Existence of weak quasi-periodic solutions for

Existence of weak quasi-periodic solutions for

a second order Hamiltonian system with damped term via a PDE approach

Xingyong Zhang

^B

and Liben Wang

Department of Mathematics, Faculty of Science, Kunming University of Science and Technology Kunming, Yunnan, 650500, P.R. China

Received 28 November 2015, appeared 27 November 2016 Communicated by Gabriele Bonanno

Abstract. In this paper, we investigate the existence of weak quasi-periodic solutions for the second order Hamiltonian system with damped term:

¨

u(t) +q(t)u˙(t) +DW(u(t)) =0, t∈_R, (HSD) whereu:R→_Rⁿ,q:R→_Ris a quasi-periodic function,W :Rⁿ→_Ris continuously differentiable, DW denotes the gradient of W, W(x) = −K(x) +F(x) +H(x) for all x∈Rⁿ andW is concave and satisfies the Lipschitz condition. Under some reasonable assumptions on q,K,F,H, we obtain that system has at least one weak quasi-periodic solution. Motivated by Berger et al. (1995) and Blot (2009), we transform the problem of seeking a weak quasi-periodic solution of system (HSD) into a problem of seeking a weak solution of some partial differential system. We construct the variational func- tional which corresponds to the partial differential system and then by using the least action principle, we obtain the partial differential system has at least one weak solution.

Moreover, we present two propositions which are related to the working space and the variational functional, respectively.

Keywords: weak quasi-periodic solution, second order Hamiltonian system, damped term, variational method, PDE approach.

2010 Mathematics Subject Classification: 37J45, 34C25, 70H05.

1 Introduction

Assume thatω = (ω₁, . . . ,ω_m)is a list of linearly independent real numbers over the rationals.

Definition 1.1([20]). u:R→_Rⁿis said to be quasi-periodic with m basic frequencies if there exists a function x → _Φ(x) ∈ _Rⁿ which is Lipschitz continuous for x ∈ _R^m and periodic of period 1 in each of its arguments, and m real numbers ω₁, . . . ,ω_m linearly independent over the rationals, such that u(t) = _Φ(ω₁t, . . . ,ω_mt).Any such choice of ω₁, . . . ,ω_m will be called a set of basic frequencies for u(t).

BCorresponding author. Email: zhangxingyong1@163.com

In this paper, we are concerned with the second order Hamiltonian system with damped term:

¨

u(t) +q(t)u˙(t) +DW(u(t)) =0, t ∈_R, (HSD) whereDW denotes the gradient ofW, q:R→_Ris a quasi-periodic function with module of frequencies generated byω= (ω₁, . . . ,ω_m), and satisfies the following condition:

(Q) q(t) =_∑^m_j=1q_j(t), where q_j(t)is continuous onRand _ω¹

j-periodic, j=1, . . . ,m;

andW : Rⁿ → _R, W(x) = −K(x) +F(x) +H(x) for all x ∈ _Rⁿ and satisfies the following condition:

(W) W ∈C¹(_Rⁿ,R), and there exists a positive constant L such that

|DW(x)−DW(y)| ≤ L|x−y|, for all x,y∈_Rⁿ.

?Next, we present the definition of weakω-quasi-periodic solution for system (HSD).

For the purpose, we need to recall some function spaces which can be seen in [15], [16], [6] and [7] for more details.

Define

AP⁰(_Rⁿ) ={u:R→_Rⁿ |uis Bohr almost periodic}

endowed with the norm kuk_∞ = sup_t∈R|u(t)|. Then AP⁰(_Rⁿ) is a Banach space. Let f ∈ L¹_loc(_R,Rⁿ), that is f is locally Lebesgue-integrable from R to Rⁿ. Then the mean value of f is the limit (when it exists) lim_T→_{∞ 2T}¹ RT

−T f(t)dt. A fundamental property of Bohr almost periodic function u is that such function has convergent mean, that is, the limit limT→_∞2T¹ RT

−Tu(t)dtexists. Whenu∈ AP⁰(_Rⁿ), define a(u,λ):= lim

T→_∞

1 2T

Z _T

−Te^−iλtu(t)dt

which is a complex vector and is called Fourier–Bohr coefficient ofu. Let Λ(u) ={λ∈_R|a(u,λ)6=0}

and Mod(u)_theZ-module generated by Λ(u)_{. Denote Z}hωiby the Z-module generated by ωinR.

