Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded (ϕ1,ϕ2) Laplacian

R E S E A R C H

Open Access

Multiple periodic solutions for a class of

nonlinear difference systems with classical or

bounded

(

φ

_

,

φ

_

)

-Laplacian

Yun Wang and Xingyong Zhang

*

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

Abstract

In this paper, we consider the multiplicity of periodic solutions for a class of difference systems involving the (

φ

1,

φ

2)-Laplacian in the cases when the gradient of the

nonlinearity has a sublinear growth. By using the variational method, some existence results are obtained. Our results generalize some recent results in (Mawhin in Discrete Contin. Dyn. Syst. 6:1065-1076, 2013).

Keywords: difference systems; periodic solutions; critical point theorem; variational method

1 Introduction and main results

LetRdenote the real numbers andZthe integers. Givena<binZ. LetZ[a,b] ={a,a+ , . . . ,b}. LetT>  andNbe fixed positive integers.

In this paper, we investigate the multiplicity of periodic solutions for the following non-linear difference systems:

⎧ ⎨ ⎩

φ(u(t– )) =∇uF(t,u(t),u(t)) +h(t),

φ(u(t– )) =∇uF(t,u(t),u(t)) +h(t),

(.)

whereF:Z×RN_×RN_{→Randφm,m= , , satisfy the following condition:}

(A) φmis a homeomorphism fromRN ontoBa⊂RN (a∈(, +∞]), such thatφm() = , φm=∇m, withm∈C(RN, [, +∞])strictly convex andm() = ,m= , .

Remark . Assumption (A) is given in [], which is used to characterize the classical homeomorphism and the bounded homeomorphism.φmis called classical whena= +∞

and bounded whena< +∞. If furthermorem:RN→Ris coercive (i.e.m(x)→+∞as

|x| → ∞), there existsδm>  such that

m(x)≥δm

|x|– , x∈RN, (.)

whereδm=min|x|=m(x),m= ,  (see []).

It is well known that the variational method has been an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contribu-tions have been obtained (for example, see [–]). However, to the best of our knowl-edge, few people investigated system (.). Recently, in [] and [], by using the variational approach, Mawhin investigated the following second order nonlinear difference systems withφ-Laplacian:

φu(n– )=∇uF

n,u(n)+h(n) (n∈Z), (.)

whereφ=∇,strictly convex, is a homeomorphism ofRN _{onto the ballB}

a⊂RN or

ofBa ontoRN. By using the variational approach, under different conditions, the author

found that system (.) has at least one orN+  geometrically distinctT-periodic solu-tions. It is interesting that Mawhin considered three kinds ofφ: ()φ:RN_→RN _{is a}

clas-sical homeomorphism, for example,φ(x) =|x|p–xfor somep>  and allx∈RN; ()φ:

RN_→B

a(a< +∞) is a bounded homeomorphism, for example,φ(x) = √x

+|x| ∈Bfor all

x∈RN_{; ()φ:B}

a⊂RN→RN is a singular homeomorphism, for example,φ(x) =√x

–|x|

for allx∈B.

For a classical and bounded homeomorphism, in [], Mawhin obtained the following multiplicity results.

Theorem A(see [], Theorem .) Assume that the following assumptions hold: (HB) φis a homeomorphism fromRN ontoRN,such thatφ() = ,φ=∇,with

∈C_(RN_{, [, +∞])strictly convex and() = .}

(HF) F∈C(Z×RN_{,R),F(n,·)∈C}_(RN_{,R),and there exist an integerT> and real}

numbersω> ,ω> , . . . ,ωN> such that

F(t+T,u+ω,u+ω, . . . ,uN+ωN) =F(t,u,u, . . . ,uN)

for allt∈Randu= (u,u, . . . ,uN)∈RN. If there existγ > and p> such that

(u) ≥γ|u|p u∈RN.

Then,for any h∈HT such that _T

T

t=h(t) =  (the definition of HTcan be seen in[]), system(.)has at least N+ geometrically distinct T -periodic solutions.

Theorem B(see [], Theorem .) Assume that assumption(HF)and the following con-dition hold:

(HB) φ is a homeomorphism fromRN _ontoB

a⊂RN (a∈(, +∞)),such thatφ() = , φ=∇,with∈C_(RN_{, [, +∞])strictly convex and() = .}

If:RN_{→Ris coercive,h∈H}

Tsuch that _T

T

t=h(t) = and|H|∞<δ,system(.)has

at least N+ geometrically distinct T -periodic solutions,whereδ> is given by()in[]

Obviously, (HF) implies thatF is periodic on all variablesu, . . . ,uN. Hence, a natural

question is that what will occur ifFis periodic on some of variablesu, . . . ,uN. For

differ-ential systems, in [] and [], the arguments on this question have been given. In [], Tang and Wu considered the second order Hamiltonian system

⎧ ⎨ ⎩ ¨

u(t) +∇F(t,u(t)) =e(t), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = , (.)

and in [], Zhang and Tang generalized and improved the results in []. They considered the following ordinaryp-Laplacian system:

⎧ ⎨ ⎩

(|u(t)|p–_{u(t)) +∇F(t,u(t)) =e(t), a.e.t∈[,T],}

u() –u(T) =u˙() –u˙(T) = . (.)

