R E S E A R C H
Open Access
Ground state homoclinic orbits of damped
vibration problems
Guan-Wei Chen
*and Jian Wang
*Correspondence:
guanweic@163.com School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, P.R. China
Abstract
In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.
MSC: 49J40; 70H05
Keywords: non-periodic damped vibration problems; ground state homoclinic orbits; superquadratic nonlinearity
1 Introduction and main results
We shall study the existence of ground state homoclinic orbits for the following non-periodic damped vibration system:
¨
u(t) +Mu˙(t) –L(t)u(t) +Hut,u(t)= , t∈R, (.)
whereMis an antisymmetricN×Nconstant matrix,L(t)∈C(R,RN×N) is a symmetric matrix,H(t,u)∈C(R×RN,R) and Hu(t,u) denotes its gradient with respect to theu
variable. We say that a solutionu(t) of (.) is homoclinic (to ) ifu(t)∈C(R,RN) such that
u(t)→ andu˙(t)→ as|t| → ∞. Ifu(t)≡, thenu(t) is called a nontrivial homoclinic solution.
If M= (zero matrix), then (.) reduces to the following second-order Hamiltonian system:
¨
u(t) –L(t)u(t) +Hut,u(t)= , t∈R. (.)
This is a classical equation which can describe many mechanic systems such as a pen-dulum. In the past decades, the existence and multiplicity of periodic solutions and ho-moclinic orbits for (.) have been studied by many authors via variational methods; see [–] and the references therein.
The periodic assumptions are very important in the study of homoclinic orbits for (.) since periodicity is used to control the lack of compactness due to the fact that (.) is set on allR. However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [] introduced a type of coercive condition on the
matrixL(t),
l(t) :=inf
|u|=
L(t)u,u→+∞ as|t| → ∞, (.)
and obtained the existence of a homoclinic orbit for non-periodic (.) under the Ambrosetti-Rabinowitz (AR) superquadratic condition:
<μH(t,u)≤Hu(t,u),u, ∀t∈R,∀u∈RN\ {},
whereμ> is a constant, (·,·) denotes the standard inner product inRNand the associated
norm is denoted by| · |.
We should mention that in the case whereM= ,i.e., the damped vibration system (.), only a few authors have studied homoclinic orbits of (.); see [–]. Zhu [] considered theperiodiccase of (.) (i.e.,L(t) andH(t,u) areT-periodic intwithT> ) and obtained the existence of nontrivial homoclinic solutions of (.). The authors [– ] considered thenon-periodiccase of (.): Zhang and Yuan [] obtained the existence of at least one homoclinic orbit for (.) whenHsatisfies thesubquadraticcondition at infinity by using a standard minimizing argument; by a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [] obtained the existence and multiplicity of homoclinic orbits for (.) whenHsatisfies the local (AR)superquadratic
growth condition:
<μH(t,u)≤Hu(t,u),u, ∀t∈R,∀|u| ≥r, (.) whereμ> andr> are two constants. Notice that the authors [, ] all used condi-tion (.). Recently, the author in [, ] obtained infinitely many homoclinic orbits for (.) whenHsatisfies thesubquadratic[] andasymptotically quadratic[] condition at infinity by the followingweakerconditions than (.):
(L) There is a constantβ> such that
meast∈R:|t|–βL(t) <bIN< +∞, ∀b> ;
(L) There is a constantγ≥such that
l(t) := inf
|u|=
L(t)u,u≥–γ, ∀t∈R,
which were firstly used in []. It is not hard to check that the matrix-valued function
L(t) := (tsint+ )IN satisfies (L
) and (L), but does not satisfy (.). We define an operator:H(R,RN)→H(R,RN) by
(u,v) :=
R
Mu(t),v˙(t)dt, ∀u,v∈HR,RN.
SinceMis an antisymmetricN×Nconstant matrix,is self-adjoint onH(R,RN). Letχ
denote the self-adjoint extension of the operator –dtd +L(t) +. We are interested in the
(J) a:=sup(σ(χ)∩(–∞, )) < <b:=inf(σ(χ)∩(,∞)).
To state our main result, we still need the following assumptions:
(H) |∇H(t,u)| ≤c( +|u|p–)for somec> andp> ,∀t∈Randu∈RN.
(H) H(t,u)≥a|u|,∀t∈Randu∈RN.