Define

QP_ω⁰(_Rⁿ)_:={u|u∈ AP⁰(_Rⁿ)_{and Mod}(_u)⊂_Zhωi}. B²_ω(_Rⁿ)is the completion ofQP_ω⁰(_Rⁿ)with respect to the inner product

(u,v)_B2 := lim

T→_∞

1 2T

Z _T

−Tu(t)v(t)dt.

For u ∈ B²_ω(_Rⁿ), if limr→0 u(t+r)−u(t)

r exists, then define ∇u = limr→0 u(t+r)−u(t)

r and ∇²u =

∇(∇u). Define

B_ω^1,2(_Rⁿ):={u|u∈ B²_ω(_Rⁿ)and∇u∈ B²_ω(_Rⁿ)}

endowed with the inner product

(u,v)_B1,2 := (u,v)_B2 + (∇u,∇v)_B2. ThenB^1,2_ω (_Rⁿ)is a Hilbert space. Define

B^2,2_ω (_Rⁿ)_:={u|u∈ B^1,2_ω (_Rⁿ)_and∇²u∈ B²_ω(_Rⁿ)}_.

Definition 1.2. If u∈ B^2,2_ω (_Rⁿ)and satisfies

∇²u(t) +q(t)∇u(t) +DW(u(t)) =0, t∈_R.

Thenuis a weakω-quasi-periodic solution of system (HSD).

Hamiltonian system is a very important model in physics and it has also extensively ap- peared in other subjects such as life science, social science, bioengineering, space science and so on. Hence, the theory of Hamiltonian system has been focused on for a long time by math- ematicians and physicists. Especially, over the past 40 years, the existence and multiplicity of various solutions have attracted lots of mathematicians. Since Hamiltonian system pos- sesses the variational structure, variational method becomes a very effective tool to deal with those problems on the existence and multiplicity of solutions for Hamiltonian systems. There have been many contributions on periodic solutions, subharmonic solutions and homoclinic solutions (for example, see [10,14,17,21–25,30] and reference therein). For the investigation about almost periodic solutions of Hamiltonian system, there are less works. Joël Blot and co-authors made some important contributions and had a list of papers (see [1–7]). We refer the reader to [8,9,11–13,18,28,29] for some other known results. Next we only recall two works which have a direct relationship with our problem investigated in this paper.

In 1995, via a PDE approach and the least action principle, Berger and Zhang [11] inves- tigated the existence of quasi-periodic solutions of fixed frequencies for the nondissipative second order Duffing equation:

¨

u(t) +au(t)−bu³(t) = f(t)_,

whereu:R→_R,a>0,b>0 and f :R→_Ris a quasi-periodic function with frequenciesω.

For system case, in 2009, Blot [7] investigated the existence of ω-quasi-periodic solution for the second order Hamiltonian system without the damped term:

¨

u(t) +DW(u(t)) =e(t), t∈_R, (HS) via a PDE viewpoint which is partially similar to [11], whereu:R→_Rⁿ,W ∈ C²(_Rⁿ)and is concave, and

kD²Wk_∞ := sup

z∈_Rⁿ

|D²W(z)|<min 1

C∗, 1 C_∗²

, (1.1)

C∗ is defined by (2.3) below ande ∈Bω^1,2(_Rⁿ)which satisfies∑ν∈_Z^m|a(e,ν·ω)|²(1+|ν|²)<_∞.

In order to obtain the ω-quasi-periodic solution of system (HS), the author first investigated the existence of weak ω-quasi-periodic solution for system (HS) via a PDE approach. To be precise, the author transformed the problem into seeking a weak solution of the partial differential system:







∑m j=1

∑m i=1

ω_iω_j ∂²U

∂x_i∂x_j +DW(U(x)) =E(x)_{, onΩ} U=0, on∂Ω

(1.2) where Ω := (−π,π)^m ⊂ _R^m, U : R^m → _Rⁿ, E : R^m → _Rⁿ andE(tω) = e(t). Furthermore, in order to obtain weak solution of system (1.2), the author first investigated the existence of weak solution for the partial differential system:







∑m j=1

∑m i=1

ω_iω_j ∂²U

∂x_i∂x_j +¹ k

∑m j=1

∂²U

∂x²_j +DW(U(x)) =E(x), onΩ U=0, on ∂Ω

(1.3)

Then by careful analysis, ask →+∞, the sequence {U_k}which consists of weak solutions of system (1.3) converges to a weak solution of system (1.2).

