Nonlinear Analysis 71 (2009) 550–557

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Existence of three solutions for a class of quasilinear elliptic systemsinvolving the (p(x), q(x))-LaplacianI

Jingjing Liu a,∗, Xiayang Shi ba Department of Mathematics, Lanzhou University, Lanzhou, 730000, PR Chinab College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, 541004, PR China

a r t i c l e i n f o

Article history:Received 20 March 2008Accepted 29 October 2008

MSC:35D0535J70

Keywords:Three solutions(p(x), q(x))-LaplacianDirichlet problemRicceri’s three-critical-points theorem

a b s t r a c t

We study the solutions of the (p(x), q(x))-Laplacian equations with Dirichlet boundarycondition on a bounded domain, and obtain three solutions under appropriate hypotheses.Our technical approach is based on the general three-critical-points theorem obtained byB. Ricceri.

Crown Copyright© 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction and main result

In recent years, the study of differential equations and variational problems with p(x)-growth conditions has been aninteresting topic, which arises from nonlinear electrorheological fluids (see [17]) and elastic mechanics (see [20]).In this paper, we consider the quasilinear elliptic systems−1p(x)u = λFu(x, u, v)+ µGu(x, u, v) inΩ,−1q(x)v = λFv(x, u, v)+ µGv(x, u, v) inΩ,u = v = 0 on ∂Ω.

(P)

where −1p(x)u = −div(|∇u|p(x)−2∇u) is the p(x)-Laplacian operator, λ, µ ∈ [0,∞), Ω ⊂ RN(N ≥ 1) is a nonemptybounded open set with a boundary ∂Ω of class C1, F ,G : Ω × R × R → R are functions such that F(·, s, t),G(·, s, t) aremeasurable inΩ for all (s, t) ∈ R × R and F(x, ·, ·) is C1 in R × R for a.e. x ∈ Ω , Fu denotes the partial derivative of F withrespect to u, Gu denotes the partial derivative of G with respect to u. And p, q ∈ C(Ω), 1 < p− = infx∈Ω p(x) ≤ p

+=

supx∈Ω p(x) < +∞, 1 < q−= infx∈Ω q(x) ≤ q

+= supx∈Ω q(x) < +∞.

Moreover,

p∗(x) =

Np(x)N − p(x)

if p(x) < N,

∞ if p(x) ≥ N.

is the critical exponent just as in many papers. Obviously, p(x) < p∗(x), q(x) < q∗(x) for all x ∈ Ω .

I The project supported by the National Natural Science Foundation of China (Nos. 10671084 and 10871217) and Innovation Project of Guangxi GraduateEducation (No. 2007105950701M04).∗ Corresponding author.E-mail address: jingjing830306@163.com (J. Liu).

0362-546X/$ – see front matter Crown Copyright© 2008 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.10.094

J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557 551

In what follows, E will denote the Cartesian product of two Sobolev spaces W 1,p(x)0 (Ω) and W 1,q(x)0 (Ω), i.e. E =W 1,p(x)0 (Ω)×W 1,q(x)0 (Ω). X will denote the Sobolev spaceW 1,p(x)0 (Ω).In recent years, many publications [2–6,11–13,18,19] have appeared concerning quasilinear elliptic systems which have

been used in a great variety of applications. Existence andmultiplicity results for quasilinear elliptic systemswith variationalstructure have been broadly investigated.In [2], L. Boccardo and D. Figueiredo studied the existence of critical points (minima and saddle points) of functionals of

the type

Φ(u, v) =1p

∫Ω

|∇u|p +1q

∫Ω

|∇v|q −

∫Ω

F(x, u, v)

where p and q are real numbers larger than 1. That is to say the solutions of the system−1pu = Fu(x, u, v),−1qv = Fv(x, u, v)

where p and q are real numbers larger than 1.In [3], using the fibering method introduced by Pohozaev, Y. Bozhkova and E. Mitidieri proved the existence of multiple

solutions for a Dirichlet problem associated with a quasilinear system involving a pair of (p, q)-Laplacian operators.In [12] (the technical approach is based on the three-critical-point theorem in [15]), when p(x) ≡ p, q(x) ≡ q, µ = 0

