doi:10.1155/2010/104625
Research Article
Large Solutions of Quasilinear Elliptic
System of Competitive Type: Existence and Asymptotic Behavior
Lin Wei
1and Zuodong Yang
1, 21Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu, Nanjing 210046, China
2College of Zhongbei, Nanjing Normal University, Jiangsu, Nanjing 210046, China
Correspondence should be addressed to Zuodong Yang,zdyang [email protected] Received 22 July 2009; Accepted 23 October 2009
Academic Editor: Wenming Zou
Copyrightq2010 L. Wei and Z. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the existence and asymptotic behavior of positive solutions for a class of quasilinear ellip- tic systems in a smooth boundary via the upper and lower solutions and the localization method.
The main results of the present paper are new and extend some previous results in the literature.
1. Introduction
This paper is concerned with the study of positive boundary blow-up solutions to a quasilinear elliptic system of competitive type:
Δpuaxuavb inΩ, Δpvbxucve inΩ, uv ∞ on∂Ω,
1.1
whereΩis a boundedC2domain ofRNandΔpstands for thep-Laplacian operator defined byΔpu div|∇u|p−2∇u, p > 1.The exponentsa, b, c, everifya, e > p−1, b, c > 0,a−p 1e−p1> bc.There existsCx, Dx∈CΩ, R, γx, ηx∈CΩ, Rsuch that
xlim→x0
ax
Cx0dxγx0 1, lim
x→x0
bx
Dx0dxηx0 1, 1.2
wherex0∈∂Ω, dx distx, ∂Ω.
We must emphasize that the weight functionsax, bxare allowed decaying to zero onΩwith arbitrary rate, depending upon the particular point of∂Ω. The boundary condition is to be understoodux → ∞, vx → ∞asdx → 0. Problems like1.1are usually known in the literature as boundary blow-up problems, and their solutions are also named large solutions or boundary blow-up solutions.
The problem of the previous form is mathematical models occuring in studies of the p-Laplace system, generalized reaction-diffusion theory, non-Newtonian fluid theory 1,2, non-Newtonian filtration 3, and the turbulent flow of a gas in porous medium. In the non- Newtonian fluid theory, the quantityp is a characteristic of the medium. Media with p >
2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p 2, they are Newtonian fluids. Whenp /2, the problem becomes more complicated since certain nice properties inherent to the casep 2 seem to be lost or at least difficult to verify. The main differences betweenp2 andp /2 can be founded in 4,5.
Whenp2, system1.1becomes
Δuaxuavb inΩ, Δvbxucve inΩ, uv ∞ on∂Ω,
1.3
for which the existence, uniqueness, and asymptotic behavior of large solutions have been investigated extensively. We list here, for example, 6–12.
This is a huge amount of literature dealing with single equation with infinite boundary conditions see, e.g., 13–34. This problem with more general nonlinearies and weight- function has been discussed by many authors recently 35–39.
Problem1.1 is considered in special case. Whenp 2, in 40, problem1.1was analyzed with ax 1, bx 1. In the same paper, some existence, uniqueness, and boundary behavior of solutions were obtained under the assumptions
ax∼C1dxk1, bx∼C2dxk2 1.4
asdx → 0for some positive constantsC1, C2and real numbersk1, k2>−2. This problem was later studied in 41with general form, where
C1dxγ1≤ax≤C2dxγ1,
C1dxγ1 ≤bx≤C2dxγ1, 1.5
for x ∈ Ω, γ1, γ2 ∈ RN, C1, C2, C1, C2 are positive constants. The author also obtained uniqueness results.
In 42, Yang extended the quasilinear elliptic system to Δpuum1vn1 inΩ, Δqvum2vn2 inΩ, uv ∞ on∂Ω,
1.6
wherem1 > p−1, n2 > q−1, m2, n1 > 0, andΩ ⊆ RNis a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: uλ, vμoruv ∞oru
∞, vμon∂Ω, whereλ, μ >0. Under several hypotheses on the parametersm1, n1, m2, n2, the author showed the existence of positive solutions and further provided the asymptotic behavior of the solutions near∂Ω.
When p /2, in 43, problem 1.1 was analyzed with ax 1, bx 1 under assumption1.4. The author obtained the existence, uniqueness, and behavior of solutions to problem1.1.