(H) For someδ> andγ∈(,b),
∇H(t,u)≤γ|u|, H(t,u)≤
∇H(t,u)· |u|, ∀|u|<δ,∀t∈R.
(H) H|(ut|,u) →+∞as|u| →+∞and there existsW(t)∈L(R,R+)such that
H(t,u)≥–W(t), ∀t∈Randu∈RN. (.)
(H) For allt∈Randu,z∈RN, there holds
H(t,u+z) –H(t,u) –r∇H(t,u),z+(r– )
∇H(t,u),u
≥–W(t), ∀r∈[, ].
Our main results read as follows.
Theorem . If(L)-(L), (J)and(H)-(H)hold,then(.) has at least one nontrivial
homoclinic orbit.
Theorem . LetMbe the collection of solutions of(.),then there is a solution that minimizes the energy functional
I(u) =
R
u˙(t)+Mu(t),u˙(t)+L(t)u(t),u(t) dt–
RH(t,u)dt, u∈E
overM,where the space E is defined in Section.In addition,if
∇H(t,u)=o|u| as|u| →
uniformly in t,then there is a nontrivial homoclinic orbit that minimizes the energy func-tional overM\ {},i.e.,a ground state homoclinic orbit.
Remark . Although the authors [] have studied (.) withsuperquadratic nonlinear-ities, our superquadratic condition (H) is weaker than (.) in []. Moreover, we study theground statehomoclinic orbit of (.). To the best of our knowledge, there has been no result published concerning theground statehomoclinic orbit of (.).
Example .
() H(t,u) =|u|p,
() H(t,u) =g(t)(|u|p+ (p– )|u|p–εsin(|u|ε ε )),
The rest of the present paper is organized as follows. In Section , we establish the varia-tional framework associated with (.), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section , we give the detailed proofs of our main results.
2 Preliminary lemmas
In the following, we use · Lp to denote the norm ofLp(R,RN) for anyp∈[,∞]. Let
W:=H(R,RN) be a Hilbert space with the inner product and norm given respectively by
u,vW=
R
˙
u(t),˙v(t)+u(t),v(t) dt, uW=u,u/E , ∀u,v∈W.
It is well known thatWis continuously embedded inLp(R,RN) forp∈[,∞). We define
an operator:W→Wby (u,v) :=
R
Mu(t),v˙(t)dt, ∀u,v∈W.
SinceMis an antisymmetricN×N constant matrix,is self-adjoint onW. Moreover, we denote byχthe self-adjoint extension of the operator –dtd +L(t) +with the domain D(χ)⊂L(R,RN).
LetE:=D(|χ|/), the domain of|χ|/. We define respectively onEthe inner product and the norm
u,vE:=
|χ|/u,|χ|/v+ (u,v) and uE=u,u/E ,
where (·,·)denotes the inner product inL(R,RN).
By a similar proof of Lemma . in [], we can prove that if conditions (L) and (L) hold, then
Eis compactly embedded intoLpR,RN, ∀p∈[, +∞]. (.)
Therefore, it is easy to prove that the spectrum σ(χ) has a sequence of eigenvalues (counted with their multiplicities)
λ≤λ≤ · · · ≤λk≤ · · · → ∞,
and the corresponding system of eigenfunctions{ek:k∈N}(χek=λkek) forms an
orthog-onal basis inL(R,RN).
By (J), we may let
k:={j:λj< }, E–:=span{e, . . . ,ek}, E +:=cl
E
span{ek+, . . .}
.
Then one has the orthogonal decomposition
E=E–⊕E+
with respect to the inner product·,·E. Now, we introduce respectively onEthe following
new inner product and norm:
whereu,v∈E=E–⊕E+withu=u–+u+ andv=v–+v+. Clearly, the norms · and · Eare equivalent (see []), and the decompositionE=E–⊕E+is also orthogonal with
respect to both inner products ·,·and (·,·). Hence, by (J),E with equivalent norms, besides, we have
–u–=χu–,u–≤au–L, ∀u–∈E– (.)
and
u+=χu+,u+ ≥bu
+
L, ∀u+∈E+, (.)
whereaandbare defined in (J).
For problem (.), we consider the following functional:
I(u) =
R
u˙(t)
+Mu(t),u˙(t)+L(t)u(t),u(t) dt–
RH(t,u)dt, u∈E.
