Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a

Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a

forcing term

Xingyong Zhang

^B

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

Received 12 April 2014, appeared 19 December 2014 Communicated by Michal Feˇckan

Abstract. In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev’s inequality and Wirtinger’s inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi- periodic solutions for the second order Hamiltonian system:

d[P(t)u˙(t)]

dt =∇F(t,u(t)) +e(t),

which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions F(_t,x) = F(t,−x) and e(t) ≡ 0 are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.

Keywords:second order Hamiltonian system, weak quasi-periodic solution, variational method, subquadratic case.

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

1 Introduction and main results

In this paper, we are concerned with the existence and multiplicity of weak-quasi periodic solutions for the second order Hamiltonian system:

d[P(t)u˙(t)]

dt =∇F(t,u(t)) +e(t)_, t∈_{R (1.1)} where u(t) = (u₁(t), . . . ,u_N(t))^τ, N > 1 is an integer, F ∈ C¹(_R×_R^N,R), ∇F(t,x) = (∂F/∂x₁, . . . ,∂F/∂xN)^τ, P(t) = (p_ij(t))_N×N is a symmetric and continuous N×N matrix- value functions onR,e: R→_R^N,(·)^τ stands for the transpose of a vector or a matrix.

BEmail: zhangxingyong1@163.com

It is well known that the variational method is a very effective tool which investigate the existence and multiplicity of periodic solutions, subharmonic solutions and homoclinic solutions for Hamiltonian systems and in these directions, lots of contributions have been obtained (for example, see [6,7,11,12,15–28,30–34] and references therein). However, the results on existence and multiplicity of almost periodic solutions for Hamiltonian systems are not often seen by using variational approach. We refer readers to [1–5,13,29]. Especially, whenP(t)≡ I_N×N ande(t)≡0, where I_N×N is the unit matrix, recently, in [13], by using the least action principle and the saddle point theorem, Kuang obtained two existence results of quasi-periodic solutions for system (1.1). Next, we recall two definitions and Kuang’s results in [13].

Definition 1.1([8]). A function f(t)is said to be Bohr almost periodic, if for anyε> 0, there is a constantlε >0, such that in any interval of lengthlε, there existsτsuch that the inequality

|f(t+τ)− f(t)|< εis satisfied for allt ∈_R.

Definition 1.2([9]). A function f ∈ C⁰(_R×_R^m,R^N)is called almost periodic in t uniformly forx∈ _R^m when, for each compact subset KinR^m, for eachε>0, there exists l>0, and for eachα∈_R, there existsτ∈[α,α+l]such that

sup

t∈_R

sup

x∈K

kf(t+τ,x)− f(t,x)k_RN < ε.

Let p > 1 be a positive integer and {T_j}^p_j=1 be rationally independent positive real con- stants. Define

Λ=∪^p_j=1Λj =∪_j^p₌₁ 2mπ

T_j

m∈_Z

, (1.2)

whereΛj =^2mπ_T

j

m∈_Z .

To be precise, in [13], Kuang obtained the following results.

Theorem 1.3([13, Theorem 2.3]). Suppose F satisfies the following conditions:

(f₁) F(t,·)∈C¹(_R×_R^N,R)and F(t,·)is almost periodic in t uniformly for x ∈_R^N; (f₂) ∇F(t,·)is almost periodic in t uniformly for x∈_R^N;

(f₃) for anyλ∈ _{R/Λ, x} ∈V,

Tlim→∞

1 2T

Z _T

−T

∇F(t,x)e^−iλtdt=0;

(f₄) there exists g∈ L¹_loc(_R), for a.e. t∈ _Rand all x∈_R^N, such that

|∇F(t,x)| ≤g(t)_;

(f₅) lim

T→_∞

1 2T

Z _T

−TF(t,x)dt→+_∞ as|x| →_∞.

Then (1.1) with P(t) ≡ I_N×N and e(t) ≡ 0 has at least a quasi periodic solution, where the definition of V can be seen in Section 2 below.