Very recently, Huang et al. 12 obtained existence, uniqueness, and asymptotic behavior of problem1.1whenp 2, and ax, bxsatisfy condition1.2. Motivated by the results of the papers 12,40,41,43, we consider the quasilinear elliptic system1.1. We modify the method developed by Huang et al. 12and extend the results to a quasilinear elliptic system1.1under condition1.2.
Throughout of this paper, set
C1min
x∈ΩCx, C2max
x∈Ω Cx, D1min
x∈ΩDx, D2max
x∈Ω Dx,
γ1max
x∈Ω γx, γ2min
x∈Ω γx, η1max
x∈Ω ηx, η2 min
x∈Ω ηx, α
x, y
px
e−p1
− py b
a−p1
e−p1
−bc , β x, y
py
a−p1
− px c
a−p1
e−p1
−bc ,
E x, y
⎛
⎝ p−1
αp−1α1e−p1 xb p−1
βp−1
β1b ye−p1
⎞
⎠
1/a−p1e−p1−bc
,
F x, y
⎛
⎝ p−1
βp−1
β1a−p1 yc p−1
αp−1α1c
xa−p1
⎞
⎠
1/a−p1e−p1−bc
,
1.7
nx0stands for the outward unit normal atx0∈∂Ω.
The paper is organized as follows. InSection 2we consider some preliminaries which will be used in proof ofTheorem 1.1. InSection 3we will give the proof of the main theorem.
By modifications of the arguments in the proof ofTheorem 1.1in 12, we obtain the following main results.
Theorem 1.1. Assume thatΩ is a boundedC2 domain ofRN,ax, bx ∈ CθΩfor someθ ∈ 0,1, ax, bx > 0 inΩ and verify 1.2, a−p 1e −p1 > bc, a, e > p− 1, b, c >
0, γx, ηx∈CΩ, Rand satisfy
b
e−p1 < pγx0
pηx0 < a−p1
c forx0∈∂Ω. 1.8
Then problem1.1has a solutionu, vif and only if b
e−p1 < pγ1
pη2 , pγ2
pη1 < a−p1
c . 1.9
And one has
xlim→x0
ux
dx−αγx0,ηx0EDx0, Cx0 1, 1.10
xlim→x0
vx
dx−βγx0,ηx0FDx0, Cx0 1. 1.11
2. Preliminaries
In this section, we will introduce some propositions.
Definition 2.1. u, vis a subsolution of
⎧⎨
⎩
Δpuaxuavb inΩ,
Δpvbxucve inΩ, provided
⎧⎨
⎩
Δpu≥axuavb inΩ,
Δpv≤bxucve inΩ. 2.1
A supersolutionu, vis defined by reversing the inequalities.
Proposition 2.2. Assume thatu, vis a subsolution andu, vis a supersolution of problem1.1, withu v u v ∞on∂Ω.Then problem1.1has at least a solutionu, vwithu ≤u ≤ u, v≥v≥vin Ω.In particularuv ∞on∂Ω.
Proposition 2.3 see 43. Assume that ax, bx satisfy 1.4, then problem 1.1 admits a positive solutionu, vwithuv ∞on∂Ωif and only ifk1, k2>−pand
b
e−p1 < pk1
pk2 < a−p1
c . 2.2
This solution is unique and satisfies
xlim→x0
ux
dx−αk1,k2EC2, C1 1, lim
x→x0
vx
dx−βk1,k2FC2, C1 1 2.3 for eachx0∈∂Ω.
Next, we are ready to study two auxiliary problems in a ball and an annuli. To this aim, for given 0< R1< Randx0∈RN, N≥1,set
BRx0
x∈RN:|x−x0|< R
, AR1,Rx0
x∈RN:R1<|x−x0|< R
. 2.4
Proposition 2.4. AssumeΩ BRx0,a−p1e−p1> bc, a, e > p−1, b, c >0, Cr, Dr∈ C 0, R, Randγ, η >0 satisfy
b
e−p1 < pγ
pη < a−p1
c . 2.5
Then the following systems
ΔpΦ CrR−rγΦaΨb inBRx0, ΔpΨ DrR−rηΦcΨe in BRx0,
Φ Ψ ∞ on∂BRx0
2.6
possess a unique radially symmetric positive solutionΦr,Ψrsatisfying
xlim→x0
Φr
EDR, CRR−r−αγ,η 1,
xlim→x0
Ψr
FDR, CRR−r−βγ,η 1,
2.7
wherer|x−x0|.