ThenIcan be rewritten as
I(u) = u
+ –
u –
–
RH(t,u)dt, u=u
–+u+∈E.
Let(u) :=RH(t,u)dt. In view of the assumptions ofH, we knowI,∈C(E,R) and the derivatives are given by
(u)v=
R
Hu(t,u),vdt, I(u)v=u+,v+–u–,v––I(u)v
for anyu,v∈E=E–⊕E+ withu=u–+u+ andv=v–+v+. By the discussion of [], the (weak) solutions of system (.) are the critical points of theCfunctionalI:E→R. Moreover, it is easy to verify that ifu≡ is a solution of (.), thenu(t)→ andu˙(t)→ as|t| → ∞(see Lemma . in []).
The following abstract critical point theorem plays an important role in proving our main result. LetEbe a Hilbert space with the norm · and have an orthogonal decom-positionE=N⊕N⊥,N⊂Eis a closed and separable subspace. There exists a norm|v|ω
satisfying|v|ω≤ vfor allv∈Nand inducing a topology equivalent to the weak topology
ofNon a bounded subset ofN. Foru=v+w∈E=N⊕N⊥withv∈N,w∈N⊥, we define |u|
ω=|v|ω+w. Particularly, ifun=vn+wnis · -bounded andun
|·|ω
→u, thenvnv
weakly inN,wn→wstrongly inN⊥,unv+wweakly inE(cf.[]). LetE:=E–⊕E+,z
∈E+withz= . LetN:=E–⊕RzandE+ :=N⊥= (E–⊕Rz)⊥. ForR> , let
Q:=u:=u–+sz:s∈R+,u–∈E–,u<R
withp=sz∈Q,s> . We define
D:=u:=sz+w+:s∈R,w+∈E+,sz+w+=s
ForI∈C(E,R), define := ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ h:
h: [, ]× ¯Q→Eis| · |ω-continuous; h(,u) =uandI(h(s,u))≤I(u) for allu∈ ¯Q;
For any (s,u)∈[, ]× ¯Q, there is a| · |ω-neighborhood U(s,u)s.t.{u–h(t,u) : (t,u)∈U(s,u)∩([, ]× ¯Q)} ⊂Efin
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ,
whereEfindenotes various finite-dimensional subspaces ofE,= sinceid∈. The variant weak linking theorem is as follows.
Lemma .([]) The family of C-functionals{Iλ}has the form
Iλ(u) :=λK(u) –J(u), ∀λ∈[,λ],
whereλ> .Assume that
(a) K(u)≥,∀u∈E,I=I;
(b) |J(u)|+K(u)→ ∞asu → ∞;
(c) Iλis| · |ω-upper semicontinuous,Iλ is weakly sequentially continuous onE.Moreover, Iλmaps bounded sets to bounded sets;
(d) sup∂QIλ<infDIλ,∀λ∈[,λ].
Then,for almost allλ∈[,λ],there exists a sequence{un}such that sup
n
un<∞, Iλ(un)→, Iλ(un)→cλ,
where cλ:=infh∈supu∈ ¯QIλ(h(t,u))∈[infDIλ,supQ¯Iλ].
In order to apply Lemma ., we shall prove a few lemmas. We pickλsuch that <λ<
min[,b/γ]. For ≤λ≤λ, we consider
Iλ(u) :=λ u
+ –
u
– +
RH
t,u(t)dt
:=λK(u) –J(u).
It is easy to see that Iλsatisfies condition (a) in Lemma .. To see (c), ifun→|·|ω uand
Iλ(un)≥c, thenu+
n→u+andu–nu–inE,un→ua.e. onR, going to a subsequence if
necessary. It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the factH(t,u) +W(t)≥ for allt∈Randu∈RN by (.) in (H) that
c≤lim sup n→∞ Iλ(un)
=lim sup n→∞
λ
u + n – u
– n + R
H(t,un) +W(t)
dt
+
RW(t)dt
≤λ u
+–lim inf
n→∞
u
– n + R
H(t,un) +W(t)
dt
+
RW(t)dt
≤λ u
+ –
u
– +
RH(t,u)dt
=Iλ(u).
Thus we getIλ(u)≥c. It implies thatIλis| · |ω-upper semicontinuous.Iλis weakly
Lemma . Under assumptions of Theorem.,then
J(u) +K(u)→ ∞ asu → ∞.