Following ideas in [11] and [7], in this paper, when W satisfies some reasonable growth conditions, we investigate the existence of weakω-quasi-periodic solutions for system (HSD) via a similar PDE approach.

?Next, we transform the problem of seeking a weakω-quasi-periodic solution of (HSD) into a problem of seeking a weak solution of partial differential system (PDS*) below.

By (Q), define ˆq: R^m → _R by ˆq(x) =_∑^m_j=1qˆ_j(x_j), where ˆq_j :R → _Rdefined by ˆq_j(x_j):= q_j(_2πω^xj

j)which satisfies

ˆ

q_j(2πω_jt) =q_j

2πωjt 2πω_j

=q_j(t). It is easy to verify that ˆq_j is 2π-periodic.

Let Q_j(t) = Rt

0 q_j(t)dt. Then Q_j ∈ C¹(_R,R) and is _ω¹

j-periodic. Define ˆQ : R^m → _R by Qˆ(x) = _∑^m_j=1Q^ˆ_j(x_j), where ˆQ_j :R →_Rdefined by ˆQ_j(x_j):= Q_j(_2πω^xj

j)which is continuously differentiable and satisfies

Qˆ_j(2πω_jt) =Q_j

2πω_jt 2πω_j

= Q_j(t). Then we have

∂Qˆ(x)

∂x_j = ^dQˆj(x_j) dx_j =

dQ_j(_2πω^xj

j) dx_j =q_j

x_j 2πω_j

1

2πω_j = ¹

2πω_jqˆ_j(x_j) and

Qˆ(2πωt) =

∑

m j=1

Qˆ_j(_2πωjt) =

∑

m j=1

Q_j(t)_, which implies that

q_j(t) = ^dQj(t) dt = ^d

Qˆ_j(2πω_jt)

dt = ^d

Qˆ_j(x_j) dx_j ·^dxj

dt = ¹

2πω_jqˆ_j(x_j)·2πω_j = qˆ_j(x_j) ifx_j =_2πωjt,j=_{1, . . . ,}m.

Consider the second order elliptic partial differential system:

∑

m j=1

∑

m i=1

(2π)²ω_iω_j ∂²U

∂x_i∂x_j +qˆ(x)

∑

m i=1

2πω_i∂U

∂x_i−DK(U(x)) +DF(U(x)) +DH(U(x)) =0, (PDS) whereU:R^m →_Rⁿ. LetU(s):=U(2πs),s∈_R^m. Then it is easy to verify thatUis 1-periodic in each of its arguments ifUis 2π-periodic in each of its arguments. Hence, ifUis a weak 2π- periodic solution of system (PDS), thenu(t) =U(2πωt) =U(tω)is a weakω-quasi-periodic solution of system (HSD). Furthermore, in order to obtain a 2π-periodic solution of system (PDS), following the idea of Blot [7], we seek a weak solution U of the Dirichlet boundary value problem















∑m j=1

∑m i=1

(2π)²ω_iω_j ∂²U

∂x_i∂x_j +qˆ(x) _∑^m

i=1

2πω_i∂U

∂x_i

−DK(U(x)) +DF(U(x)) +DH(U(x)) =0, on Ω U =0, on∂Ω

(PDS*)

where Ω := (−π,π)^m ⊂ _R^m, and the solution can be extendable into a 2π-periodic solution Uof system (PDS) onR^m.

We organize our paper as follows. In Section 2, we introduce the working spaces. In Sec- tion 3, we present the variational functional which corresponds to system (PDS*) and then by using the least action principle, give two existence theorems. Finally, we present two proposi- tions which are related to the working space and the variational functional, respectively.

2 Working spaces

In this section, we present the working spaces which were established in [6] and [7].

