t)dt→ asn→ ∞. (.)
From (.), we have
←ϕ(un) –ϕ(u),un–u
= ˙un–u˙+
Ra(t)e
Q(t)u
n(t)p–un(t) –u(t)p–u(t)
un(t) –u(t)
–
Re
Q(t)∇Wt,u n(t)
–∇Wt,u(t)
,un(t) –u(t)
dt
≥ ˙un–u˙+C
Ra(t)e
Q(t)u
n(t) –u(t) p
dt
–
Re
Q(t)∇Wt,u n(t)
–∇Wt,u(t)
,un(t) –u(t)
dt, n→ ∞, (.)
whereCis a positive constant. It follows from (.) and (.) that
˙un→ ˙u asn→ ∞ (.)
and
Ra(t)e
Q(t)u n(t)
p
dt→
Ra(t)e
Q(t)u (t)
p
dt asn→ ∞. (.)
Hence,un→uinEby (.) and (.). This shows thatϕsatisfies (PS)-condition.
Step . From (W), there existsδ∈(, ) such that
∇W(t,x)≤ a(t)|x|
p– for|t| ≥R,|x| ≤δ. (.)
By (.) andW(t, ) = , we have
W(t,x)≤ pa(t)|x|
p for|t| ≥R,|x| ≤δ. (.)
Let
C=sup
W(t,x)
a(t)
t∈[–R,R],x∈R,|x|=
. (.)
Setσ=min{/(pC+ )/(μ–p),δ}andu= √
eσ:=ρ, it follows from Lemma . that |u(t)| ≤σ≤δ< fort∈R. From Lemma .(i) and (.), we have
R
–R
eQ(t)W
t,u(t)dt≤
{t∈[–R,R]:u(t)=}
eQ(t)W
t, u(t)
|u(t)|
u(t)μdt
≤C
R
–R
a(t)eQ(t)u(t)μdt
≤Cσμ–p
R
–R
a(t)eQ(t)u(t)pdt
≤
p R
–R
a(t)eQ(t)u(t)pdt. (.)
By (W), (.), and (.), we have
ϕ(u) =
Re
Q(t)u˙(t)
dt+
R a(t)
p e
Q(t)u(t)p
dt–
Re
Q(t)Wt,u(t)dt
= ˙u
+
pu
p p,a–
R\(–R,R)
eQ(t)Wt,u(t)dt–
R
–R
≥
˙u
+
pu
p p,a–
R\(–R,R)
eQ(t)Wt,u(t)dt–
R
–R
eQ(t)W
t,u(t)dt
≥
˙u
+
pu
p p,a–
p
R\(–R,R)
a(t)eQ(t)u(t)pdt
– p
R
–R
a(t)eQ(t)u(t)pdt
= ˙u
+
pu
p p,a.
Therefore, we can choose a constantα> depending onρ such thatϕ(u)≥α for any
u∈Ewithu=ρ.
Step . From Lemma .(ii) and (.), we have for anyu∈E
–
eQ(t)W
t,u(t)dt
=
{t∈[–,]:|u(t)|>}e
Q(t)W
t,u(t)dt+
{t∈[–,]:|u(t)|≤}e
Q(t)W
t,u(t)dt
≤
{t∈[–,]:|u(t)|>}e
Q(t)W
t, u(t)
|u(t)|
u(t)dt+
–
eQ(t)max
|x|≤W(t,x)dt
≤ u
∞
–
eQ(t)max
|x|=W(t,x)dt+
–
eQ(t)max
|x|≤W(t,x)dt ≤
√
e
u
–
eQ(t)max
|x|=W(t,x)dt+
–
eQ(t)max
|x|≤W(t,x)dt
=Cu+C, (.)
whereC= (√e
)
–eQ(t)max|x|=W(t,x)dt,C=
–eQ(t)max|x|≤W(t,x)dt. Takeω∈
Esuch that
ω(t)=
for|t| ≤,
for|t| ≥, (.)
and|ω(t)| ≤ for|t| ∈(, ]. Fors> , from Lemma .(i) and (.), we get
–
eQ(t)W
t,sω(t)dt≥sμ
–
eQ(t)W
t,ω(t)dt=Csμ, (.)