Chun Li and Chun-Lei Tang ensured the existence of three solutions for the problem−1pu = λFu(x, u, v) inΩ,−1qv = λFv(x, u, v) inΩ,u = v = 0 on ∂Ω

where p > N , q > N and the F satisfies suitable assumptions.The aim of this paper is to prove the following result:

Theorem 1. Assume that there exist two positive constants c, d and two functionsγ (x), β(x) ∈ C(Ω)with1 < γ− < γ+ < p−,1 < β− < β+ < q− such that(j1) F(x, s, t) ≥ 0 for a.e. x ∈ Ω and all (s, t) ∈ [0, d] × [0, d];(j2) ∃ p1(x), q1(x) ∈ C(Ω) and p+ < p−1 ≤ p1(x) < p

∗(x), q+ < q−1 ≤ q1(x) < q∗(x) such that:

lim sup(s,t)→(0,0)

supx∈Ω

F(x, s, t)|s|p1(x) + |t|q1(x)

< +∞;

(j3) |F(x, s, t)| ≤ C(1+ |s|γ (x) + |t|β(x)) for a.e. x ∈ Ω and all (s, t) ∈ R× R;(j4) F(x, 0, 0) = 0 for a.e. x ∈ Ω . Then, there exist an open interval Λ ⊆ [0,+∞) and a positive real number r with the

following property: for each λ ∈ Λ and each function G:Ω × R× R→ R, measurable inΩ , C1 in R× R and satisfying

sup(x,s,t)∈Ω×R×R

|G(x, s, t)|1+ |s|p2(x) + |t|q2(x)

< +∞

where p2, q2 ∈ C(Ω) and p2(x) < p∗(x), q2(x) < q∗(x) for all x ∈ Ω , there exists δ > 0 such that, for each µ ∈ [0, δ],problem (P) has at least three weak solutions whose norms in W 1,p(x)0 (Ω)×W 1,q(x)0 (Ω) are less than r.

This paper is divided into three sections. In Section 2, we recall some basic facts about the variable exponent Lebesgueand Sobolev spaces, and recall B. Ricceri’s three-critical-points theorem. In Section 3, we give the proof of the main result.

2. Preliminaries

LetΩ be a bounded domain of RN(N ≥ 1)with a smooth boundary ∂Ω . SetC+(Ω) = h|h ∈ C(Ω), h(x) > 1 for all x ∈ Ω.L∞+(Ω) = p ∈ L∞(Ω) : ess inf

x∈Ωp(x) > 1.

For p ∈ L∞+(Ω), denote

p− = p−(Ω) = ess infx∈Ωp(x), p+ = p+(Ω) = ess sup

x∈Ωp(x).

Define

Lp(x)(Ω) =u|u : Ω → R is measurable and

∫Ω

|u(x)|p(x) dx <∞

with the norm

‖u‖Lp(x)(Ω) = |u|p(x) = inf

λ > 0 :

∫Ω

∣∣∣∣u(x)λ∣∣∣∣p(x) dx ≤ 1

.

552 J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557

DefineW 1,p(x)(Ω) =

u ∈ Lp(x)(Ω) : |∇u| ∈ Lp(x)(Ω)

with the norm

‖u‖W1,p(x)(Ω) = |u|p(x) + |∇u|p(x) .

Denote byW 1,p(x)0 (Ω) the closure of C∞0 (Ω) inW1,p(x)(Ω).

On the basic properties of the spaces Lp(x)(Ω),W 1,p(x)(Ω) andW 1,p(x)0 (Ω)we refer the reader to [7–10]. Here we displaysome facts which will be used later.

Proposition 2.1 (See [9]).(i) The spaces Lp(x) (Ω), W 1,p(x) (Ω) and W 1,p(x)0 (Ω) are separable and reflexive Banach spaces;(ii) If q ∈ C+(Ω) and q(x) < p∗(x) for any x ∈ Ω , then the imbedding from W 1,p(x) to Lq(x)(Ω) is compact and continuous;(iii) There is a constant C > 0, such that |u|p(x) ≤ C |∇u|p(x)∀ u ∈ W

1,p(x)0 (Ω).