Theorem 1.4([13, Theorem 2.4]). Suppose that F satisfies(f₁)–(f₄)and

(f₆) lim

T→_∞

1 2T

Z _T

−TF(t,x)dt→ −_∞ as|x| →_∞.

Then (1.1) with P(t) ≡ IN×N and e(t) ≡ 0has at least one quasi-periodic solution by saddle point theorem.

Obviously, (f₄) implies that |∇F| is bounded, which makes lots of functions eliminated.

For example, a simple function

F(t,x)≡ ±|x|³², ∀ t∈_R (1.3) which does not satisfy (f₄). However, in this paper, we obtain that system (1.1) still has quasi-periodic solution for such potential Flike (1.3). To be precise, in this paper, inspired by [10,13,15,24,28,32], we obtain the following results.

(I) Existence of weak quasi-periodic solution

By using the least action principle and the saddle point theorem, we obtain that system (1.1) has at least one weak quasi-periodic solution.

Theorem 1.5. Suppose that (f₁)–(f3)hold. If

(P) p_ij(t), i,j=1, 2, . . . ,N are Bohr almost periodic and there exists m> ¹₂ such that (P(t)x,x)>m|x|², for all(t,x)∈_R× {_R^N\ {0}}; (E) e is Bohr almost periodic and

Tlim→_∞ Z _T

−Te(t)dt=_0;

(W) there exist constants c₀ > 0, k₁ > 0, k₂ > 0, α ∈ [0, 1) and a nonnegative function w ∈ C([_0,+_∞)_,[_0,+_∞))with the properties:

(i) w(s)≤w(t), ∀s ≤t,s,t ∈[0,+_∞),

(ii) w(s+t)≤c₀(w(s) +w(t)), ∀s,t∈[0,+_∞), (iii) 0≤w(t)≤k₁t^α+k₂, ∀t ∈[0,+_∞),

(iv) w(t)→+_∞, as t →_∞;

(f₄)⁰ there exist g,h∈ L¹_loc(_R,R⁺)such that

|∇F(t,x)| ≤g(t)w(|x|) +h(t), for a.e. t∈_R;

(f5)^{0 1}

w²(|x|)T^lim→_∞

1 2T

Z _T

−TF(t,x)dt>

c²₀

∑p j=1

T_j² 12

2m

Tlim→_∞

1 2T

Z _T

−Tg(t)dt 2

, as|x| →_∞, then system(1.1)has at least one weak quasi-periodic solution.

Theorem 1.6. Suppose that (P),(E),(W),(f₁)–(f₃)and(f₄)⁰ hold. If (f₅)⁰⁰

1

w²(|x|)T^lim→_∞

1 2T

Z _T

−TF(t,x)dt

<−

c²₀(kPk+2m)

∑p j=1

T_j² 12

2(2m−1)

Tlim→_∞

1 2T

Z _T

−Tg(t)dt 2

as|x| →_∞,

where

kPk= sup

t∈[0,T]

|x|=max1,x∈_R^N|P(t)x|

= _sup

t∈[0,T]

max q

λ(t)_:λ(t)is the eigenvalue of P^τ(t)P(t)

, then system(1.1)has at least one weak quasi-periodic solution.

Remark 1.7. Obviously, Theorem 1.5 and Theorem1.6 generalize and improve Theorem1.3 and Theorem 1.4, respectively. It is easy to verify that F(t,x) ≡ |x|^3/2 and F(t,x) ≡ −|x|^3/2 satisfy Theorem1.5and Theorem 1.6, respectively, but do not satisfy Theorem 1.3and Theo- rem1.4. Moreover, similar to the argument of Remark 2.5 in [13], whenP(t)≡ I_N×N,e(t)≡0, V only contains a frequency 2π/T and F(t,x) is periodic in t with period T, in some sense, Theorem1.5 and Theorem1.6improve the corresponding results in [15] because of the pres- ence of (W) and (f₄)⁰. (W) and (f₄)⁰ were given by Wang and Zhang in [28], which present some advantages compared to the well known condition: there existg,h∈ L¹([0,T];R⁺)and α∈ [0, 1)such that