whereC=
–eQ(t)W(t,ω(t))dt> . From (W), (.), (.), (.), and (.), we get for
s>
ϕ(sω) = s
˙ω
+
sp
pω
p p,a+
Re
Q(t)W
t,sω(t)–W
t,sω(t) dt
≤ s
˙ω
+
sp
pω
p p,a+
–
eQ(t)W
t,sω(t)dt–
–
eQ(t)W
t,sω(t)dt
≤ s
˙ω
+
sp
pω
p
Sinceμ>>pandC> , it follows from (.) that there existss> such thatsω>ρ
andϕ(sω) < . Sete=sω(t), thene∈E,e=sω>ρ, andϕ(e) =ϕ(sω) < . It is easy
to see thatϕ() = . By Lemma .,ϕhas a critical valuec>αgiven by
c=inf g∈smax∈[,]ϕ
g(s), (.)
where
=g∈C[, ],E:g() = ,g() =e.
Hence, there existsu∗∈Esuch that
ϕu∗=c,
ϕu∗= .
The functionu∗is the desired solution of problem (.). Sincec> ,u∗is a nontrivial fast homoclinic solution. The proof is complete.
Proof of Theorem. In the proof of Theorem ., the conditionW(t,x)≥ in (W) is
only used in the proofs of (.) and Step . Therefore, we only need to prove that (.) and Step still hold if we use (W)and (W)instead of (W) and (W). We first prove that (.) holds. From (W), (W), (.), (.), and (.), we have
C+
Cμ
un ≥ϕ(un) –
ϕ(un),un
= –
˙un
+
Re
Q(t)
W
t,un(t)
–
∇W
t,un(t)
,un(t)
dt
–
Re
Q(t)
W
t,un(t)
–
∇W
t,un(t)
,un(t)
dt
+
p–
Ra(t)e
Q(t)u n(t)
p
dt
≥–
˙un +
p–
unpp,a,
which implies that there exists a constantC> such that (.) holds. Next, we prove that
Step still holds. From (W), there existsδ∈(, ) such that
∇W(t,x)≤ a(t)|x|
p– fort∈R,|x| ≤δ. (.)
By (.) andW(t, ) = , we have
W(t,x)≤ pa(t)|x|
Letu=√eδ:=ρ, it follows from Lemma . that|u(t)| ≤δ. It follows from (.) and
(.) that
ϕ(u) =
Re
Q(t)u˙(t)dt+
R
a(t)eQ(t)
p u(t)
p
dt–
Re
Q(t)Wt,u(t)dt
≥
˙u
+
pu
p p,a–
R
pa(t)e
Q(t)u(t)pdt
= ˙u
+
pu
p p,a.
Therefore, we can choose a constantα> depending onρ such thatϕ(u)≥α for any
u∈Ewithu=ρ. The proof of Theorem . is complete.
Proof of Theorem. Condition (W) shows thatϕis even. In view of the proof of The-orem ., we know thatϕ∈C(E,R) and satisfies (PS)-condition and assumption (i) of
Lemma .. Now, we prove that (iii) of Lemma .. LetEbe a finite dimensional subspace ofE. Since all norms of a finite dimensional space are equivalent, there existsd> such that
u ≤du∞. (.)
Assume thatdimE=mand{u,u, . . . ,um}is a basis ofEsuch that
ui=d, i= , , . . . ,m. (.)
For anyu∈E, there existsλi∈R,i= , , . . . ,msuch that
u(t) =
m
i=
λiui(t) fort∈R. (.)
Let
u∗=
m
i=
|λi|ui. (.)
It is easy to see that · ∗is a norm ofE. Hence, there exists a constantd> such that
du∗≤ u. Sinceui∈E, by Lemma ., we can chooseR>Rsuch that
ui(t)<
dδ
+m, |t|>R,i= , , . . . ,m, (.)
whereδis given in (.). Let
=
m
i=
λiui(t) :λi∈R,i= , , . . . ,m; m
i= |λi|=
=u∈E:u∗=d. (.)
Hence, foru∈, lett=t(u)∈Rsuch that
Then by (.)-(.), (.), and (.), we have
d=u∗≤u
d ≤ d du∞=
d du(t)
= d d m i=
λiui(t)
≤d m i=
|λi|ui(t), u∈. (.)