LetT^m :=_R^m/2πZ^m be the m-dimensional torus and ¯Ω:= [−π,π]^m ⊂_R^m. Define L²(_T^m):={U:R^m →_R|U(x+2πν) =U(x)for allx ∈_R^m and for allν∈_Z^m,

and|U|²is locally Lebesgue-integrable onR^m and

L² := L²(_T^m)ⁿ:=

n

z }| {

L²(_T^m)× · · · ×L²(_T^m) with the inner product

(U,V)_L2 =

Z

T^mU·Vdx, for allU,V ∈ L², and

kUk_L2 =

Z

T^m|U|²dx, for allU ∈ L², whereU= (U¹, . . . ,Uⁿ),V= (V¹, . . . ,Vⁿ),U·V =_∑ⁿ_j=1U^jV^j and

Z

T^mU(x)dx= ¹ (2π)^m

Z

Ω¯ U(x)dx, for allU ∈ L². (2.1) Define

H¹:=

U= (U¹, . . . ,Uⁿ)

U∈ L²and ∂U^j

∂x_k ∈ L²(_T^m)for all j=1, . . . ,n,k =1, . . . ,m

and H²_:=

U= (U¹, . . . ,Uⁿ)U∈ H¹_and ^∂2Uj

∂x_l∂x_k ∈ L²(_T^m)_{for all} j=_{1, . . . ,}n,l,k=_{1, . . . ,}m

. ForU ∈ H¹, define

kUk²_H1 :=kUk²_L2+

∑

m k=1

∂U

∂x_k

2 L²

. ForU ∈ L², define

DωU:=lim

s→0

U(x+sω)−U(x)

s .

Then

D_ωU=

∑

m j=1

ω_j∂U

∂x_j. (2.2)

By (2.2), it is easy to obtain that

D_ω²U := D_ω(D_ωU) =

∑

m j=1

∑

m i=1

ω_jω_i ∂²U

∂x_i∂x_j if it exists. Let

H¹_ω :=U|U∈ L²andDωU∈ L² with the inner product

(U,V)_H1

ω = (U,V)_L2+ (DωU,DωV)_L2. ThenH¹_ω is a Hilbert space and

kUk_H1

ω =kUk_L2+kD_ωUk_L2. Let

H²_ω := ⁿU|U∈ H¹_ω andD²_ωU∈ L^2o with the inner product

(U,V)_H2

ω = (U,V)_L2+ (D_ωU,D_ωV)_L2 + (D²_ωU,D²_ωV)_L2. H²_ωis also a Hilbert space.

Let

Ω:= (−π,π)^m =

m

z }| {

(−π,π)× · · · ×(−π,π)⊂_R^m. Define

L²(_Ω):=

U:Ω→_R

Z

Ω|U(x)|²dx<+_∞

and

L²(_Ω):=L²(_Ω)ⁿ:=

n

z }| {

L²(_Ω)× · · · ×L²(_Ω) with the inner product

(U,V)_L2(_Ω) =

Z

ΩU·Vdx, for allU,V∈ L²(_Ω) and the norm

kUk_L2(_Ω)=

Z

Ω|U|²dx, for allU∈ L²(_Ω). Obviously,L²(_Ω) =L²_.

Define

C⁰(_Ω)_:=C⁰₀(_Ω)ⁿ

= {U :Ω→_Rⁿ |Uis continuous and has a compact support included inΩ} and for integer 1≤k <+_{∞, define}

C^k(_Ω):= C^k(_Ω)ⁿ =ⁿU:Ω→_Rⁿ|Uis of classC^k onΩo .

Let C₀^k(_Ω) = C⁰(_Ω)∩ C^k(_Ω) and then define H¹₀(_Ω) = C₀¹(_Ω) which is the closure of C₀¹(_Ω) with the inner product

(U,V)_H1

0 = (U,V)_L2(_Ω)+

∑

m j=1

∂U

∂x_j,∂V

∂x_j

L²(_Ω)

. Then (H¹₀(_Ω),(·,·)_H1

0)is a Hilbert space. Following Blot [7], one can extend a function U ∈ H¹₀(_Ω) to a function ˜U ∈ H¹ and by a trace theorem, one can give sense to U = 0 on ∂Ωif U ∈ H₀¹(_Ω)so that H¹₀(_Ω) ⊂ H¹ and for U ∈ H¹₀(_Ω), the following inequality holds: there existsC∗ >0 such that

kUk_L2(_Ω)≤ C∗kDωUk_L2(_Ω), for allU∈ H¹₀(_Ω). (2.3) Let CLωH¹₀(_Ω) be the closure of H¹₀(_Ω) in H¹_ω with the norm k · k_H1

ω. Then, obviously, (CL_ωH¹₀(_Ω)_,(·,·)_H1

ω) is also a Hilbert space. We refer the reader for more details about the above working spaces to [6] and [7].