|∇F(t,x)| ≤g(t)|x|^α+h(t). (1.4) Finally, one can also compare Theorem1.5and Theorem1.6with the corresponding results in [32], in which, Zhang and Tang investigated the existence of T-periodic solution under (W) and the following condition: there exist g ∈ L²([0,T];R⁺), h ∈ L¹([0,T];R⁺) and α ∈ [0, 1) such that

|∇F(t,x)| ≤ g(t)w(|x|) +h(t), (1.5) whereg∈ L²([0,T];R⁺)is demanded from proofs of their theorems. In our Theorem1.5and Theorem1.6, when P(t) ≡ IN×N, e(t) ≡ 0, V only contains a frequency 2π/T and F(t,x)is periodic int with periodT, we only demand that g ∈ L¹([0,T];R⁺). Hence, our results are different from those in [32].

(II) Multiplicity of weak quasi-periodic solutions

Moreover, by using a critical point theorem due to Ding in [6], we obtain the following multiplicity results.

Theorem 1.8. Suppose that(P),(W),(f₁)–(f₃),(f₄)⁰ and(f₅)⁰ hold. If (E)⁰ e(t)≡0, ∀t∈_R;

(f8) F(t, 0)≡0 and F(t,x) =F(t,−x) for all(t,x)∈_R×_R^N;

(f₉) lim

|x|→0

F(t,x)

|x|² =−_∞ uniformly for all t∈_R, then system(1.1)has infinitely many weak quasi-periodic solutions.

Theorem 1.9. Suppose that(P),(E)⁰,(W),(f₁)–(f3),(f₄)⁰,(f5)⁰⁰,(f8)and(f9)hold. Then system (1.1)has infinitely many weak quasi-periodic solutions.

2 Preliminaries

In this section, we need to make some preliminaries. Some knowledge and statements below come from [3,4,8,9,13] .

Define

AP⁰(_R^N) ={u:R→_R^N |uis Bohr almost periodic},

endowed with the normkuk_∞ =_supt∈R|u(t)|_{. Then}(AP⁰(_R^N)_,k · k_∞)is a Banach space.

Define

AP¹(_R^N) =ⁿu∈ AP⁰(_R^N)∩C¹(_R,R^N)

u⁰(t)∈ AP⁰(_R^N)^o, endowed with the norm

kuk=kuk_∞+ku⁰k_∞. Then(AP¹(_R^N),k · k)is also a Banach space.

Let f ∈ L¹_loc(_R,R^N), that is f is locally Lebesgue integrable fromRtoR^N. Then the mean value of f is the limit (when it exists)

Tlim→_∞

1 2T

Z _T

−T f(t)dt.

A fundamental property of almost periodic functions is that such functions have conver- gent means, that is, the limit

Tlim→_∞

1 2T

Z _T

−Tu(t)dt exists.

Let p ∈ _Z⁺. B^p(_R^N) is the completion of AP⁰(_R^N)into L¹_loc(_R,R^N)with respect to the norm

kuk_p =

Tlim→_∞

1 2T

Z _T

−T

|u(t)|^pdt 1/p

.

The elements of these spacesB^p(_R^N)are called Besicovitch almost periodic functions.

Foru∈ B^p(_R^N), if

limr→0

u(t+r)−u(t) r

exists, then define

∇u=lim

r→0

u(t+r)−u(t)

r .

For u,v ∈ B^p(_R^N), ifku−vk_p = 0, then we say thatu,v belong to a class of equivalence.

We will identify the equivalence classuwith its continuous representant u(t) =

Z _t

0

∇u(t)dt+c.