This shows that|u(t)| ≥dand there existsi∈ {, , . . . ,m}such that|ui(t)| ≥d/m,
which together with (.), implies that|t| ≤R. LetR=R+ and
γ =min
eQ(t)W(t,x) : –R≤t≤R,
d
√
≤ |x| ≤
d
√
e
. (.)
SinceW(t,x) > for allt∈Randx∈RN\{}, andW∈C(R×RN,R), it follows that γ > . For anyu∈E, from Lemma . and Lemma .(i), we have
R
–R
eQ(t)W
t,u(t)dt
=
{t∈[–R,R]:|u(t)|>}
eQ(t)W
t,u(t)dt
+
{t∈[–R,R]:|u(t)|≤}
eQ(t)W
t,u(t)dt
≤
{t∈[–R,R]:|u(t)|>}
eQ(t)W
t, u(t)
|u(t)|
u(t)dt
+
R
–R
eQ(t)max
|x|≤W(t,x)dt ≤ u∞
R
–R
eQ(t)max
|x|=W(t,x)dt+
R
–R
eQ(t)max
|x|≤W(t,x)dt ≤
√
e
u R
–R
eQ(t)max
|x|=W(t,x)dt+
R
–R
eQ(t)max
|x|≤W(t,x)dt
=Cu+C, (.)
where C= (√e)
R
–Re
Q(t)max
|x|=W(t,x)dt, C =
R
–Re
Q(t)max
|x|≤W(t,x)dt. Since ˙
ui∈L(eQ(t)),i= , , . . . ,m, it follows that there existsε∈(, ((d)e)/(md)) such that
t–ε
t+ε
u˙i(s)ds=
t–ε
t+ε e–Q(s)e
Q(s)
u˙i(s)ds
≤√
e
t–ε
t+ε eQ(s)u˙
i(s)ds ≤√
e
(ε)/
t–ε
t+ε
eQ(s)u˙i(s) ds / ≤ ε e / ˙ui
≤ d
Then foru∈with|u(t)|=u∞ andt∈[t–ε,t+ε], it follows from (.), (.),
(.), (.), and (.) that
u(t) =u(t) + t t ˙
u(s),u(s)ds
≥u(t)
–
t+ε
t–ε
u(s)u˙(s)ds
≥u(t)
– u(t)
t+ε
t–ε
u˙(s)ds
≥u(t)
– u(t) m
i= |λi|
t+ε
t–ε
u˙i(s)ds
≥ (d)
. (.)
On the other hand, sinceu ≤dforu∈, then
u(t)≤ u∞≤ d
√
e
, t∈R,u∈. (.)
Therefore, from (.), (.), and (.), we have
R
–R
eQ(t)W
t,u(t)dt≥ t+ε
t–ε
eQ(t)W
t,u(t)dt≥εγ foru∈. (.)
By (.) and (.), we have
u(t)≤
m
i=
|λi|ui(t)≤δ for|t| ≥R,u∈. (.)
By (.), (.), (.), (.), (.), and Lemma ., we have foru∈andr>
ϕ(ru) = r
˙u
+
rp pu
p p,a+
Re
Q(t)W
t,ru(t)–W
t,ru(t) dt
≤ r
˙u
+
rp
pu
p p,a+r
Re
Q(t)W
t,u(t)dt–rμ
Re
Q(t)W
t,u(t)dt
= r
˙u
+
rp
pu
p p,a+r
R\(–R,R)
eQ(t)W
t,u(t)dt
–rμ
R\(–R,R)
eQ(t)W
t,u(t)dt+r R
–R
eQ(t)W
t,u(t)dt
–rμ R
–R
eQ(t)W
t,u(t)dt
≤ r
˙u
+
rp pu
p p,a–r
R\(–R,R)
eQ(t)Wt,u(t)dt
–rμ R
–R
eQ(t)W
t,u(t)dt+r R
–R
eQ(t)W
≤ r
˙u
+
rp
pu
p p,a+
r
p
R\(–R,R)
a(t)eQ(t)u(t)pdt
+rCu+C
– εγrμ
≤ r
˙u
+
rp pu
p p,a+
r
pu
p p,a+r
Cu+C
– εγrμ
≤ r
d
+rp
pd
p+ r
pd
p+C
(rd)+Cr– εγrμ. (.)