Proof By the definition ofI(u) and (H), we have
J(u) +K(u) = u
++ u
–+
RH
t,u(t)dt
≥ u
–
RW(t)dt→+∞ asu → ∞,
which is due toW(t)∈L(R,R+).
Therefore, Lemma . implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.
Lemma . Under assumptions of Theorem.,there are two positive constants,ρ>
such that
Iλ(u)≥, u∈E+,u=ρ,λ∈[,λ].
Proof By (H), (H), (.) and the Sobolev embedding theorem, for allu∈E+,
Iλ(u)≥ u
–
RH
t,u(t)dt
= u
–
{t∈R:|u|<δ}
Ht,u(t)dt–
{t∈R:|u|≥δ}
Ht,u(t)dt
≥ u
– γ
{t∈R:|u|<δ}|u| dt–c
{t∈R:|u|≥δ}
|u|p+|u|dt
≥ u
–γ
b
u
–Cup
= u
–γ
b – Cu p–
, ≤γ <b,
whereCis a positive constant. It implies the conclusion if we takeusufficiently small.
Lemma . Under assumptions of Theorem.,then there is an R> such that
Iλ(u)≤, u∈∂QR,λ∈[,λ],
where QR:={u:=v+sz:s≥,v∈E–,z∈E+withz= ,u ≤R}.
Proof Suppose by contradiction that there existRn→ ∞,λn∈[,λ] andun=vn+snz∈
∂QRsuch thatIλn(un) > . Ifsn= , then by (H) and (.), we have
Iλn(vn) = – vn
–
RH(t,vn)dt≤–
vn
– avn
Therefore,sn= andun=vn+s
n=Rn. Letun˜ =uunn=˜snz+˜vn, then
˜un=˜vn+˜sn= .
It follows fromIλn(un) > and the definition ofIthat
< Iλn(un) un
=
λn˜sn–˜vn
–
R
H(t,un) |un|
|˜un|dt
=
(λn+ )s˜n– –
R
H(t,un) |un| |˜un|
dt. (.)
There are renamed subsequences such that˜sn→ ˜s,λn→λ, and there is a renamed
sub-sequence such thatun˜ = un
un=˜snz+vn˜ u˜inEandun˜ → ˜ua.e. onR.
We claim that
lim inf n→∞
R
H(t,un) |un|
|˜un|dt≥. (.)
Case . Ifu˜ ≡. Letbe the subset ofRwhereu˜= , then for all t∈we have |un|=|˜un| · un → ∞. It follows from (H) andW(t)∈L(R,R+) that
R
H(t,un) |un| |˜un|
dt≥
H(t,un) |un| |˜un|
dt–
R\
W(t)
undt→+∞ asn→ ∞.
Case. Ifu˜≡, then by (H) andW(t)∈L(R,R+), we have
R
H(t,un)
|un| |˜un| dt=
R
H(t,un)
un dt≥–
R W(t)
undt→ asn→ ∞.
Therefore, Cases and imply that (.) holds. Therefore, by (.), (.) and the facts ˜
sn→ ˜s,λn→λ, we have
(λ+ )˜s– ≥, that is,˜s≥
+λ≥
+λ> . Thus,u˜= . It follows from (H) that
R
H(t,un) |un| |˜un|
dt→+∞ asn→ ∞,
which contradicts (.). The proof is finished.
Therefore, Lemmas . and . imply that condition (d) of Lemma . holds. Applying Lemma ., we soon obtain the following fact.
Lemma . Under assumptions of Theorem.,for almost allλ∈[,λ],there exists a
sequence{un}such that
sup n
un<∞, Iλ(un)→, Iλ(un)→cλ,
Lemma . Under assumptions of Theorem.,for almost allλ∈[,λ],there exists a
uλ∈E such that
Iλ(uλ) = , Iλ(uλ) =cλ.
Proof Let{un}be the sequence obtained in Lemma .. Since{un} is bounded, we can assumeunuλinEandun→uλa.e. onR. By (H), (H), (.) and Theorem A. in [], we have
R
∇H(t,un),undt→
R
∇H(t,uλ),uλdt (.)
and
RH(t,un)dt→
RH(t,uλ)dt. (.)
By Lemma . and the fact thatIλ is weakly sequentially continuous, we have
Iλ(uλ)ϕ= lim n→∞I
λ(un)ϕ= , ∀ϕ∈E.