When p=2, B²(_R^N)is a Hilbert space with its normk · k₂ and the inner product hu,vi₂= lim

T→_∞

1 2T

Z _T

−T

(u(t),v(t))dt.

When u∈B²(_R^N), define

a(u,λ):= lim

T→_∞

1 2T

Z _T

−Te^−iλtu(t)dt

which are complex vector and are called Fourier–Bohr coefficients ofu. LetΛ(u) = {λ∈ _R| a(u,λ)6=0}.

Define

B^1,2(_R^N) =ⁿu∈ B²(_R^N)∇uexists and∇u∈ B²(_R^N)^o, endowed with the inner product

hu,vi=hu,vi₂+h∇u,∇vi₂

= lim

T→_∞

1 2T

Z _T

−T

(u(t),v(t))dt+ lim

T→_∞

1 2T

Z _T

−T

(∇u(t),∇v(t))dt, (2.1) and the corresponding norm

kuk=

Tlim→_∞

1 2T

Z _T

−T

|u(t)|²dt+ lim

T→_∞

1 2T

Z _T

−T

|∇u(t)|²dt 1/2

Define

V= ⁿu∈ B^1,2(_R^N)

Λ(u)⊂ _Λ^o.

ThenV is a linear subspace ofB^1,2(_R^N)_and(V,h·_,·i)is a Hilbert space.

Inspired by [13] and [16], we present the following two lemmas:

Lemma 2.1. If u∈V, then

u(t) =

∑

p j=1

u_j(t)∈ AP⁰(_R^N), where

u_j(t) =

+_∞ m=−

∑

_∞

a(u,λ⁽_m^j))e^iλ(mj)t, λ⁽_m^j):= ^2mπ T_j ∈ _Λj, and

kuk_∞ ≤ v u u tp²+

∑

p j=1

T_j²

12kuk (2.2)

Proof. SinceV⊆ B^1,2(_R^N)⊆B²(_R^N), then u(t)∼

+_∞ m=−

∑

_∞

a(u,λm)e^iλmt, λm ∈_Λ and

∇u(t)∼

+_∞ m=−

∑

_∞

iλ_ma(u,λ_m)e^iλmt, λ_m ∈_Λ.

Combining (1.2), we obtain that u(_t)∼

∑

p j=1

u_j(_t)_, ∇u(_t)∼

∑

p j=1

∇u_j(_t)_{. (2.3)}

By Parseval’s equality, we have kuk²₂ = lim

T→_∞

1 2T

Z _T

−T

|u(t)|²dt=

+_∞ m=−

∑

_∞

|a(u,λ_m)|², (2.4) k∇uk²₂ = lim

T→_∞

1 2T

Z _T

−T

|∇u(t)|²dt=

+_∞ m=−

∑

_∞

λ²_m|a(u,λm)|². (2.5)

Hence

+_∞ m=−

∑

_∞

|a(u,λ_m)|² =

∑

p j=1

+_∞ m=−

∑

_∞

|a(u,λ⁽_m^j))|² <+_∞.