Sinceμ>>p> , we deduce that there existsr=r(d,d,C,C,R,R,ε,γ) =r(E) >
such that
ϕ(ru) < foru∈andr≥r.
It follows that
ϕ(u) < foru∈Eandu ≥dr,
which shows that (iii) of Lemma . holds. By Lemma ., ϕ possesses an unbounded sequence {cn}∞n= of critical values withcn=ϕ(un), whereun is such thatϕ(un) = for
n= , , . . . . If{un}is bounded, then there existsC> such that
un ≤C forn∈N. (.)
In a similar fashion to the proof of (.) and (.), for the given δin (.), there exists
R>Rsuch that
un(t)≤δ for|t| ≥R,n∈N. (.)
Hence, by (.), (.), (.), (.), and (.), we have
˙un
+
pun
p p,a
=cn+
Re
Q(t)Wt,u n(t)
dt
=cn+
R\[–R,R]
eQ(t)Wt,un(t)
dt+
R
–R
eQ(t)Wt,un(t)
dt
≥cn–
p
R\[–R,R]
a(t)eQ(t)un(t)pdt–
R
–R
eQ(t)Wt,un(t)dt
≥cn–
pun
p p,a–
R
–R
eQ(t) max |x|≤√eC
W(t,x)dt,
which, together with (.), implies that
cn≤
˙un
+
pun
p p,a+
R
–R
max |x|≤√eC
eQ(t)W(t,x)dt< +∞.
This contradicts the fact that{cn}∞n=is unbounded, and so{un}is unbounded. The proof
Proof of Theorem. In view of the proofs of Theorem . and Theorem ., the conclu-sion of Theorem . holds. The proof is complete.
4 Examples
Example . Consider the following system:
¨
u(t) +tu˙(t) –a(t)u(t)/u(t) +∇Wt,u(t)= , a.e.t∈R, (.)
whereq(t) =t,p= /,t∈R,u∈RN,a∈C(R, (,∞)), andasatisfies (A). Let
W(t,x) =a(t)
m
i=
ai|x|μi– n
j=
bj|x|j
,
whereμ>μ>· · ·>μm>>>· · ·>n> /,ai,bj> ,i= , . . . ,m,j= , . . . ,n. Let
W(t,x) =a(t) m
i=
ai|x|μi, W(t,x) =a(t) n
j=
bj|x|j.
Then it is easy to check that all the conditions of Theorem . are satisfied withμ=μm
and=. Hence, problem (.) has an unbounded sequence of fast homoclinic solutions. Example . Consider the following system:
¨
u(t) +t+tu˙(t) –a(t)u(t)u(t) +∇Wt,u(t)= , a.e.t∈R, (.)
whereq(t) =t+t,p= ,t∈R,u∈RN,a∈C(R, (,∞)), andasatisfies (A). Let
W(t,x) =a(t)a|x|μ+a|x|μ–b
(cost)|x|–b|x| ,
whereμ>μ>>> ,a,a> ,b,b> . Let
W(t,x) =a(t)
a|x|μ+a|x|μ, W
(t,x) =a(t)
b(cost)|x|+b|x| .
Then it is easy to check that all the conditions of Theorem . are satisfied withμ=μ
and=. Hence, by Theorem ., problem (.) has an unbounded sequence of fast
homoclinic solutions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1College of Science, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China.2College of Science, Hunan
University of Technology, Zhuzhou, Hunan 412000, P.R. China.3School of Mathematical Sciences and Computing
Acknowledgements
Qiongfen Zhang was supported by the NNSF of China (No. 11301108), Guangxi Natural Science Foundation (No. 2013GXNSFBA019004) and the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093). Qi-Ming Zhang was supported by the NNSF of China (No. 11201138). Xianhua Tang was supported by the NNSF of China (No. 11171351).
Received: 3 January 2014 Accepted: 23 April 2014 Published:06 May 2014
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10.1186/1687-2770-2014-89