Proof. At first, we consider the following systems Φp−2Φ
N−1 r
Φp−2Φ
CrR−rγΦaΨb in0, R, Ψp−2Ψ
N−1 r
Ψp−2Ψ
DrR−rηΦcΨe in0, R, ΦR ΨR ∞, Φ0 Ψ0 0.
2.8
We will show that problem2.8has a solutionΦr,Ψr,which provide a positive radially symmetric solution to problem2.6. Indeed, any positive solutionΦr,Ψrof the integral equation system
Φr l r
0
t1−N
t
0
sN−1CsR−sγΦaΨbds 1/p−1
dt, 0< r < R,
Ψr m r
0
t1−N
t
0
sN−1DsR−sηΦcΨeds 1/p−1
dt, 0< r < R,
2.9
provides a solution of2.8, whereΦ0 l,Ψ0 m,ΦR ∞,ΨR ∞.
DefineΦ0r l,Ψ0r mfor all 0< r < R, let{Φk},{Ψk}be the function sequences given by
Φkr l r
0
t1−N
t
0
sN−1CsR−sγΦak−1Ψbk−1ds 1/p−1
dt, 0< r < R,
Ψkr m r
0
t1−N
t
0
sN−1DsR−sηΦck−1Ψek−1ds 1/p−1
dt, 0< r < R,
2.10
subject toΦk0 l,Ψk0 m,ΦkR ΨkR k.
We remark that{Φk},{Ψk}are nondecreasing sequences. In fact,
Φ1r l
lamb1/p−1r
0
t1−N
t
0
sN−1CsR−sγds 1/p−1
dt
l
lamb1/p−1
Ar≥l Φ0r,
Ψ1r m lcme1/p−1 r
0
t1−N
t
0
sN−1DsR−sηds 1/p−1
dt m lcme1/p−1Br≥m Ψ0r,
2.11
where
Ar r
0
t1−N
t
0
sN−1CsR−sγds 1/p−1
dt,
Br r
0
t1−N
t
0
sN−1DsR−sηds 1/p−1
dt.
2.12
Proceeding by the same manner, we conclude that
l≤Φk≤Φk1, m≤Ψk≤Ψk1. 2.13
We now prove that{Φk},{Ψk}are bounded in0, R. To prove this, we consider ΔpΥ
CrR−rγDrR−rη
Υab Υce
, 2.14
problem2.14has a large radially symmetric solutionΥr, and
Υr Υ0 r
0
t1−N
t
0
sN−1
CsR−sγDsR−sη
Υab Υce ds
1/p−1 dt,
2.15
whereΥ0 lm. It follows that
Φ1r l
lamb1/p−1r
0
t1−N
t
0
sN−1CsR−sγds 1/p−1
dt
≤Υ0 r
0
t1−N
t
0
sN−1
CsR−sγDsR−sη
Υab Υce ds
1/p−1 dt Υr.
2.16
Similarly, we haveΨ1≤Υr.
Arguing as before, we obtain Φk ≤ Υr,Ψk ≤ Υr. Therefore, we show that {Φk},{Ψk} are nondecreasing and bounded sequences in 0, R, which implies that the following limit holds
Φ,Ψ lim
k→ ∞Φk,Ψk, 2.17
we deduce thatΦ,Ψis a positive solution of2.8. ThenΦx,Ψx Φr,Ψris a positive radially symmetric solution to problem2.6and
ΦR lim
r→RΦr ∞, ΨR lim
r→RΨr ∞. 2.18
Secondly, it is clear that
C1R−rγ ≤CrR−rγ ≤C2R−rγ, D1R−rη≤DrR−rη≤D2R−rη. 2.19
By2.5andProposition 2.3, we have
ED1, C2≤ lim
r→R
Φr
R−r−αγ,η ≤ED2, C1, FD2, C1≤ lim
r→R
Ψr
R−r−βγ,η ≤FD1, C2.