By (iii) of Proposition 2.1, we know that |∇u|p(x) and ‖u‖ are equivalent norms on W1,p(x)0 (Ω). We will use |∇u|p(x) to

replace ‖u‖ in the following discussions.

Proposition 2.2 (See [7]). Set ρ(u) =∫Ω|u(x)|p(x) dx. For u, uk ∈ Lp(x) (Ω), we have

(1) For u 6= 0, |u|p(x) = λ⇔ ρ( uλ) = 1;

(2) |u|p(x) < 1(= 1;> 1)⇔ ρ(u) < 1(= 1;> 1);(3) If |u|p(x) > 1, then |u|

p−

p(x) ≤ ρ (u) ≤ |u|p+

p(x);

(4) If |u|p(x) < 1, then |u|p+

p(x) ≤ ρ (u) ≤ |u|p−

p(x);(5) limk→∞ |uk|p(x) = 0 ⇐⇒ limk→∞ ρ(uk) = 0;(6) |uk|p(x) →∞ ⇐⇒ ρ(uk)→∞.

In this paper, the space E will be endowed with the following equivalent norm:‖(u, v)‖ = ‖u‖ + ‖v‖,

here

‖u‖ = inf

λ > 0 :

∫Ω

∣∣∣∣∇uλ∣∣∣∣p(x) dx ≤ 1

, ‖v‖ = inf

µ > 0 :

∫Ω

∣∣∣∣∇vµ∣∣∣∣q(x) dx ≤ 1

.

Similar to Proposition 2.2, we have

Proposition 2.3. Set φ(u) =∫Ω|∇u|p(x)dx. For u, uk ∈ W 1,p(x)(Ω), we have

(1) For u 6= 0, ‖u‖ = λ⇔ φ( uλ) = 1;

(2) ‖u‖ < 1(= 1;> 1)⇔ φ(u) < 1(= 1;> 1);(3) If ‖u‖ > 1, then ‖u‖p

−

≤ φ (u) ≤ ‖u‖p+

;

(4) If ‖u‖ < 1, then ‖u‖p+

≤ φ (u) ≤ ‖u‖p−

;

(5) limk→∞ ‖uk‖ = 0 ⇐⇒ limk→∞ φ(uk) = 0;(6) ‖uk‖ → ∞ ⇐⇒ φ(uk)→∞.

Let G(u) =∫Ω

1p(x) |∇u|

p(x)dx, u ∈ X . We denote L = G′ : X → X∗, then

(L(u), v) =∫Ω

|∇u|p(x)−2∇u∇vdx ∀ u, v ∈ X .

Proposition 2.4 (see [9]).(i) L : X → X∗ is a continuous, bounded and strictly monotone operator;(ii) L is a mapping of type (S+), i.e. if un u in X and limn→∞((L(un)− L(u), un − u)) ≤ 0, then un → u in X;(iii) L : X → X∗ is a homeomorphism.

Proposition 2.5 (See [16]). Let X be a reflexive real Banach space; I ⊆ R an interval; Φ : X → R a sequentially weakly lowersemi-continuous C1 functional whose derivative admits a continuous inverse on X∗; J : X → R a C1 functional with compactderivative. In addition,Φ is bounded on each bounded subset of X. Assume that

lim‖x‖→+∞

(Φ(x)+ λJ(x)) = +∞ (2.1)

for all λ ∈ I , and that there exists ρ ∈ R such that

sup infλ∈Ix∈X

(Φ(x)+ λ(J(x)+ ρ)) < inf supx∈Xλ∈I

(Φ(x)+ λ(J(x)+ ρ)). (2.2)

Then, there exist a nonempty open set A ⊆ I and a positive real number r with the following property: for every λ ∈ A and everyC1 functional Ψ : X → R with compact derivative, there exists δ > 0 such that, for each µ ∈ [0, δ], the equation

J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557 553

Φ ′(x)+ λJ ′(x)+ µΨ ′(x) = 0

has at least three solutions in X whose norms are less than r.