Inspired by [, , ] and [], in this paper, we investigate system (.), which is dif-ferent from (.), and consider the case thatF(t,x,x) is periodic on some of the vari-ables x()_ , . . . ,x()_N and some of the variables x()_ , . . . ,x()_N , where x = (x() , . . . ,x

()

N)τ and x= (x() , . . . ,x

()

N)

τ_{. We generalize Theorem A and Theorem B.}

Next, in order to present our main results, we consider two decompositionsRN_=R

⊕

SandRN=R⊕Swith

R=spanei, . . . ,eir, S=spaneir_+, . . . ,eiN,

R=spanej, . . . ,ejr, S=spanejr_+, . . . ,ejN,

whereeik andejs are the canonical basis ofR

N _{for ≤k≤N, ≤s≤N, ≤r}

≤N, and ≤r≤N.

In this paper, we make the following assumptions:

(A) Letp> ,q> ,β∈[,p),andβ∈[,q).Assume that there exist positive constants

γ,γ,γ,γsuch that

(x)≥γ|x|p–γ|x|β, (y)≥γ|y|q–γ|y|β, ∀x,y∈RN.

(A) There exist positive constantsd,d,d,dwithd> _p and d> _q,β∈[,p),and

β∈[,q)such that

φ(x),x≥d|x|p–d|x|β_{, φ(x),x}≥_d|_x|q_–d|_x|β_, ∀_x∈RN_.

(A) There exist constantscm> ,km> ,km> ,α∈[,p– ),α∈[,q– ),and two nonnegative functionswm∈C([, +∞), [, +∞)),wherem= , ,with the properties:

(i) wm(s)≤wm(t)∀s≤t,s,t∈[, +∞),

(ii) wm(s+t)≤cm(wm(s) +wm(t))∀s,t∈[, +∞),

(F) F:Z×RN_×RN _−→RN_{,(t,x,x)−→F(t,x,x)isT-periodic intfor all(x,x)∈} RN_×RN _{and continuously differentiable in(x,x)for everyt∈Z[,T],wherex}

= (x()_ , . . . ,x()_N)τ_,x

= (x() , . . . ,x ()

N )τ.

(F) F(t,x,x)isTi()k -periodic inx ()

ik,wherex ()

ik is a component of vectorx andT ()

ik > , ≤k≤r,andT_j()_s -periodic inx()_js ,wherex()_js is a component of vectorxandTj()s > , ≤s≤r.

(F) There existfm,gm:Z[,T]→R,m= , ,such that

_∇xF(t,x,x) ≤f(t)w

|x|+g(t), _∇xF(t,x,x) ≤f(t)w

|x|+g(t)

for all(x,x)∈RN×RN andt∈Z[,T]. (E)

T

t=

h(t) =

T

t=

h(t) = .

Remark . A condition similar to (A) and (F) was given first in [] for the second order Hamiltonian systems

⎧ ⎨ ⎩ ¨

u(t) =∇F(t,u(t)),

u() –u(T) =u˙() –u˙(T). (.)

The condition presented some advantages over the following subquadratic condition: there existα∈[, ) andf,g∈L_([,T];RN_{) such that}

_∇F(t,x) ≤f(t)|x|α+g(t).

We refer readers to [] for more details.

Moreover, assume thatp >  andq >  satisfying /p+ /p =  and /q+ /q = . Let

Cp=min

(T– )(p+)/p

T ,

(T+ )p+_{– }

Tp_{(p + )}

/p

, (.)

Cq=min

(T– )(q+)/q

T ,

(T+ )q+_{– }

Tq_{(q + )}

/q

, (.)

Cp,p=min

(T– )p–

Tp– ,

Tp–_(p,p) (p + )p/p

, (.)

Cq,q=min

(T– )q–

Tq– ,

Tq–_(q,q) (q + )q/q

, (.)

p,p=

T

t=

t T

p+ +

 – t

T +



T

p+

– 

Tp+ p/p

,

q,q=

T

t=

t T

q+ +

 – t

T +



T

q+

– 

Tq+ q/q

Next, we present our main results.