2.20
Denote by
l lim
r→R
Φr
R−r−αγ,η, k lim
r→R
Ψr
R−r−βγ,η. 2.21
By usingγp a−p1αγ, η bβγ, η, ηp e−p1βγ, η cαγ, ηandLHopital rule, we obtain
l lim
r→R
Φ0 r
0
t1−Nt
0sN−1CsR−sγΦaΨbds1/p−1 dt R−r−α
lim
r→R
r1−Nr
0tN−1CtR−tγΦaΨbdt1/p−1
αR−r−α−1
rlim→R
r1−Nr
0tN−1CtR−tγΦaΨbdt αR−r−α1p−1
1/p−1
rlim→R
1−Nr−Nr
0tN−1CtR−tγΦaΨbdtCrR−rγΦaΨb αα1
p−1
R−r−αpα−p
1/p−1
CR αα1
p−1lim
r→RR−raαbβΦaΨb 1−N
αα1
p−1lim
r→R
r−Nr
0tN−1CtR−tγΦaΨbdt R−r−αpα−p
1/p−1
CR αα1
p−1lakb 1−N αα1
p−1lim
r→R
r−Nr
0tN−1CtR−tγΦaΨbdt R−r−αpα−p
1/p−1 .
2.22 We note that
0≤ lim
r→R
r−Nr
0tN−1CtR−tγΦaΨbdt R−r−αpα−p
≤ lim
r→R
r−1r
0CtR−tγΦaΨbdt R−r−αpα−p lim
r→R
CrR−rγΦaΨbdt R
−αpα−p
R−r−αpα−p1 CR
R
−αpα−plim
r→RR−rγαp−αp1ΦaΨb CR
R
−αpα−plakblim
r→RR−r 0.
2.23
This implies that
lp−1 CR
αα1
p−1lakb. 2.24
Similarly, we obtain
kp−1 DR
β β1
p−1lcke. 2.25
Since
αp−1α1 p−1
CR EDR, CRa−p1FDR, CRb,
βp−1 β1
p−1
DR EDR, CRcFDR, CRe−p1.
2.26
If 0< α <1,1< p≤2, then
Ea−p1Fb αp−1α1 p−1
CR ≥ αα1
p−1
CR la−p1kb, 2.27
therefore, we getE≥l, F ≥k. If 0< α < 1, p >2, we getE≤l, F ≤k. So, when 0< α <1, we getEl, Fk.
Similarly, whenα≥1, we also getEl, Fk.
By 2.24 and 2.25, we conclude that l EDR, CR, k FDR, CR, this completes the proof.
Proposition 2.5. Assumea−p1e−p1> bc, a, e > p−1, b, c >0, γ >0, η >0,and b
e−p1 < pγ
pη < a−p1
c , 2.28
Dr, Cr ∈ C R1, R, R are the reflection around R0 R1 R/2 of some functions Cr, Dr∈C R0, R, R. Then the following system
ΔpΦ CrdxγΦaΨb in AR1,Rx0, ΔpΨ DrdxηΦcΨe inAR1,Rx0,
Φ Ψ ∞ on∂AR1,Rx0
2.29
has a unique radially symmetric positive solutionΦr,Ψrsuch that
dxlim→0
Φr
EDR, CRdx−αγ,η 1,
dxlim→0
Ψr
FDR, CRdx−βγ,η 1,
2.30
where
dx dx, ∂AR1,Rx0
⎧⎨
⎩
R− |x−x0|, if R0≤ |x−x0| ≤R,
|x−x0| −R1, if R1≤ |x−x0| ≤R0. 2.31
Proof. The proof is similarl to the proof ofProposition 2.4, so we omit it here.
3. Proof of Theorem 1.1
We are now ready to proveTheorem 1.1, whose proof will be split into the following several lemmas.