Proposition 2.6 (See [14]). Let X be a nonempty set andΦ , J two real functionals on X. Assume that there are γ > 0, u0, u1 ∈ X,such that

Φ(u0) = J(u0) = 0, Φ(u1) > γ ,

supu∈Φ−1((−∞,γ ])

J(u) < γJ(u1)Φ(u1)

. (2.3)

Then, for each ρ satisfying

supu∈Φ−1((−∞,γ ])

J(u) < ρ < γJ(u1)Φ(u1)

,

one has

sup infλ≥0u∈X

(Φ(u))+ λ(ρ − J(u)) < inf supu∈Xλ≥0

(Φ(u)+ λ(ρ − J(u))).

3. Proof of the main result

Definition 3.1. We say that (u, v) ∈ E is a weak solution of problem (P) if∫Ω

(|∇u|p(x)−2 ∇u∇ξ + |∇v|q(x)−2 ∇v∇η)dx− λ∫Ω

(Fuξ + Fvη)dx− µ∫Ω

(Guξ + Gvη)dx = 0

for ∀ (ξ , η) ∈ E. The corresponding energy functional of problem (P) is

H(u, v) = Φ(u, v)+ λJ(u, v)+ µΨ (u, v)

=

∫Ω

1p(x)|∇u|p(x) +

1q(x)|∇v|q(x)dx− λ

∫Ω

F(x, u, v)dx− µ∫Ω

G(x, u, v)dx,

where

Φ(u, v) =∫Ω

1p(x)|∇u|p(x) +

1q(x)|∇v|q(x)dx;

J(u, v) = −∫Ω

F(x, u, v)dx; Ψ (u, v) = −∫Ω

G(x, u, v)dx.

Then, H(u, v) is a C1 functional and the critical points of it are weak solutions of problem (P).

Proof of Theorem 1. SetΦ , J , Ψ as above. So, for each u, v, ξ , η ∈ E, one has

Φ ′(u, v)(ξ, η) =∫Ω

|∇u|p(x)−2 ∇u∇ξdx+ |∇v|q(x)−2 ∇v∇ηdx

J ′(u, v)(ξ, η) = −∫Ω

Fu(x, u, v)ξdx−∫Ω

Fv(x, u, v)ηdx

Ψ ′(u.v)(ξ , η) = −∫Ω

Gu(x, u, v)ξdx−∫Ω

Gv(x, u, v)ηdx.

Hence, the weak solutions of problem (P) are exactly the solutions of the equation

Φ ′(u, v)+ λJ ′(u, v)+ µΨ ′(u.v) = 0.

From Proposition 2.4 (or [9] for details), of course, Φ is a continuously Gâteaux differentiable and sequentially weaklylower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on E∗, moreover, J and Ψ arecontinuously Gâteaux differentiable functional whose Gâteaux derivative are compact. Obviously, Φ is bounded on eachbounded subset of X under our assumptions.From Proposition 2.3, let G(u) =

∫Ω

1p(x) |∇u|

p(x)dx just as before, we have: if ‖u‖ ≥ 1, then

1p+‖u‖p

−

≤ G(u) ≤1p−‖u‖p

+

; (3.1)

554 J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557

if ‖u‖ < 1, then

1p+‖u‖p

+

≤ G(u) ≤1p−‖u‖p

−

. (3.2)

In fact, when ‖u‖ < 1 we can set C0 ≥ 1p+ ‖u‖

p−−

1p+ ‖u‖

p+≥ 0, then we can get

G(u) =∫Ω

1p(x)|∇u|p(x)dx ≥

1p+‖u‖p

−

− C0.

It follows that

G(u) =∫Ω

1p(x)|∇u|p(x)dx ≥

1p+‖u‖p

−

− C0 ∀u ∈ X .

So there exists a constant C1 ≥ 0, such that

Φ(u, v) =∫Ω

1p(x)|∇u|p(x) +

1q(x)|∇v|q(x)dx

≥1p+‖u‖p

−

+1q+‖v‖q

−

− C1

holds for any (u.v) ∈ E.