(I) For classical homeomorphism

Theorem . Assume that(A)with a= +∞, (A), (A), (F)-(F),and(E)hold. As-sume that F satisfies the following condition:

(F) For(x,x)∈S×S,

lim

|x|+|x|→+∞

T

t=F(t,x,x)

wp_(|x|) +wq_(|x|)> max

[cC(p)]p [pγ]p–p

_T

t=

f(t) p

,

[cC(q)]q [qγ]q–_q

_T

t=

f(t) q

.

Then system(.)has at least r+r+ geometrically distinct solutions inH,where the

definition ofHis given in Sectionbelow.

Theorem . Assume that(A)with a= +∞, (A), (A), (A), (F)-(F),and(E)hold.

Assume that F satisfies the following condition:

(A) Letθ∈[,p)andθ∈[,q).Assume that there exist positive constantsζ,ζ,ζ,ζ such that

(x)≤ζ|x|p+ζ|x|θ_{, (y)}≤_ζ|_y|q_+ζ|_y|θ_, ∀_x,y∈RN_;

(F) For(x,x)∈S×S,

lim

|x|+|x|→+∞

T

t=F(t,x,x)

wp_ (|x|) +wq_(|x|)

< –max

[C(p)c]p p

 +pζ

dp– 

+  +qζ

dq–  + 

T

t=

f(t) p

,

[C(q)c]q q

 +pζ

dp– 

+ +qζ

dq–  + 

T

t=

f(t) q

.

Then system(.)has at least r+r+ geometrically distinct solutions inH.

(II) For bounded homeomorphism

Theorem . Assume that(A)with a< +∞,m:RN →Rare coercive,m= , , (F),

(F),and(E)hold.Assume that F satisfies the following conditions:

(F) There exists a nonnegativebm:Z[,T]→R+,m= , ,such that

_∇xF(t,x,x) ≤b(t), _∇xF(t,x,x) ≤b(t)

(F) For(x,x)∈S×S,

lim

|x|+|x|→+∞

T

t=

F(t,x,x) = +∞;

(F)

T

t=

b(t) +

T

t=

h(t) <

δ

C(p),

T

t=

b(t) +

T

t=

h(t) < δ

C(q),

whereδm,m= , are given in(.).Then system(.)has at leastr+r+  geomet-rically distinct solutions inH.

Theorem . Assume that(A)with a< +∞,m:RN →Rare coercive,m= , , (F),

(F), (F),and(E)hold.If F satisfies the following conditions:

(F) For(x,x)∈S×S,

lim

|x|+|x|→+∞

T

t=

F(t,x,x) = –∞;

(F)

Cp T

t=

b(t) +Cp T

t=

h(t) +Cp,p+ /p<δ,

Cq T

t=

b(t) +Cq T

t=

_{h(t) +}

Cq,q+ /q<δ,

whereδm,m= , are given in(.),then system(.)has at leastr+r+  geometri-cally distinct solutions inH.

2 Preliminaries

First, we present some basic notations. We use| · |to denote the usual Euclidean norm inRN_{. Define}

V=u= (u,u)τ=

u(t)|u(t) =u(t),u(t) τ

∈RN_,

um=

um(t)

,um(t)∈RN,m= , ,t∈Z

.

His defined as a subspace ofVby

H=u=u(t)∈V|u(t+T) =u(t),t∈Z.

Define

Hm=

um=

um(t)

|um(t+T) =um(t),um(t)∈RN,t∈Z

ThenH=H×H. Forum∈Hm, set

umr=

_T

t= um(t)

r

/r

and um∞= max t∈Z[,T]

um(t) , m= , ,r> .

Obviously, we have

um∞≤ um, m= , . (.)

For  <p,q< +∞, onH, we define

up=

_T

t=

u(t) p+

T

t= u(t) p

/p

and, onH, we define

uq=

_T

t=

u(t) q+

T

t= _u(t) q

/q

.

Foru= (u,u)τ∈H, we define

u=up+uq.

Let

W=

u= (u,u)τ ∈H um() =· · ·=um(T) =



T T

t=

um(t),m= , 

and

˜

H=

u= (u,u)τ ∈H

T

t=

um(t) = ,m= , 

.

ThenHcan be decomposed into the direct sumH=W⊕ ˜H. So, for anyu∈H,ucan be expressed in the formu=u˜+u¯, whereu˜= (u˜,u˜)τ_{∈Vandu¯= (u¯,u¯)}τ_{∈W. Obviously,} um=u˜m+u¯m,m= , .