3 Main results

In [26], Wu and Chen directly construct a variational functional which corresponds to the second order Hamiltonian system like (HSD) in order to investigate the existence of periodic solutions. Motivated by [26], we define a functionalJ :CLωH¹₀(_Ω)→_Rby

J(U) =

Z

Ωe^Qˆ(x)



 1 2

∑

m i=1

2πω_i∂U

∂x_i

!2

−W(U(x))



dx

=

Z

Ωe^Qˆ(x)



 1 2

∑

m i=1

2πω_i∂U

∂x_i

!2

+K(U(x))−F(U(x))−H(U(x))



dx.

When (Q) and (W) hold, a standard argument can be made easily so thatJ is of classC¹and hJ⁰(U),Vi=

Z

Ωe^Qˆ(x)

"

∑

m i=1

2πω_i∂U

∂x_i,

∑

m j=1

2πω_j∂V

∂x_j

!

−(DW(U(x)),V(x))

# dx

=

Z

Ωe^Qˆ(x)

"

∑

m i=1

2πω_i∂U

∂x_i,

∑

m j=1

2πω_j∂V

∂x_j

!

+ (DK(U(x)),V(x))

−(DF(U(x)),V(x))−(DH(U(x)),V(x))

#

dx (3.1)

for allV ∈CL_ωH₀¹(_Ω).

Remark 3.1. When n = 1, in [19] and [27], there have been more general functionals which correspond to more general partial differential equations. In some sense, when n = 1, the functionalJ(U)can be seen as a special case of those in [19] and [27] if we choosea_i,j(x,u)≡ e^Qˆ(x),i,j=1, . . . ,m, where the details ofa_i,j(x,u)can be seen in [19] and [27].

Lemma 3.2. Assume thatJ⁰(U^∗) = 0 for some U^∗ ∈ CL_ωH¹₀(_Ω). Then u^∗(t) := U^∗(2πωt)is a weakω-quasi-periodic solution of system(HSD).

Proof. For any givenV ∈CL_ωH₀¹(_Ω), there exists a sequence{V_k} ⊂ H¹₀(_Ω)such that kV_k−Vk_H1

ω →0 which implies that

kDωV_k−DωVk_L2 →0 and kV_k−Vk_L2 →0, ask→_∞ (3.2) and soV_k(x)→V(x)for a.e.x∈_{Ω. Let}

M(V_k):=

Z

Ωe^Qˆ(x)

∑

m i=₁

2πω_i∂U^∗

∂x_i ,

∑

m j=₁

2πω_j∂V

∂x_j −

∑

m j=₁

2πω_j∂V_k

∂x_j

! dx.

Then it follows from Hölder’s inequality and (3.2) that M(V_k)→0 ask→_∞,

Z

Ωe^Qˆ(x)

∑

m j=1

∑

m i=1

(_2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j,V_k(x)

! dx

→

Z

Ωe^Qˆ(x)

∑

m j=1

∑

m i=1

(_2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j,V(x)

! dx

(3.3)

and

Z

Ωe^Qˆ(x)

∑

m j=1

qˆ_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i ,V_k(x)

! dx

→

Z

Ωe^Qˆ(x)

∑

m j=1

qˆ_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i ,V(x)

! dx.