That is,Iλ(uλ) = . By Lemma ., we have
Iλ(un) – I
λ(un)un=
R
∇H(t,un),un–H(t,un)
dt→cλ.
It follows from (.), (.) and the factIλ(uλ) = that
Iλ(uλ) =Iλ(uλ) – I
λ(uλ)uλ=
R
∇H(t,uλ),uλ–H(t,uλ)
dt=cλ.
The proof is finished.
Applying Lemma ., we soon obtain the following fact.
Lemma . Under assumptions of Theorem.,for everyλ∈[,λ],there are sequences {un} ⊂E andλn∈[,λ]withλn→λsuch that
Iλn(un) = , Iλn(un) =cλn.
Lemma . Under assumptions of Theorem.,then
R
H(t,u) –H(t,rw) +r∇H(t,u),w– +r
∇H(t,u),udt≤C,
where u∈E,w∈E+, ≤r≤and the constant C:=R|W(t)|dt does not depend on u,
w,r.
Lemma . The sequences given in Lemma.are bounded.
Proof Writeun=u+
n+u–n, whereu±n∈E±. Suppose that
un → ∞ asn→ ∞.
Let vn:= un
un, then v+n= u+n un, v–n =
u–n
un,vn=v+n+v–n= andv+n ≤. Thus v+
nv+inEandv+n→v+a.e. onR, after passing to a subsequence. Case. Ifv+≡. Let
be the subset ofRwherev+= . Thenv= and|un|=|vn| · un → ∞on. It follows from (H) andW(t)∈L(R,R+) that
R
H(t,un) |un| |vn|
dt≥
H(t,un) |un| |vn|
dt–
R\
W(t)
undt→+∞ asn→ ∞,
which together with Lemmas . and . andv±n →v±inLq(R,RN) for all ≤q≤ ∞(by (.)) implies that
≤ cλn un =
Iλn(un)
un =
λn
v +
n
–
v – n – R
H(t,un)
|un| |vn|
dt→–∞ asn→ ∞.
It is a contradiction.
Case. Ifv+≡. We claim that there is a constantCindependent ofu
nandλnsuch that Iλn
ru+n–Iλn(un)≤C, ∀r∈[, ]. (.)
Since
I
λn(un)ϕ= λn
u+n,ϕ+–
u–n,ϕ––
R
∇H(t,un),ϕdt= , ∀ϕ∈E,
it follows from the definition ofIthat
Iλn
ru+n–Iλn(un) = λn
r– u+n+ u
– n + R
H(t,un) –Ht,ru+ndt
+ λn
u+n,ϕ+–
u–n,ϕ––
R
∇H(t,un),ϕ
dt. (.)
Takeϕ:= (r+ )u–
n– (r– )u+n= (r+ )un– ru+nin (.), then it follows from Lemma .
that
Iλn
ru+n–Iλn(un) = –
r u
– n + R
H(t,un) –Ht,ru+n+r∇H(t,un),u+n
– +r
∇H(t,un),undt
≤C.
Thus (.) holds.
LetC> be a fixed constant and take
rn:= C un
Therefore, (.) implies that
Iλn
rnu+n
–Iλn(un)≤C.
It follows fromv+
n= u+
n
un and Lemma . that
Iλn
Cv+n
≤C. (.)
Note that Lemmas . and . and (H) imply that
≤ cλn un
=Iλn(un) un
=λn v
+
n
–
v –
n
–
RH(t,un)
un dt
≤λ v
+
n
–
v –
n
+
RW(t)dt un .
It follows from the fact
RW(t)dt
un → asn→ ∞due toW(t)∈L(R,R+) that
λ v
+
n
–
v –
n
+ε≥, ∀ε> (.)
for all sufficiently largen. We takeε=, by (.) andvn=v+n+v–n= , we have v+n≥
( +λ)
(.)
for all sufficiently largen. By (H) and (H), we have
RH
t,Cv+n
dt
≤ γC
{t∈R:|Cv+n|<δ} v+
n
dt+ c
{t∈R:|Cv+n|≥δ}
Cv+n+C p
v+n p
dt
≤ γC
{t∈R:|Cv+n|<δ}
v+ndt+CCp
{t∈R:|Cv+n|≥δ}
v+npdt. (.)