Foru= (u,u)τ∈ ˜H, let

umr=

_T

t=

um(t) r

/r

,

wherem= , ,r> . It is easy to verify that

u=up+uq

Lemma .(see []) Let u= (u,u)∈ ˜H.Then

Lemma . Let L:Z[,T]×RN_×RN_×RN_×RN_{−→R, (t,x,x,y,y)−→L(t,x,x,y,} y)and assume that L is continuously differential in(x,x,y,y)for all t∈Z[,T].Then the functionϕ:H→Rdefined by

ϕ(u) =ϕ(u,u) =

T

t=

Lt,u(t),u(t),u(t),u(t)

is continuously differentiable onHand

ϕ(u),v=ϕ(u,u), (v,v)

=

T

t=

DxL

t,u(t),u(t),u(t),u(t),v(t)

+DyL

t,u(t),u(t),u(t),u(t),v(t)

+DxL

t,u(t),u(t),u(t),u(t)

,v(t)

+DyL

t,u(t),u(t),u(t),u(t),v(t),

where u,v∈H.

Proof DefineG: [–, ]×Z[,T]→R, (λ,t)→G(λ,t) by

G(λ,t) =Lt,u(t) +λv(t),u(t) +λv(t),u(t) +λv(t),u(t) +λv(t).

SinceLis continuously differential in (x,x,y,y) for allt∈Z[,T],G(λ,t) is differential inλand

G(λ,t)

=DxL

t,u(t) +λv(t),u(t) +λv(t),u(t) +λv(t),u(t) +λv(t),v(t)

+DxL

t,u(t) +λv(t),u(t) +λv(t),

u(t) +λv(t),u(t) +λv(t),v(t)

+DyL

t,u(t) +λv(t),u(t) +λv(t),

u(t) +λv(t),u(t) +λv(t),v(t)

+DyL

t,u(t) +λv(t),u(t) +λv(t),

u(t) +λv(t),u(t) +λv(t),v(t).

Hence,

ϕ(u),v= lim λ→

ϕ(u+λv) –ϕ(u)

λ

= lim λ→

T

t=G(λ,t) – T

t=G(,t)

λ

=

T

t=

lim λ→

G(λ,t) –G(,t)

=

T

t=

G(,t)

=

T

t=

DxL

t,u(t),u(t),u(t),u(t),v(t)

+DyL

t,u(t),u(t),u(t),u(t)

,v(t)

+DxL

t,u(t),u(t),u(t),u(t)

,v(t)

+DyL

t,u(t),u(t),u(t),u(t)

,v(t)

.

The proof is complete.

Let

L(t,x,x,y,y) =(y) +(y) +F(t,x,x) +

h(t),x

+h(t),x

.

Then

ϕ(u) =ϕ(u,u)

=

T

t=



u(t)

+

u(t)

+Ft,u(t),u(t)

+h(t),u(t)+h(t),u(t). (.)

It follows from (A), (F), and Lemma . that

ϕ(u),v=ϕ(u,u), (v,v)

=

T

t=

φ

u(t),v(t)+φ

u(t),v(t)

+∇uF

t,u(t),u(t)

,v(t)

+∇uF

t,u(t),u(t)

,v(t)

+h(t),v(t)+h(t),v(t), ∀u,v∈H. (.)

By Lemma ., it is easy to see that the critical points ofϕinHare periodic solutions of system (.).

Next, we recall a definition. LetGbe a discrete subgroup of a Banach spaceXand let

π:X→X/Gbe the canonical surjection. A subsetAofXisG-invariant ifπ–_{(π(A)) =A.} A functionf defined onXisG-invariant iff(u+g) =f(u) for everyu∈Xand everyg∈G

(see []).

Definition .(see [], Definition .) AG-invariant differentiable functionalϕ:X→R

satisfies the (PS)G condition, if for every sequence{uk}inXsuch thatϕ(uk) is bounded

andϕ(uk)→, the sequence{π(uk)}contains a convergent subsequence.

Lemma .(see [], Theorem .) Letϕ∈C_{(x,R)be a G-invariant functional}

satis-fying the(PS)Gcondition.Ifϕis bounded from below and if the dimension N of the space generated by G is finite,thenϕhas at least N+ critical orbits.

Lemma .(see []) Let X be a Banach space and have a decomposition:X=Y+Z where Y and Z are two subspaces of X withdimY< +∞.Let V be a finite-dimensional,compact C_{-manifold without boundary.Let f :X×V→Rbe a C}_{-function and satisfy the(PS)}

condition.Suppose that f satisfies

inf

u∈Z×Vf(u)≥a, _u∈sup_S×Vf(u)≤b<a,

where S=∂D,D={u∈Y|u ≤R},R,a,and b are constants.Then the function f has at least cuplength(V) + critical points.