(3.4)

By integration by parts and noting thatV_k =0 on∂Ω, we have Z

Ωe^Qˆ(x)

∑

m i=1

2πω_i∂U^∗

∂x_i ,

∑

m j=1

2πω_j∂V

∂x_j

! dx

=

Z

Ωe^Qˆ(x)

∑

m i=1

2πω_i∂U^∗

∂x_i ,

∑

m j=1

2πω_j∂V

∂x_j −

∑

m j=1

2πω_j∂V_k

∂x_j +

∑

m j=1

2πω_j∂V_k

∂x_j

! dx

=

Z

Ωe^Qˆ(x)

∑

m i=1

2πω_i∂U^∗

∂x_i ,

∑

m j=1

2πω_j∂V_k

∂x_j

!

dx+M(V_k)

= −

Z

Ω

∑

m j=1

2πω_j

∑

m i=1

2πω_ie^Qˆ(x)∂U^∗

∂x_i 0

xj

,V_k(x)

!

dx+M(V_k)

= −

Z

Ω

∑

m j=1

2πω_j

∑

m i=1

2πω_ie^Qˆ(x) ∂²U^∗

∂x_i∂x_j

! ,V_k(x)

! dx

−

Z

Ω

∑

m j=1

2πω_j

∑

m i=1

2πω_ie^Qˆ(x)∂Qˆ(x)

∂x_j

∂U^∗

∂x_i

! ,V_k(x)

!

dx+M(V_k)

= −

Z

Ω

∑

m j=1

2πω_j

∑

m i=1

2πω_ie^Qˆ(x) ∂²U^∗

∂x_i∂x_j

! ,V_k(x)

! dx

−

Z

Ω

∑

m j=1

2πω_j

∑

m i=1

2πω_ie^Qˆ(x) 1

2πω_jqˆ_j(x_j)^∂U

∗

∂x_i

! ,V_k(x)

!

dx+M(V_k)

= −

Z

Ωe^Qˆ(x)

∑

m j=1

∑

m i=1

(2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j,V_k(x)

! dx

−

Z

Ωe^Qˆ(x)

∑

m j=1

ˆ q_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i ,V_k(x)

!

dx+M(V_k). (3.5)

Letk →∞. (3.3), (3.4) and (3.5) imply that Z

Ωe^Qˆ(x)

∑

m i=1

2πω_i∂U^∗

∂x_i ,

∑

m j=1

2πω_j∂V

∂x_j

! dx

= −

Z

Ωe^Qˆ(x)

∑

m j=1

∑

m i=1

(2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j,V(x)

! dx

−

Z

Ωe^Qˆ(x)

∑

m j=1

ˆ q_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i ,V(x)

! dx

for allV ∈CLωH₀¹(_Ω). IfJ⁰(U^∗) =0, then (3.1) and the above equality imply that 0=−

Z

Ωe^Qˆ(x)

∑

m j=1

∑

m i=1

(2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j,V(x)

! dx

−

Z

Ωe^Qˆ(x)

∑

m j=1

ˆ q_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i ,V(x)

! dx

+

Z

Ωe^Qˆ(x)[(DK(U^∗(x)),V(x))−(DF(U^∗(x)),V(x))−(DH(U^∗(x)),V(x))]dx

(3.6)

for allV ∈CL_ωH₀¹(_Ω). Following the idea of Blot [7], (3.6) implies that

−

∑

m j=1

∑

m i=1

(2π)²ω_jω_i ∂²U^∗

∂x_i∂x_j −

∑

m j=1

ˆ q_j(x_j)

∑

m i=1

2πω_i∂U^∗

∂x_i

+DK(U^∗(x))−DF(U^∗(x))−DH(U^∗(x)) =0 (3.7) in D⁰(_Ω)ⁿ, where D⁰(_Ω) denotes the space of the distributions in the sense of Schwartz on Ωand D⁰(_Ω)ⁿ is the n-times product of D⁰(_Ω). Note that DK(U^∗), DF(U^∗), DH(U^∗), and

∑^mi=₁ω_i^∂U

∗

∂xi (= DωU^∗)belong toL²(_Ω)(= L²)and ˆq_j ∈ L²(_Ω), j= 1, . . . ,m. Hence, D²_ωU^∗ :=

∑^mj=1∑^mi=1ω_jω_i ^∂2U

∗

∂x_i∂x_j ∈ L²(_Ω)(=L²). Hence, (3.7) holds inL², which shows thatU^∗ is a weak 2π-periodic solution of system (PDS) andU^∗ ∈ H²_ω. Thenu^∗(t):=U^∗(2πωt)∈ B^2,2_ω (_Rⁿ)and u^∗(t)is a weakω-quasi-periodic solution of system (HSD).

Lemma 3.3(see [21]). If ϕis weakly lower semi-continuous on a reflexive Banach space X and has a bounded minimizing sequence, then ϕhas minimum on X.