Then by [14, Theorem 3.5-2], we have u(t) =

+_∞ m=−

∑

_∞

a(u,λm)e^iλmt, λm ∈_Λ (2.6) and

∇u(t) =

+_∞ m=−

∑

_∞

iλ_ma(u,λ_m)e^iλmt, λ_m ∈_Λ. (2.7) Since

+_∞ m=−

∑

_∞

m6=0

1 m² = ^π

2

3 , then

|u(t)| ≤

∑

p j=1

|u_j(t)|

≤

∑

p j=1

+_∞ m=−

∑

_∞

|a(u,λ⁽_m^j))||e^iλ(mj)t|

=

∑

p j=1

+_∞ m=−

∑

_∞

|a(u,λ⁽_m^j))|

=

∑

p j=1

|a(u,λ₀^(j))|+

∑

p j=1

+_∞ m=−

∑

_∞

m6=0

|a(u,λ⁽_m^j))|

=

∑

p j=1

|a(u, 0)|+

∑

p j=1

+_∞ m=−

∑

_∞

m6=0

1

|λ⁽_m^j)||λ⁽_m^j)a(u,λ⁽_m^j))|

=

∑

p j=1

|a(u, 0)|+

∑

p j=1

+_∞ m=−

∑

_∞

m6=0

T_j

2π|m||λ⁽_m^j)a(u,λ⁽_m^j))|

≤ p lim

T→_∞

1 2T

Z _T

−T

|u(t)|dt+

∑

p j=₁







+_∞ m=−

∑

∞

m6=0

T_j² 4π²m²







1/2





+_∞ m=−

∑

∞

m6=0

|λ⁽_m^j)a(u,λ⁽_m^j))|²







1/2

≤ p lim

T→_∞

√1 2T

Z _T

−T

|u(t)|²dt 1/2

+

∑

p j=1

s T_j² 12







+_∞ m=−

∑

_∞

m6=0

|λ⁽_m^j)a(u,λ⁽_m^j))|²







1/2

≤ pkuk₂+

∑

p j=1

T_j² 12

!1/2





∑

p j=1

+_∞ m=−

∑

_∞

m6=0

|λ⁽_m^j)a(u,λ⁽_m^j))|²







1/2

≤ p²+

∑

p j=1

T_j² 12

!1/2

kuk²₂+k∇uk²₂^1/2. (2.8)

Hence (2.2) holds. (2.2) implies that the embedding fromV into AP⁰(_R^N)is continuous. So u∈ AP⁰(_R^N). Thus we complete the proof.

Lemma 2.2. If u∈V and

Tlim→_∞

1 2T

Z _T

−Tu(t)dt=0, (2.9)

then

kuk_∞ ≤ v u u t

∑

p j=1

T_j²

12k∇uk₂ (2.10)

and

kuk₂ ≤max T_j

2π

j=1, . . . ,p

k∇uk₂. (2.11)

Proof. By (2.8) and (2.9), it is obvious that (2.10) holds. Moreover, by (2.5) and (2.9), we have k∇uk²₂=

+_∞ m=−

∑

_∞

λ²_m|a(u,λm)|²

=

∑

p j=1

+_∞ m=−

∑

_∞

m6=0

|λ⁽_m^j)a(u,λ⁽_m^j))|²

=

∑

p j=₁

+_∞ m=−

∑

_∞

m6=0

4m²π² T_j²

a(u,λ⁽_m^j))

2

≥

∑

p j=1

4π² T_j²

+_∞ m=−

∑

_∞

m6=0

a(u,λ⁽_m^j))

2

≥min (4π²

T_j²

j=1, . . . ,p )

kuk²₂. Hence, (2.11) holds.

Remark 2.3. A version of Lemma 2.1 and (2.10) has been given in [13] (see [13, Lemma 3.1 and Lemma 3.3]), where the author obtained that there exists a constantC>0 such that

kuk_∞≤ (C+1)kuk, ∀u ∈V, and when (2.9) holds,

kuk_∞ ≤Ck∇uk₂.

However, the value ofC are not given. Our Lemma2.1 and Lemma2.2 present the value of C, which will play an important role in our main results and their proofs. Moreover, we also present the inequality (2.11). One can compare (2.10) and (2.11) with Sobolev’s inequality and Wirtinger’s inequality in [16] which investigate periodic functions u ∈ W_T^1,2. It is easy to see that whenV only contains a frequency 2π/T, (2.11) reduces to Wirtinger’s inequality.

Lemma 2.4([13, Lemma 3.2]). For any{un} ⊂V, if the sequence{un}converges weakly to u, then {u_n}converges uniformly to u on any compact subset ofR.