Let

ˆ

um(t) =Pmu¯m+Qmu¯m+u˜m, m= , ,

where

Pu¯=

N

k=r+

(¯u,eik)eik, Qu¯=

r

k=

(u¯,eik) –mikTik

eik,

Pu¯=

N

s=r+

(¯u,ejs)ejs, Qu¯=

r

s=

(¯u,ejs) –mjsTjs

ejs,

andmik,mjs are the unique integers such that

≤(u¯,eik) –mikTik<Tik, ≤k≤r,

≤(u¯,ejs) –mjsTjs<Tjs, ≤s≤r.

Let

G=

g=

g

g

∈RN×RN

g

g

= r

k=mikTikeik r

s=mjsTjsejs

,

mikandmjsare integers, ≤k≤r, ≤s≤r

. (.)

LetZ=H˜,Y=S×S,X=Y+Z, andV is the quotient space (R×R)/Gwhich is isomorphic to the torusTr+r_{. Now define:X}×_Tr+r→R_by

y+z(t),v=ϕy+v+z(t), ∀(y,z,v)∈Y×Z×Tr+r_{, (.)}

that is,

y+z(t),v=ϕy+v+z(t),y+v+z(t)

where

y= (y,y)τ∈Y, v= (v,v)τ∈V, z=z(t) =z(t),z(t)τ∈Z,

y+v+z(t) =y+v+z(t),y+v+z(t) τ

.

It is easy to verify thatis continuously differentiable and that

y[]+z[](t),v[],y[]+z[](t),v[]

=ϕy[]+v[]+z[](t),y[]+v[]+z[](t)

=ϕy[]_ +v[]_ +z_[](t),y[]_ +v[]_ +z[]_ (t),

y[] +v[] +z[] (t),y[] +v[] +z[] (t)

,

∀y[m],z[m],v[m]∈Y×Z×Tr+r_{,m= , . (.)}

Then (F) implies that

F(t+T,x+g,x+g) =F(t,x,x), ∀t∈Zand∀g∈G.

Hence, we have

Ft,u(t),u(t)=F

t,uˆ(t) +

r

k=

mikTikeik,uˆ(t) +

r

s=

mjsTjsejs

=Ft,uˆ(t),uˆ(t), (.)

∇Ft,u(t),u(t)

=∇F

t,uˆ(t) +

r

k=

mikTikeik,uˆ(t) +

r

s=

mjsTjsejs

=∇Ft,uˆ(t),uˆ(t)

(.)

and, by (E), we have

T

t=

h(t),u(t)=

T

t=

h(t),uˆ(t) +

r

k=

mikTikeik

=

T

t=

h(t),uˆ(t), (.)

T

t=

h(t),u(t)=

T

t=

h(t),uˆ(t) +

r

k=

mikTikeik

=

T

t=

h(t),uˆ(t). (.)

3 Proofs

For the sake of convenience, we denote byCijandDij,i= , ,j= , , . . . ,  below the

vari-ous positive constants, byCij(ε) andDij(ε),i= , ,j= , , . . . ,  below the various positive

constants depending onεand

–C+

wp_|Pu¯|

+wq_|Pu¯|

F(t,Pu¯,Pu¯)

wp_ (|Pu¯|) +wq(|Pu¯|)

–max

Mp_cp_ap_/pC(p)

p ,

Mq_cq_aq_/qC(q)

q

–Cp

_T

t=

u(t)

p



pT

t= h(t)

–Cq

_T

t=

u(t) q 

qT

t=

h(t) . (.)

It follows from (.), (.),a>C_pγ(p_), anda>C_qγ(q_)thatϕis bounded from below. LetGbe a discrete subgroup ofHdefined by (.) and letπ:H→H/Gbe the canonical surjection. By (.)-(.), it is easy to verify thatϕisG-invariant. In what follows, we show that the functionalϕsatisfies the (PS)Gcondition, that is, for every sequence{um}inHsuch that

{ϕ(un)}is bounded andϕ(un)→, the sequence{π(un)}has a convergent subsequence. In

fact, the boundedness ofϕ(un), (.), (.), and the facts thata>C_pγ(p_)anda>C_qγ(q_)imply that (Pu¯n) and

T

t=|un(t)|are bounded. Furthermore, by Lemma ., we know that (˜un)

is also bounded. Hence{ˆun}is bounded inH. SincedimH<∞, we know that{ˆun}has a

convergent subsequence. Sinceπ(un) =π(ˆun),{π(un)}also has a convergent subsequence.

Thus, by Lemma ., we know thatϕhasr+r+  critical orbits. Hence, system (.) has at leastr+r+  geometrically distinct solutions inH. The proof is complete.