λJ(u, v) = −λ∫Ω

F(x, u, v)dx

≥ −λ

∫Ω

C(1+ |u|γ (x) + |v|β(x))dx

≥ −λC(|Ω| + |u|γ+

γ (x) + |u|γ−

γ (x) + |v|β+

β(x) + |v|β−

β(x))

≥ −C2(1+ |u|γ+

γ (x) + |v|β+

β(x))

≥ −C3(1+ ‖u‖γ+

+ ‖v‖β+

)

holds for any (u.v) ∈ E, where constants C2 ≥ 0, C3 ≥ 0. Here, we used conditions (j3) and (ii) of proposition (2.1).Combining the two inequalities above, we can get

Φ(u, v)+ λJ(u, v) ≥1p+‖u‖p

−

+1q+‖v‖q

−

− C3(1+ ‖u‖γ+

+ ‖v‖β+

)− C1,

because of γ+ < p−, β+ < q−, it follows that

lim‖(u,v)‖→+∞

(Φ(u, v)+ λJ(u, v)) = +∞ ∀(u, v) ∈ E, λ ∈ [0,+∞).

Then assumption (2.1) of Proposition 2.5 is satisfied.Next, we will prove that assumption (2.2) is also satisfied. It suffices to verify the conditions of Proposition 2.6. Let

(u0, v0) = (0, 0), we can easily have

Φ(u0, v0) = −J(u0, v0) = 0.

Now we claim that there exist γ > 0 and (u1, v1) ∈ E such thatΦ(u1, v1) > γ and (2.3) is satisfied.There is a point x0 ∈ Ω sinceΩ is a nonempty bounded open set. Let r2 > r1 > 0, put

w(x) =

0 x ∈ Ω \ B(x0, r2)

dr2 − r1

r2 −√√√√ N∑i=1

(xi − x0i )2

x ∈ B(x0, r2) \ B(x0, r1)

d x ∈ B(x0, r1)

here, B(x, r) stands for the open ball in RN of radius r centered at x.Let (u1(x), v1(x)) = (w(x), w(x)), then, thanks to (j1)we can obtain that

−J(u1, v1) = −J(w,w) =∫Ω

F(x, w,w) > 0.

J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557 555

From (j2), ∃ η ∈ [0, 1], C1 > 0, such that

F(x, s, t) < C1(|s|p1(x) + |t|q1(x))

< C1(|s|p−

1 + |t|q−

1 ) ∀(s, t) ∈ [−η, η] × [−η, η] a.e. x ∈ Ω.

From (j3), there are nine positive real numbers Mi (i = 1, 2, . . . , 9) according to |s|, |t| larger or smaller than η and 1. Forexample, when |s| > 1, |t| < η some

Mi = sup|s|>1,|t|<η

C(1+ |s|γ+

+ |t|β−

)

|s|p−

1 + |t|q−

1.

LetM = maxC1,M1, . . . ,M9, then

F(x, s, t) < M(|s|p−

1 + |t|q−

1 ) ∀(s, t) ∈ R× R a.e x ∈ Ω.

Consequently, fix γ such that 0 < γ < 1. And when 1p+ ‖u‖

p++

1q+ ‖v‖

q+≤ γ < 1, by the Sobolev embedding theorem

(X → Lp−

1 (Ω) is continuous), we have (for suitable positive constants C2, C3)

−J(u, v) =∫Ω

F(x, u, v)dx < M∫Ω

(|u|p−

1 + |v|q−

1 )dx

≤ C2(‖u‖p−

1 + ‖v‖q−

1 )

≤ C3(γp−1p+ + γ

q−1q+ ).

Since p−1 > p+, q−1 > q

+, we have

limγ→0+

sup 1p+‖u‖p++ 1

q+‖v‖q

+≤γ−J(u, v)

γ= 0. (3.3)

We choose w(x) ∈ X as above such that −J(w,w) > 0. Fix γ0 such that 0 < γ < γ0 < min 1p+ ,1q+ · min‖w‖

p++

‖w‖q+

, ‖w‖p−

+ ‖w‖q−

, 1 ≤ 1. Then, we divide the proof into two cases.(i) When ‖w‖ < 1, from (3.2) we have

Φ(u1, v1) = Φ(w,w)

=

∫Ω

1p(x)|∇w|p(x) +

1q(x)|∇w|q(x)dx

≥ min1p+,1q+

·

∫Ω

|∇w|p(x) + |∇w|q(x)dx

≥ min1p+,1q+

· (‖w‖p

+

+ ‖w‖q+

)

≥ γ0 > γ .