Lemma 2.5. Suppose F satisfies(f₁)–(f₃), then the functional ϕ: V→R, defined by ϕ(u) = lim

T→_∞

1 2T

Z _T

−T

1

2(P(t)∇u(t),∇u(t)) +F(t,u(t)) + (e(t),u(t))

dt (2.12) is continuously differentiable on V, andϕ⁰(u)is defined by

hϕ⁰(u),vi= lim

T→_∞

1 2T

Z _T

−T

h

(P(t)∇u(t),∇v(t)) + (∇F(t,u(t)),v(t)) + (e(t),v(t))ⁱdt (2.13) for v∈ V. Moreover, if u is a critical point ofϕin V, then

∇(P(t)∇u(t)) =∇F(t,u(t)) +e(t). (2.14) Proof. The proof withP(t)≡ IN×N ande(t)≡0 can be seen in [13, Theorem 2.1]. With the aid of the conditions (P) and (E), it is easy to see that the proof is the essentially same as Theorem 2.1 of [13]. So we omit the details. we refer readers to Theorem 2.1 and its proof in [13].

Definition 2.6. When usatisfies (2.14), we say thatuis a weak solution of system (1.1).

3 Existence

In this section, we will use the least action principle (see [16, Theorem 1.1]) to prove Theo- rem1.5and use the saddle point theorem (see [19]) to prove Theorem1.6.

Define

V˜ =

u∈V lim

T→_∞

1 2T

Z _T

−Tu(t)dt=₀

and

V¯ ={u|u∈ V∩_R^N}.

ThenV =V^˜ ⊕V. For^¯ u∈V,ucan be written asu=u¯+u, where˜

¯

u= lim

T→_∞

1 2T

Z _T

−Tu(t)dt∈V.^¯ It is easy to obtain that

Tlim→_∞

1 2T

Z _T

−Tu˜(t)dt=_0.

Then ˜u∈V. For the sake of convenience, we denote^˜ M₁= lim

T→_∞

1 2T

Z _T

−Tg(t)dt, M2 = lim

T→_∞

1 2T

Z _T

−Th(t)dt, M3= _lim

T→_∞

1 2T

Z _T

−T

|e(_t)|dt, C^∗ =_max T_j

2π

j=_{1, . . . ,p}

.

Proof of Theorem1.5. Since V is a Hilbert space, then V is reflexive. Note that P is positive definite. Then ¹₂(P(t)∇u(t),∇u(t))and (e(t),u(t))are convex and continuous. Then by the proof of [13, Theorem 2.3], we know that ϕis weakly lower semi-continuous. Condition (f₅)⁰ implies that there exists a₁ > _m¹ _∑^p_j=1(T_j²/12)such that

|xlim|→_∞

1

w²(|x|)T^lim→_∞

1 2T

Z _T

−TF(t,x)dt

> ^a1c

20M²₁

2 . (3.1)

It follows from (W), (f₄)⁰ and Lemma2.2that

Tlim→_∞

1 2T

Z _T

−T

[F(t,u(t))−F(t, ¯u)]dt

= lim

T→_∞

1 2T

Z _T

−T

[F(t,u(t))−F(t, ¯u)]dt

= lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

(∇F(t, ¯u+su˜(t)), ˜u(t))ds dt

≤ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

|∇F(t, ¯u+su˜(t))||u˜(t)|ds dt

≤ ku˜k_{∞ lim}

T→_∞

1 2T

Z _T

−T

Z ₁

0

|∇F(t, ¯u+su˜(t)|ds dt

≤ ku˜k_{∞ lim}

T→_∞

1 2T

Z _T

−T

Z ₁

0

[g(t)w(|u¯+su˜(t)|) +h(t)]ds dt

≤ ku˜k_∞ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

[c0g(t)w(|u¯|) +c0g(t)w(|su˜(t)|) +h(t)]ds dt

≤c0ku˜k_∞w(|u¯|) lim

T→_∞

1 2T

Z _T

−Tg(_t)_dt +c0ku˜k_∞ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0 g(t)[k₁|su˜(t)|^α+k2]ds dt+ku˜k_∞ lim