Proof of Theorem . First, we prove that defined by (.) satisfies the (PS)

con-dition. Assume that {(y[n] _+z[n]_,v[n]_)}∞

n= ⊂X×Tr+r is (PS) sequence for , that is, {((y[n]_+z[n]_,v[n]_{))}is bounded and((y}[n]_+z[n]_,v[n]_{))→, wherey}[n]_{= (y}[n]

 ,y [n]  )τ∈Y,

z[n]_=z[n]_{(t) = (z}[n]  (t),z

[n]

 (t))τ∈Z,v[n]= (v [n]  ,v

[n]

 )τ∈Tr+rforn= , , . . . . Let

u[n]=y[n]+v[n]+z[n]=y[_n]+v_[n]+z[_n],y[_n]+v_[n]+z[_n]τ, n= ,  . . . .

Then it is easy to see that

y_m[n]=Pmu¯m[n], v[mn]=Qmu¯m[n], z[mn](t) =u˜[mn](t), m= , ,n= ,  . . . .

By (.) and (.), we find that{ϕ(u[_n],u[_n])}is bounded andϕ(u[_n],u[_n])→. Then there exists a positive constantDsuch that

ϕu[n],u [n]

 ≤D, ϕ

u[n],u [n]

 ≤D, ∀n∈R. (.)

By (F), there exista>C(p) anda>C(q) such that

lim

|x|+|x|→∞

F(t,x,x)

wp_ (|x|) +wq_(|x|)

< –max

[aMc]p

p

 +pζ

dp– 

+ +qζ

dq–  + 

,

[aMc]q

q

 +pζ

dp– 

+  +qζ

dq–  + 

–[aMc]

q

q w

q

 Pu¯[n]

–C(ε) _T

t=

u[_n](t) q /q

–C(ε) _T

t=

u[_n](t) q (α+)/q

–MC

p

_T

t=

u[_n](t) p /p

–MC

q

_T

t=

u[_n](t) q /q

(.)

for alln∈R. Moreover, by Lemma ., we have

Du˜[n]p+u˜

[n]  q

≤D

Cp,p+ /p _T

t=

u[_n](t) p /p

+D

Cq,q+ /q _T

t=

u[_n](t) q /q

. (.)

Then (.) and (.) imply that

[aMc]p

p w

p

 Pu¯[n] +

[aMc]q

q w

q

 Pu¯[n]

≥

d– 

p

T

t=

u[_n](t) p+

d– 

q

T

t=

u[_n](t) q+D, (.)

where

D= min

s∈[,+∞)



p–

[C(p)]p pap_

sp–dT–

β_

p_sβ_{–C(ε)s}α+

–C(ε) +MC

ps

+ min

s∈[,+∞)



q–

[C(q)]q qaq_

sp–dT–

β

q _sβ_{–C(ε)s}α+

–C(ε) +MC

qs

.

Note thata>C(p),a>C(q),α,β∈[,p), andα,β∈[,q). Hence (.) implies that there exist positive constantsDandDsuch that

T

t=

u[_n](t) p≤p[aMc]

p

(dp– )p

wp_ Pu¯[n]

+p[aMc]

q

(dp– )q

T

t=

u[_n](t) q≤q[aMc]

p

(dq– )p

wp_Pu¯[n]

+q[aMc]

q

(dq– )q

w_q Pu¯[n] +D. (.)

Then it is easy to see that –∞<D< . By (.), we know that

T

t=

Ft,u[_n](t),u[_n](t)–Ft,Pu¯[n],Pu¯[n]

≤C(p)

pa

T

t=

u[_n](t) p+M

p

c

p

a

p/p

 C(p)

p w

p

 Pu¯[n] +C

+C(ε)Cpα+

_T

t=

u[_n](t) p α+

p

+MMcw Pu¯[n]

+C(ε)

Cpα

_T

t=

u[_n](t) p α

p

+C(ε)Cp

_T

t=

u[_n](t) p 

p

+C(q)

qa

T

t=

u[_n](t) q+M

q

c

q

a

q/q

 C(q)

q w

q

 Pu¯[n] +C

+MMcw Pu¯[n] +C(ε)

Cqα+

_T

t=

u[_n](t) q α+

q

+C(ε)Cq

_T

t=

u[_n](t) q 

q

+C(ε)Cqα

_T

t=

u[n](t)

q

α

q

. (.)