From (3.3), we know that

sup 1p+‖u‖p++ 1

q+‖v‖q

+≤γ−J(u, v) ≤

γ

2·

−J(u1, v1)max 1p− ,

1q− · (‖w‖

p− + ‖w‖q−)

≤γ

2·−J(u1, v1)Φ(u1, v1)

< γ ·−J(u1, v1)Φ(u1, v1)

.

(ii) When ‖w‖ ≥ 1, then from (3.1) we have

Φ(u1, v1) = Φ(w,w)

=

∫Ω

1p(x)|∇w|p(x) +

1q(x)|∇w|q(x)dx

≥ min1p+,1q+

·

∫Ω

|∇w|p(x) + |∇w|q(x)dx

556 J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557

≥ min1p+,1q+

· (‖w‖p

−

+ ‖w‖q−

)

≥ γ0 > γ .

From (3.3), we know that

sup 1p+‖u‖p++ 1

q+‖v‖q

+≤γ−J(u, v) ≤

γ

2·

−J(u1, v1)max 1p− ,

1q− · (‖w‖

p+ + ‖w‖q+)

≤γ

2·−J(u1, v1)Φ(u1, v1)

< γ ·−J(u1, v1)Φ(u1, v1)

.

For any (u, v) ∈ Φ−1((−∞, γ ]), we can getΦ(u, v) ≤ γ , i.e.∫Ω

1p(x)|∇u|p(x) +

1q(x)|∇v|q(x)dx ≤ γ .

Then, we can get

min1p+,1q+

·

∫Ω

|∇u|p(x) + |∇v|q(x)dx ≤ γ .

So, ∫Ω

|∇u|p(x) + |∇v|q(x)dx < γ ·1

min 1p+ ,1q+

< γ0 ·1

min 1p+ ,1q+

< 1.

This inequality implies∫Ω

|∇u|p(x)dx < 1∫Ω

|∇v|q(x)dx < 1,

i.e.

‖u‖ < 1 ‖v‖ < 1.

It follows that1p+‖u‖p

+

+1q+‖v‖q

+

<

∫Ω

1p(x)|∇u|p(x) +

1q(x)|∇v|q(x)dx ≤ γ .

So we can get that

Φ−1((−∞, γ ]) ⊂

(u, v) : (u, v) ∈ E,

1p+‖u‖p

+

+1q+‖v‖q

+

< γ

.

Then

sup(u,v)∈Φ−1((−∞,γ ])

−J(u, v) ≤ sup1p+‖u‖p++ 1

q+‖v‖q

+<γ

−J(u, v) < γ ·−J(u1, v1)Φ(u1, v1)

,

that is

sup(u,v)∈Φ−1((−∞,γ ])

−J(u, v) < γ ·−J(u1, v1)Φ(u1, v1)

.

So we can find γ > 0, u1 = v1 = w andΦ(w,w) ≤ γ satisfying (2.3). Also we can find ρ satisfying

sup(u,v)∈Φ−1((−∞,γ ])

−J(u, v) < ρ < γ ·−J(u1, v1)Φ(u1, v1)

.

Set I = [0,+∞), moreover,Φ(u, v) and−J(u, v) satisfy the assumption of Proposition 2.6, so using Proposition 2.6, wecan easily obtain that (2.2) is satisfied.Thus,Φ , J and Ψ satisfy all the assumptions of Proposition 2.5, and the proof is complete.

J. Liu, X. Shi / Nonlinear Analysis 71 (2009) 550–557 557

Remark. Applying ([1], Theorem 2.1) in the proof of Theorem 1, an upper bound of the interval of parameters λ for which(P) has at least three weak solutions is obtained. To be precise, in the conclusion of Theorem 1 one has

Λ ⊆

0, hγ

inf(u,v)∈Φ−1((−∞,γ ])

J(u, v)− γ · J(u1,v1)Φ(u1,v1)

for each h > 1 and (u1, v1) as in the proof of Theorem 1 (namely, u1 = v1 = w).

Acknowledgments

The authors would like to thank professor Xianling Fan for clear suggestions and the reviewer(s) for clear valuablecomments and suggestions.

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