T→_∞

1 2T

Z _T

−Th(t)dt

≤ ku˜k²_∞ 2a₁ + ^a1c

20M²₁w²(|u¯|)

2 + ^c0M1k1

α+1 ku˜k^α_∞⁺¹+c₀k₂M₁ku˜k_∞+M₂ku˜k_∞

≤ ¹ 2a₁

∑

p j=1

T_j² 12

!

k∇uk²₂+ ^a1c

20M₁²w²(|u¯|)

2 + ^c0M1k1 α+1

∑

p j=1

T_j² 12

!^α+₂¹

k∇uk₂^α+1

+ (c₀k₂M₁+M₂)

∑

p j=1

T_j² 12

!¹₂

k∇uk₂.

(3.2)

It follows from (3.2) and Lemma2.2that ϕ(u) = lim

T→_∞

1 2T

Z _T

−T

1

2(P(t)∇u(t),∇u(t)) +F(t,u(t))−F(t, ¯u) +F(t, ¯u) + (e(t),u(t))

dt

≥ ^m 2 lim

T→_∞

1 2T

Z _T

−T

|∇u(t)|²dt− ¹ 2a₁

∑

p j=₁

T_j² 12

!

k∇uk²₂− ^a1c

20M₁²w²(|u¯|) 2

− ^c0M1k1 α+1

∑

p j=1

T_j² 12

!^α+₂¹

k∇uk^α₂⁺¹−(c0k2M₁+M2)

∑

p j=1

T_j² 12

!¹₂

k∇uk₂

+ lim

T→_∞

1 2T

Z _T

−TF(t, ¯u)dt+ lim

T→_∞

1 2T

Z _T

−T

(e(t), ˜u(t))dt

≥ ^m

2 − ¹ 2a₁

∑

p j=1

T_j² 12

!

k∇uk²₂− ^c0M1k1 α+1

∑

p j=1

T_j² 12

!^α+₂¹

k∇uk₂^α+1

−(c₀k₂M₁+M₂+M₃)

∑

p j=1

T_j² 12

!¹₂

k∇uk₂ (3.3)

+w²(|u¯|)

w⁻²(|u¯|) lim

T→_∞

1 2T

Z _T

−TF(t, ¯u)dt− ^a1c

20M²₁ 2

.

Note thata₁> _m¹ _∑^p_j=1(T_j²/12). Sincekuk →_∞if and only if

|u¯|²+ lim

T→_∞

1 2T

Z _T

−T

|∇u(t)|²dt 1/2

→_∞,

(3.3), (3.1) and (W)(iv) imply that

ϕ(u)→+_∞, askuk →_∞.

Then by the least action principle (see [16, Theorem 1.1]), we know that ϕ has at least one critical pointu^∗ which minimizesϕ. Thus we complete the proof.

Proof of Theorem1.6. It follows from (f₅)⁰⁰ that there existsa₂>_∑^p_j=1(T_j²/12)such that

|xlim|→_∞

1

w²(|x|)T^lim→_∞

1 2T

Z _T

−T

F(t,x)dt<−



 a₂kPk 4m−2 + ^m

√a₂ 2m−1

v u u t

∑

p j=1

T_j² 12



c²₀M²₁. (3.4) At first, we prove that ϕ satisfies (PS) condition. Assume that {u_n} ⊂ V such that ϕ(u_n) _is bounded and ϕ⁰(un)→0 asn→∞. Then there exists a constant D₀ such that

|ϕ(u_n)| ≤D₀, kϕ⁰(u_n)k ≤D₀. (3.5) Similar to the argument of (3.2), we have

Tlim→_∞

1 2T

Z _T

−T

Z ₁

0

(∇F(t,un(t)), ˜un(t))ds dt

≤ _lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

|∇F(t,u_n(t))| |u˜_n(t)|ds dt

≤ ku˜_nk_{∞ lim}

T→_∞

1 2T

Z _T

−T

Z ₁

0

|∇F(t,u_n(t))|ds dt

≤ ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

[g(_t)_w(|u¯n+u˜n(_t)|) +h(_t)]_{ds dt}

≤ ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0

[c0g(t)w(|u¯n|) +c0g(t)w(|u˜n(t)|) +h(t)]ds dt

≤c0ku˜nk_∞w(|u¯n|) lim

T→_∞

1 2T

Z _T

−Tg(t)dt+c0ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−Tg(t)[k₁|u˜n(t)|^α+k2]dt +ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−Th(t)dt