For all sufficiently largen, by (.) and (.), it follows fromλn→λandv+n→v+≡ in Lq(R,RN) for all ≤q≤ ∞(by (.)) that
Iλn
Cv+n
=
λnC v+n
–
RH
t,Cv+n
dt
≥ λnC
( +λ)
– γC
{t∈R:|Cv+n|<δ}
v+ndt
–CCp
{t∈R:|Cv+n|≥δ} v+npdt
→ λC ( +λ)
asn→ ∞.
This implies thatIλn(Cv+n)→ ∞asC→ ∞, contrary to (.).
3 Proofs of the main results
Proof of Theorem. From Lemma ., there are sequences <λn→ and{un} ⊂Esuch
thatIλn(un) = andIλn(un) =cλn. By Lemma ., we know that{un}is bounded inE. Thus we can assumeunuinE,un→ua.e. onR. Therefore,
Iλn(un)ϕ=λn
u+n,ϕ–u–n,ϕ–
R
∇H(t,un),ϕdt= , ∀ϕ∈E.
Hence, in the limit,
I(u)ϕ=u+,ϕ–u–,ϕ–
R
∇H(t,u),ϕdt= , ∀ϕ∈E.
ThusI(u) = . Note that
Iλn(un) – I
λn(un)un=
R
∇H(t,un),un–H(t,un)
dt=cλn≥c. (.)
Similar to (.) and (.), we know
R
∇H(t,un),un–H(t,un)
dt→
R
∇H(t,u),u–H(t,u)
dt asn→ ∞.
It follows fromI(u) = , (.) and Lemma . that
I(u) =I(u) – I
(u)u=
R
∇H(t,u),u–H(t,u)
dt
= lim n→∞
R
∇H(t,un),un–H(t,un)
dt≥c≥> .
Therefore,u= .
Proof of Theorem. By Theorem .,M=∅, whereMis the collection of solutions of (.). Let
α:= inf u∈MI(u).
Ifuis a solution of (.), then by Lemma . (taker= ),
I(u) =I(u) – I
(u)u=
R
∇H(t,u),u–H(t,u)
dt≥–C= –
R
W(t)dt.
Thusα> –∞. Let{un}be a sequence inMsuch that
I(un)→α. (.)
By Lemma ., the sequence{un}is bounded inE. Therefore,unuinE,un→ua.e. on Randun→uinLp(R,RN) for allp∈[, +∞] (by (.)), after passing to a subsequence.
Therefore,
I(un)ϕ=u+n,ϕ–u–n,ϕ–
R
Hence, in the limit,
I(u)ϕ=u+,ϕ–u–,ϕ–
R
∇H(t,u),ϕdt= , ∀ϕ∈E.
ThusI(u) = . Similar to (.) and (.), we have
I(un) –
I
(u
n)un =
R
∇H(t,un),un
–H(t,un)
dt
→
R
∇H(t,u),u–H(t,u)
dt asn→ ∞.
It follows fromI(u) = and (.) that
I(u) =I(u) – I
(u)u=
R
∇H(t,u),u–H(t,u)
dt
= lim n→∞
R
∇H(t,un),un–H(t,un)
dt
= lim
n→∞I(un) =α. Now suppose that
∇H(t,u)=o|u| as|u| →.
It follows from (H) that for anyε> , there is a constantCε> such that
∇H(t,u)≤ε|u|+Cε|u|p–. (.)
Let
β:= inf u∈MI(u),
whereM:=M\ {}. Let{un}be a sequence inM\ {}such that
I(un)→β. (.)
Note that
=I(un)un+=u+n–
R
∇H(t,un),u+ndt,
which together with (.), Hölder’s inequality and the Sobolev embedding theorem implies
u+
n
=
R
∇H(t,un),u+
n
dt
≤ε
T
|un| ·u+ndt+Cε
T
|un|p–u+ndt
≤εun ·un++Cεun p–
≤εun ·un++Cεun p–
Lp un ·u+n
≤εun+Cεun p–
Lp un. (.)
Similarly, we have
u–n≤εun+Cεun p–
Lp un
. (.)
From (.) and (.), we get
un≤εun+ Cεun p–
Lp un,
which meansunLp≥Cfor some constantC> . Sinceun→uinLp(R,RN), we know
u= . As before,I(un)→I(u) =βasn→ ∞.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by G-WC and G-WC prepared the manuscript initially and JW performed a part of steps of the proofs in this research. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).
Received: 6 January 2014 Accepted: 24 April 2014 Published:09 May 2014
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