Lemma 3.1. Assume a − p 1e − p 1 > bc, a, e > p − 1, b, c > 0, Cx, Dx ∈ CΩ, ax, bx>0in Ωand1.2holds,γx, ηx>0 and satisfy
b
e−p1 < pγx0
pηx0 < a−p1
c , 3.1
for eachx0∈∂Ω, then problem1.1has a solutionu, vif b
e−p1 < pγ1
pη2, pγ2
pη1 < a−p1
c . 3.2
Proof. By3.2andProposition 2.3, the following system ΔpΦ C1dxγ1ΦaΨb inΩ, ΔpΨ D2dxη2ΦcΨe inΩ,
Φ Ψ ∞ on∂Ω
3.3
possesses a positive solutionu1, v1. Next we will show that
u, v
⎛
⎝
mn m−nb/a−p1
1/p−1
u1,
m−n mnc/e−p1
1/p−1
v1
⎞
⎠ 3.4
is a supersolution of1.1, ifmis sufficiently large and 0< m−n <1, wherem, n∈Rand m > n. In fact, by
γ1p
a−p1 α
γ1, η2
bβ γ1, η2
, η2p
e−p1 β
γ1, η2
cα γ1, η2
. 3.5
We haveu, vis a supersolution of1.1provided
C1dxγ1≤axmne−p1a−p1−bc/e−p1p−1, D2dx≥bxm−na−p1e−p1−bc/a−p1p−1.
3.6
Sinceax, bx∈CΩ, choosingmis large enough, andm−n >0 is sufficiently small, we can prove that
u, v
⎛
⎝
m−n mnb/a−p1
1/p−1
u2,
mn m−nc/e−p1
1/p−1
v2
⎞
⎠ 3.7
is a subsolution of1.1, whereu2, v2is a solution of the following problem:
ΔpΦ C2dxγ2ΦaΨb inΩ, ΔpΨ D1dxη1ΦcΨe inΩ,
Φ Ψ ∞ on∂Ω.
3.8
Then byProposition 2.2, problem1.1has a solution.
Lemma 3.2. Assume that problem1.1has a solutionu, v, then1.9holds.
Proof. In fact, if1.9does not hold, it will lead to a contradiction. FromLemma 3.1, we find that ifmis large enough andm−n >0 is sufficiently small, we have
u≤u
mn m−nb/a−p1
1/p−1
u1, v≤v
mn m−nc/e−p1
1/p−1
v2. 3.9
On the other hand, by2.3, there existsε >0 such that forx∈Ωε {x∈Ω:dx, ∂Ω≤ε}, we get
u≤
mn m−nb/a−p1
1/p−1
u1≤
mn m−nb/a−p1
1/p−1
ED2, C1dx−αγ1,η2. 3.10
Thus, if
b
e−p1 ≥ pγ1
pη2, 3.11
by the definition ofαγ1, η2, we obtainαγ1, η2≤ 0. By3.10, it implies thatuis bounded forx∈Ωε, which is impossible sinceux ∞asdx distx, ∂Ω → 0. If
pγ2
pη2 ≥ a−p1
c , 3.12
it is similarly proved thatvis bounded near∂Ω, which is also a contradiction. The proof of Lemma 3.2is complete.
Lemma 3.3. Letu, vbe a positive solution of1.1, then1.10and1.11hold.
Proof. Fixτ∈0,1, by1.2, there exitsσ∈0,1such that, ifdx, x0< σ,
ax≥1−τCx0dxγx0, bx≤1τDx0dxηx0, 3.13
wherex0∈∂Ω. For a fixedx0∈∂Ω, set
Σ Bσ/2
∂Ω 3.14
and chooseR >0 small enough such that
K
y∈Σ
BR
y−Rny
⊂Bσx0
Ω, 3.15
wherenystands for the outward unit normal aty∈∂Ω.
Forx∈Bσx0∩Ω, we get
ax≥1−τCx0dxγx0, bx≤1τDx0dxηx0. 3.16
SinceΩis ofC2bounded domain, there exitR >0 andσ0>0 such that BRx0−Rσnx0⊂Ω, BRx0−Rnx0
∂Ω {x0}, 3.17
for eachσ∈0, σ0.
LetuB,σ, vB,σbe any positive radially symmetric solution to the following system:
Δpu 1−τCx0R− |x−x0|γx0uavb inBRx0−Rσnx0, Δpv 1τDx0R− |x−x0|ηx0ucve in BRx0−Rσnx0,
uv ∞ on∂BRx0−Rσnx0.
3.18
It is easy to see thatuσ, vσ u, v|BRx0−Rσnx0 is a positive smooth subsolution of3.18, whereu, vis a positive solution of1.1.