≤ ku˜_nk²_∞ 2a₂ + ^a2c

20M²₁w²(|u¯n|)

2 +c₀M₁k₁ku˜_nk^α_∞⁺¹+c₀k₂M₁ku˜_nk_∞+M₂ku˜_nk_∞

≤ ¹ 2a2

∑

p j=1

T_j²

12k∇u_nk²₂+ ^a2c

20M²₁w²(|u¯n|)

2 +c₀M₁k₁

∑

p j=1

T_j² 12

!^α+₂¹

k∇u_nk^α₂⁺¹

+ (c₀k₂M₁+M₂)

∑

p j=1

T_j² 12

!¹₂

k∇u_nk₂. (3.6)

Hence, by (3.5), (3.6), (P) and Lemma2.2, we have D₀ku˜_nk

≥ hϕ⁰(un), ˜uni

= lim

T→_∞

1 2T

Z _T

−T

h

(P(t)∇u_n(t),∇u_n(t)) + (∇F(t,u_n(t)), ˜u_n(t)) + (e(t), ˜u_n(t))ⁱdt

≥ m− ¹ 2a₂

∑

p j=1

T_j² 12

!

k∇u_nk²₂− ^a2c

20M₁²w²(|u¯n|) 2

−c₀M₁k₁

∑

p j=1

T_j² 12

!^α+₂¹

k∇u_nk^α₂⁺¹−(c₀k₂M₁+M₂+M₃)

∑

p j=1

T_j² 12

!¹₂

k∇u_nk₂.

(3.7)

Moreover, by Lemma2.2, D₀ku˜_nk= D₀

Tlim→_∞

1 2T

Z _T

−T

|u˜_n(t)|²dt+ lim

T→_∞

1 2T

Z _T

−T

|∇u_n(t)|²dt 1/2

≤ D₀ (C^∗)²+11/2

k∇unk₂.

(3.8)

Note thatm> ¹₂. So (3.7) and (3.8) imply that 1

2m−1a₂c²₀M₁²w²(|u¯_n|)≥ k∇u_nk²₂+D₁, (3.9) where

D₁= min

s∈[0,+_∞)





 1 2− ¹

2a₂

∑

p j=₁

T_j² 12

!

s²−c₀M₁k₁

∑

p j=₁

T_j² 12

!^α+₂¹ s^α+1

−



(c0k2M₁+M2+M3)

∑

p j=1

T_j² 12

!¹₂

−D0 (C^∗)²+11/2



s





 .

Sincea₂ > _∑^p_j=1(T_j²/12), then 0> D₁ > −∞. Hence, there exists a positive constant D₂ such that

k∇unk₂≤

r a2

2m−1c0M₁w(|u¯n|) +D2. (3.10) Similar to (3.2), we have

Tlim→_∞

1 2T

Z _T

−T

[F(t,u_n(t))−F(t, ¯u_n)]dt

≤c₀ku˜nk_∞w(|u¯n|)lim

T→_∞

1 2T

Z _T

−Tg(t)dt+c₀ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−T

Z ₁

0g(t)[k₁|su˜n(t)|^α+k₂]ds dt +ku˜nk_∞ lim

T→_∞

1 2T

Z _T

−Th(t)dt

≤ ku˜nk²_∞ 2√

a₂ s p

j∑=1 T²_j 12

+

√a₂ s p

∑

j=1 T_j²

12c²₀M²₁w²(|u¯n|)

2 (3.11)