Theorem 3.4. Suppose that W is concave, (Q), (W), and the following conditions hold:

(K) there exists a positive constant b such that

K(x)≥ b|x|²_, ∀x ∈_Rⁿ_;

(F) there exist positive constants d₁,d₂andα∈[0, 2)such that F(x)≤ d₁|x|^α+d₂, ∀x∈_Rⁿ; (H) there exists a positive constant l< ^2π_C²

∗ +b such that

|DH(x)−DH(y)| ≤l|x−y|, for all x,y ∈_Rⁿ. Then system(HSD)has at least one weakω-quasi-periodic solution u^∗ ∈ B_ω^2,2(_Rⁿ). Proof. By (H), there existsθ ∈(_{0, 1})_{such that}

|H(_x)| − |H(₀)| ≤ |H(_x)−H(₀)|

≤ |DH(θx)||x|

≤(|DH(0)|+l|θx|)|x|

≤ |DH(0)||x|+l|x|² (3.8) for allx∈ _R^m. Then it follows from (K), (F), (3.8) and Hölder’s inequality that

J(U) =

Z

Ωe^Qˆ(x)



 1 2

∑

m i=1

2πω_i∂U

∂x_i

!2

+K(U(x))−F(U(x))−H(U(x))



dx

≥

Z

Ωe^Qˆ(x)



 1 2

∑

m i=1

2πω_i∂U

∂x_i

!2

+b|U(x)|²−d₁|U(x)|^α−d₂

− |H(0)| − |DH(0)||U(x)| −l|U(x)|²



dx

≥e^M∗h

min{2π²,b−l}kUk²_H1 ω

−d₁(_mesΩ)^2−2αkUk^α_L2

−(d₂+|H(0)|)mesΩ− |DH(0)|√

mesΩkUk_L2

i

≥e^M∗h

min{2π²,b−l}kUk²_H1 ω

−d₁(_mesΩ)^2−2αkUk^α_H1 ω

−(d₂+|H(0)|)mesΩ− |DH(0)|√

mesΩkUk_H1 ω

i

(3.9) for allU ∈CL_ωH¹₀(_Ω), where M∗ =min_x∈Ω¯ Qˆ(x). Note thatα∈ [0, 2). The above inequality implies thatJ(U)is coercive, that is,

J(U)→+_∞ askUk_H1

ω →+_∞. (3.10)

If{U_k}is a minimizing sequence inCL_ωH¹₀(_Ω), that is, J(U_k)→ inf

x∈CLωH¹₀(_Ω)

J(U), ask→_∞,

then there exists a positive constant C such that |J(U_k)| ≤ C, which, together with (3.10), implies that{U_k}is bounded inCL_ωH¹₀(_Ω)_{. Since}W is concave,J is convex. Moreover, note that J is of class C¹. Hence, J is lower semi-continuous on CLωH¹₀(_Ω). By Theorem 1.2 in [21], J is weakly lower semi-continuous on CL_ωH¹₀(_Ω). Then Lemma 3.3 implies that J has a critical pointU^∗ inCL_ωH¹₀(_Ω). Finally, by Lemma3.2, we complete the proof.

Example 3.5. Let K(x) = b₁|x|² with b₁ ≥ b, F(x) = d(1+|x|)^3/2 with d > 0 and H(x) = cln(1+|x|²)withc>0 and−b₁+c+³₈d <0. It is easy to verify thatK,FandHsatisfy those assumptions in Theorem 3.1.

Whenα=2, by (3.9), it is easy to obtain the following theorem.

Theorem 3.6. Suppose that W is concave, (Q), (W), (K), (H) and the following condition holds:

(F)⁰ there exist constants0< d₁ <min{2π²,b−l}, and d₂>0such that F(x)≤d₁|x|²+d₂, ∀x ∈_Rⁿ.

Then system(HSD)has at least one weakω-quasi-periodic solution u^∗ ∈B^2,2ω (_Rⁿ).

Next, we present two propositions which are related to the working space and the varia- tional functional, respectively.

Proposition 3.7. For U ∈CLωH¹₀(_Ω),(2.3)also holds.