By (A), (.), (.), (.), and Lemma ., we have

ϕu[n]

=ϕu[n],u [n] 

=

T

t=



u[_n](t)+

u[_n](t)+Ft,u_[n](t),u[_n](t)

+h(t),u[n](t)

+h(t),u[n](t)

=

T

t=



u[_n](t)+

u[_n](t)+Ft,u[_n](t),u_[n](t)–Ft,Pu¯[n],Pu¯[n]

+Ft,Pu¯[n],Pu¯[n]

+h(t),u[n](t)

+h(t),u[n](t)

≤

T

t=

ζ u[n](t)

p

+ζ u[n](t)

q

+ζ u[n](t)

θ

+ζ u[n](t)

Then (.) and (A) imply that {Pu¯[n]}, {Pu¯[n]}, {w(Pu¯[n])}, and {w(Pu¯[n])} are bounded. Furthermore, (.), (.), and (.) imply that{˜u[n]}and{˜u

[n]

 }are bounded. Then {u[n]_{} is bounded in H. SincedimH<∞, {u}[n]_{} has a convergent subsequence.} Hence,satisfies the (PS) condition.

In order to use Lemma ., next we prove the following conclusions: (i) inf{((z,v))|(z,v)∈Z×Tr+r}> –∞;

(ii) ((y,v))→–∞uniformly for(y,v)∈Y×Tr+r_as|_y| → ∞.

For (z,v)∈Z×Tr+r_{, setu=u(t) =z(t) +v= (z}

(t) +v,z(t) +v)τ. Thenzm(t) =u˜m(t), vm=Qmu¯m, andum(t) =zm(t) +vm,m= , . By (F) and Lemma ., we have

T

t=

Ft,u(t),u(t)–F(t, , )

≤

T

t=

Ft,u(t),u(t)

–Ft, ,u(t) +

T

t=

Ft, ,u(t)

–F(t, , )

≤

T

t= 



∇xF

t,sQu¯+u˜(t)

,u(t),Qu¯+u˜(t)

ds

+

T

t=

_∇xF

t, ,sQu¯+u˜(t)

,Qu¯+u˜(t)

ds

≤|Qu¯|+˜u∞ T

t= 

 _∇x

F

t,sQu¯+u˜(t)

,u(t) ds

+|Qu¯|+˜u∞

T

t= 



_∇xFt, ,sQu¯+u˜(t) ds

≤M+˜u∞

T

t= 



f(t)w s

Qu¯+u˜(t) +g(t)

ds

+M+˜u∞ T

t= 



f(t)w s

Qu¯+u˜(t) +g(t)

ds

≤M+˜u∞

T

t= 



kf(t) s

Qu¯+u˜(t)

α

+kf(t) +g(t)

ds

+M+˜u∞

T

t= 



kf(t) s

Qu¯+u˜(t)

α

+kf(t) +g(t)

ds

≤M+˜u∞ T

t=

kf(t) Qu¯+u˜(t)

α

+kf(t) +g(t)

+M+˜u∞

T

t=

kf(t) Qu¯+u˜(t)

α

+kf(t) +g(t)

≤DC

It is easy to see that conclusion (i) holds from (.).

com-+ Theorem ., we can prove thatϕisG-invariant and satisfies the (PS)Gcondition. Then

Proof of Theorem. First, we prove that defined by (.) satisfies the (PS) condi-tion. Assume that {(y[n]_+z[n]_,v[n]_)}∞

n= ⊂X×Tr+r is a (PS) sequence for , that is, {((y[n]_+z[n]_,v[n]_{))}is bounded and((y}[n]_+z[n]_,v[n]_{))→, wherey}[n]_{= (y}[n]

 ,y [n]  )τ∈Y,

z[n]_=z[n]_{(t) = (z}[n]  (t),z

[n]

 (t))τ∈Z,v[n]= (v [n]  ,v

[n]

 )τ∈Tr+rforn= , , . . . . Let

u[n]=y[n]+v[n]+z[n]=y[_n]+v_[n]+z[_n],y[_n]+v_[n]+z[_n]τ, n= , , . . . .

Then it is easy to see that

y_m[n]=Pmu¯m[n], v[mn]=Qmu¯m[n], z[mn](t) =u˜[mn](t), m= , ,n= , , . . . .

By (.) and (.), we find that{ϕ(u[_n],u[_n])}is bounded andϕ(u[_n],u[_n])→. Then there exists a positive constantGsuch that

ϕu[_n],u_[n] ≤G, ϕu[_n],u[_n]≤G, ∀n∈N. (.)

It follows from (F), Lemma ., and Young’s inequality that, for all (u,u)∈H,

T

t=

∇xF

t,uˆ(t),uˆ(t),u˜(t)+

T

t=

∇xF

t,uˆ(t),uˆ(t)

,u˜(t)

≤

T

t=

∇xF

t,Pu¯+Qu¯+u˜(t),uˆ(t)

,u˜(t)

+ T

t=

∇xF

t,uˆ(t),Pu¯+Qu¯+u˜(t)

,u˜(t)

≤

T

t=

b(t) u˜(t) +

T

t=

b(t) u˜(t)

≤ ˜u∞

T

t=

b(t) +˜u∞

T

t=

b(t)

≤Cp

_T

t=

u(t) p /_{p T}

t=

b(t)

+Cq

_T

t=

u(t) q /_{q T}

t=

b(t). (.)