Then we get
uσ u|BRx0−Rσnx
0≤uB,σ, vσ v|BRx0−Rσnx
0≥vB,σ. 3.19
LetuB, vBbe any positive solution to the following system:
Δpu 1−τCx0R− |x−x0|γx0uavb in BRx0−Rnx0, Δpv 1τDx0R− |x−x0|ηx0ucve inBRx0−Rnx0,
uv ∞ on∂BRx0−Rnx0.
3.20
ByProposition 2.3,uB, vBsatisfies
rlim→R
uB
E1τDx0,1−τCx0R−r−αγx0,ηx0 1, 3.21
rlim→R
vB
F1τDx0,1−τCx0R−r−βγx0,ηx0 1, 3.22 wherer|x−x0|.
Taking into account that, forx∈BRx0−Rσnx0,
uB,σx uBxσnx0, vB,σx vBxσnx0, 3.23
by3.19, for eachx∈BRx0−Rσnx0andσ∈0, σ0, we have
ux≤uBxσnx0, vx≥vBxσnx0. 3.24
Letσ → 0, we have
ux≤uBx, vx≥vBx. 3.25
It follows immediately from3.21,3.22that
rlim→R
u
ER−r−α ≤ lim
r→R
uB
ER−r−α 1, 3.26
rlim→R
v
FR−r−β ≥ lim
r→R
vB
FR−r−β 1, 3.27
whereEE1τDR,1−τCR, FF1τDR,1−τCR.
We next have to prove the inverse inequalities. Similarly, there exitsR > R1 > 0 and σ0 > 0 such that Ω ⊂
0<σ<σ0AR1,Rx0 R σnx0 and AR0,Rx0 R1nx0∂Ω {x0}.
Fix a sufficiently smallτ, there exit radially symmetric functionsa:AR1,Rx0R1nx0 → R andb:AR1,Rx0R1nx0 → Rsuch thata≥a, b≤binΩ, and
max
AR1,Rx0R1nx0
a≤max
Ω a1, max
Ω b1≤ max
AR1,Rx0R1nx0
b, 3.28
and for eachx∈AR1,Rx0R1nx0
ax a1|x−x0−R1nx0| d
x, ∂AR1,Rx0R1nx0 γx0
,
bx b1|x−x0−R1nx0| d
x, ∂AR1,Rx0R1nx0 ηx0
,
3.29
wherea1, b1∈C R1, R, R, satisfing
a1R1 Cx0 τ, b1R1 Dx0−τ. 3.30
We now consider the system
Δpuaxuavb inAR1,Rx0R1nx0, Δpvbxucve inAR1,Rx0R1nx0, uv ∞ on∂AR1,Rx0R1nx0.
3.31
ByProposition 2.5, problem3.31possesses a solutionuA, vA. But for the system
Δpuaxuavb inAR1,Rx0 R1σnx0, Δpvbxucve inAR1,Rx0 R1σnx0, uv ∞ on∂AR1,Rx0 R1σnx0,
3.32
it has a solutionuA,σ, vA,σ, and for eachx∈AR1,Rx0 R1σnx0, we have
uA,σx, vA,σx uAx−σnx0, vAx−σnx0. 3.33
It is also clear thatuAx, vAx uA,σx, vA,σx|Ω is a subsolution of problem1.1.
Thus for eachx∈AR1,Rx0 R1σnx0, we getuAx−σnx0≤ux, vAx−σnx0≥vx.
Letσ → 0, we haveuAx≤ux, vAx≥vx. Thus forx∈K, we get
1 lim
|x| →R
uAx E
ax, bx
R− |x|−αγx0,ηx0
≤ lim
dx→0
ux E
ax, bx
R− |x|−αγx0,ηx0,
3.34
1 lim
|x| →R
vAx F
ax, bx
R− |x|−βγx0,ηx0
≥ lim
dx→0
vx F
ax, bx
R− |x|−βγx0,ηx0
3.35
but we have limτ→0K{x0}. Therefore, by3.26,3.27,3.34, and3.35, we finish1.10 and1.11. The proof ofLemma 3.3is complete. FromLemma 3.1toLemma 3.3, we finish the proof ofTheorem 1.1.
Acknowledgments
This paper was supported by the National Natural Science Foundation of ChinaGrant no.
10871060by the Natural Science Foundation of the Jiangsu Higher Education Institutions of ChinaGrant no. 8KJB110005.
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