Proof. Since CLωH¹₀(_Ω) is the closure of H¹₀(_Ω) in H¹_ω, for U ∈ CLωH¹₀(_Ω), there exists a sequence{U_k} ⊂ H¹₀(_Ω)such thatkU−U_kk_H1

ω →0 which implies thatkD_ωU−D_ωU_kk_L2 →0 and then by (2.1),kDωU−DωU_kk_L2(_Ω)→0. Note thatU_k ∈ H¹₀(_Ω). Hence, (2.3) implies that

C∗kDωUk_L2(_Ω) =C∗kDωU−DωU_k+DωU_kk_L2(_Ω)

≥C∗kDωU_kk_L2(_Ω)−C∗kDωU−DωU_kk_L2(_Ω)

≥ kU_kk_L2(_Ω)−C∗kDωU−DωU_kk_L2(_Ω). Letk →_{∞. We have}

C∗kDωUk_L2(_Ω) ≥ kUk_L2(_Ω).

Proposition 3.8. Assume thatJ⁰(U^∗) =0for some U^∗ ∈ H¹₀(_Ω). Then U^∗ is a critical point ofJ in CL_ωH¹₀(_Ω), that is,hJ⁰(U^∗)_,Vi= _0,for all V ∈ CL_ωH¹₀(_Ω).

Proof. For an arbitrary V ∈ CL_ωH¹₀(_Ω), there exists a sequence {V_k} ⊂ H₀¹(_Ω) such that kV_k−Vk_H1

ω →0. Then by Hölder’s inequality and (2.1), we have

|hJ⁰(U^∗),V−V_ki|

= Z

Ωe^Qˆ(x)

"

∑

m i=1

2πω_i∂U^∗

∂x_i ,

∑

m j=1

2πω_j ∂V

∂x_j − ^∂Vk

∂x_j !

−(DW(U^∗(x)),V(x)−V_k(x))

# dx

≤e^M∗









 Z

Ω

∑

m i=1

2πω_i∂U^∗

∂x_i

2

dx





1/2

 Z

Ω

∑

m j=1

2πω_j

∂(V−V_k)

∂x_j

2

dx





1/2

+ Z

Ω|DW(U^∗)|²dx

1/2Z

Ω|V−V_k|² 1/2

dx







=e^M∗h

4π²kD_ωU^∗k_L2(_Ω)kD_ωV−D_ωV_kk_L2(_Ω)+kDW(U^∗)k_L2(_Ω)kV−V_kk_L2(_Ω)

i

=e^M∗h

(2π)^(2m+2)kDωU^∗k_L2kDωV−DωV_kk_L2 + (2π)^(2m)kDW(U^∗)k_L2kV−V_kk_L2

i

≤e^M∗h

(2π)^(2m+2)kD_ωU^∗k_L2 + (2π)^(2m)kDW(U^∗)k_L2

ikV−V_kk_H1

ω →0 ask→_{∞, (3.11)}

where M^∗ = _{maxx∈Ω¯ e}^Qˆ(x). Note that hJ⁰(_U^∗)_,Vi = _{0 for all V} ∈ H¹₀(_Ω). Hence, by (3.11), we have

|hJ⁰(U^∗),Vi| ≤ |hJ⁰(U^∗),V−V_ki|+|hJ⁰(U^∗),V_ki|=|hJ⁰(U^∗),V−V_ki| →0 ask→_∞.

Remark 3.9. Proposition3.7 and Proposition3.8 are maybe useful for one to seek the critical points of the functionalJ by using those abstract critical point theorems with Palais–Smale condition. We try to do such things by using Ekeland variational principle so that the restric- tion on concavity of W can be deleted. However, we come across a difficulty whether the embeddingCLωH¹₀(_Ω) ,→ L²(_Ω) is compact. Note that the embeddingH¹₀(_Ω) ,→ L²(_Ω)is compact. So maybe one can reduce our problem fromCLωH¹₀(_Ω)toH¹₀(_Ω)by Proposition3.8.

However, a new difficulty whether (PS) sequence ofJ is bounded in H¹₀(_Ω)appears. This is a problem that is worthy of consideration.

Acknowledgements

The authors would like to thank the referee very much for his/her valuable suggestions.

Moreover, this work is supported by the National Natural Science Foundation of China (No: 11301235) and Tianyuan Fund for Mathematics of the National Natural Science Foun- dation of China (No: 11226135).

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