Hence we have

_u˜[n]  p+u˜

[n]  q

≥ ϕu[_n],u[_n],u˜[_n],u˜[_n]

= T

t=

φ

u[_n](t),u[_n](t)+φ

u[_n](t),u[_n](t)

+∇xF

t,u[_n](t),u[_n](t),u˜[_n](t)+∇xF

+h(t),u˜[_n](t)+h(t),u˜[_n](t)

≥

T

t=

δ u[n](t) – 

+

T

t=

δ u[n](t) – 

–Cp

_T

t=

u[_n](t) p /_{p T}

t=

b(t) –C

q

_T

t=

u[_n](t) q /_{q T}

t=

b(t)

–MC

p

_T

t=

u[_n](t) p /p

–MC

q

_T

t=

u[_n](t) q /q

≥δ _T

t=

u[_n](t) p /p

+δ _T

t=

u[_n](t) q /q

– (δ+δ)T

–Cp

_T

t=

u[_n](t) p /_{p T}

t=

b(t) –C

q

_T

t=

u[_n](t) q /_{q T}

t=

b(t)

–MC

p

_T

t=

u[_n](t) p /p

–MC

q

_T

t=

u[_n](t) q /q

(.)

for largen∈Nby the factϕ(u[_n],u[_n])→ asn→ ∞. Moreover, by Lemma ., we have

u_˜[_n]_p+u˜[_n]_q

≤Cp,p+ /p _T

t=

u[_n](t) p /p

+Cq,q+ /q _T

t=

u[_n](t) q /q

. (.)

Then (F), (.), and (.) imply that there exists a positive constantGsuch that

T

t=

u[_n](t) p≤G,

T

t=

u[_n](t) q≤G, ∀n∈N. (.)

By (.) and the above inequality, we know that there exists a positive constantGsuch that

T

t=

Ft,u[_n](t),u[_n](t)–Ft,Pu¯[n],Pu¯[n]

≤MM+C

pM _T

t=

u[_n](t) p /p

+MM+C

qM _T

t=

u[_n](t) q /q

By (A) and (.), there exists a positive constantGsuch that



u[_n](t)≤G, 

u[_n](t)≤G. (.)

Then it follows from (.), (.), (.), (.), and Lemma . that

–G≤ϕ

u[n]

=ϕu[_n],u[_n]

=

T

t=



u[_n](t)+

u[_n](t)+Ft,u[_n](t),u_[n](t)–Ft,Pu¯[n],Pu¯[n]

+Ft,Pu¯[n],Pu¯[n]

+h(t),u[_n](t)+h(t),u[_n](t)

≤GT+G+

T

t=

Ft,Pu¯[n],Pu¯[n]

+M˜u∞+M˜u∞

≤GT+G+

T

t=

Ft,Pu¯[n],Pu¯[n]

+MC

pG/_p+MC

qG/_q. (.)

Then (F) implies that{Pu¯[n]}and{Pu¯[n]}are bounded. Then (.) implies that{u[n]} is bounded inH. SincedimH<∞,{u[n]_{}has a convergent subsequence. Hence,satisfies} the (PS) condition.

Next we prove the following conclusions: (i) inf{((z,v))|(z,v)∈Z×Tr+r}_{> –}∞_;

(ii) ((y,v))→–∞uniformly for(y,v)∈Y×Tr+r_as|_y| → ∞.

For (z,v)∈Z×Tr+r_{, setu=u(t) =z(t) +v= (z(t) +v,z(t) +v)}τ_{. Thenz}

m(t) =u˜m(t), vm=Qmu¯m, andum(t) =zm(t) +vm,m= , . By (F) and Lemma ., we have

T

t=

Ft,u(t),u(t)–F(t, , )

≤

T

t= _F

t,u(t),u(t)–Ft, ,u(t) +

T

t= _F

t, ,u(t)–F(t, , )

≤

T

t=

_∇xF

t,sQu¯+u˜(t)

,u(t),Qu¯+u˜(t)

ds

+

T

t=

_∇xF

t, ,sQu¯+u˜(t)

,Qu¯+u˜(t)

ds

≤|Qu¯|+˜u∞ T

t= 

 _∇x

F

t,sQu¯+u˜(t)

,uˆ(t) ds

+|Qu¯|+˜u∞

T

t= 



_∇xFt, ,sQu¯+u˜(t) ds

≤M+˜u∞ T

t=

b(t) +

M+˜u∞ T

t=