Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 526026, 13 pages
doi:10.1155/2011/526026
Research Article
Multiplicity of Solutions for Nonlocal Elliptic
System of p, q-Kirchhoff Type
Bitao Cheng,1 Xian Wu,2 and Jun Liu1
1
College of Mathematics and Information Science, Qujing Normal University, Qujing,
Yunnan 655011, China
2
Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China
Correspondence should be addressed to Bitao Cheng, chengbitao2006@126.com
Received 16 May 2011; Accepted 23 June 2011
Academic Editor: Josip E. Pečarić
Copyright q 2011 Bitao Cheng et al. 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.
This paper is concerned with the following nonlocal elliptic system of p,q-Kirchhoff type
p−1
q−1
−M1 Ω |∇u|p Δp u λFu x, u, v, in Ω, −M2 Ω |∇v|q Δq v λFv x, u, v, in Ω, u v 0,
on ∂Ω. Under bounded condition on M and some novel and periodic condition on F, some new
results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic
system are obtained by means of Bonanno’s multiple critical points theorems without the PalaisSmale condition and Ricceri’s three critical points theorem, respectively.
1. Introduction and Preliminaries
We are concerned with the following nonlocal elliptic system of p, q-Kirchhoff type:
p−1
− M1
Δp u λFu x, u, v,
|∇u|p
in Ω,
q−1
Δq v λFv x, u, v,
− M2
|∇v|q
in Ω,
Ω
Ω
u v 0;
1.1
on ∂Ω,
where Ω ⊂ RN N ≥ 1 is a bounded smooth domain, λ ∈ 0, ∞, p > N, q > N, Δp is the
p-Laplacian operator
Δp u div |∇u|p−2 ∇u ,
and Mi : R
→ R, i 1, 2, are continuous functions with bounded conditions.
1.2
2
Abstract and Applied Analysis
M There are two positive constants m0 , m1 such that
m0 ≤ Mi t ≤ m1 ,
∀t ≥ 0, i 1, 2.
1.3
Furthermore, F : Ω × R × R → R is a function such that Fx, s, t is measurable in
x for all s, t ∈ R × R and Fx, s, t is C1 in s, t for a.e. x ∈ Ω, and Fu denotes the
partial derivative of F with respect to u. Moreover, Fx, s, t satisfies the following.
F1 Fx, 0, 0 0 for a.e. x ∈ Ω.
F2 There exist two positive constants γ < p, β < q and a positive real function αx ∈
L∞ Ω such that
|Fx, s, t| ≤ αx 1 |s|γ |t|β ,
for a.e. x ∈ Ω and all s, t ∈ R × R.
1.4
The system 1.1 is related to a model given by the equation of elastic strings
∂2 u
ρ 2 −
∂t
E
P0
h 2L
L 2 2
∂u dx ∂ u 0
∂x ∂x2
1.5
0
which was proposed by Kirchhoff 1 as an extension of the classical D’Alembert’s wave
equation for free vibrations of elastic strings, where the parameters in 1.5 have the following
meanings: ρ is the mass density, P0 is the initial tension, h is the area of the cross-section, E is
the Young modulus of the material, and L is the length of the string. Kirchhoff’s model takes
into account the changes in length of the string produced by transverse vibrations.
Later, 1.5 was developed to the form
utt − M
Ω
|∇u|2 Δu fx, u
in Ω,
1.6
where M : R
→ R is a given function. After that, many people studied the nonlocal elliptic
boundary value problem
−M
Ω
|∇u|2 Δu fx, u
in Ω, u 0 on ∂Ω,
1.7
which is the stationary counterpart of 1.6. It is pointed out in 2 that 1.7 models several
physical and biological systems, where u describes a process which depends on the average of
itself e.g., population density. By using the methods of sub and supersolutions, variational
methods, and other techniques, many results of 1.7 were obtained, we can refer to 2–12
and the references therein. In particular, Alves et al. 2, Theorem 4 supposes that M satisfies
bounded condition M and fx, t satisfies condition AR, that is, for some ν > 2 and R > 0
such that
0 < ν Fx, t ≤ fx, tt, ∀|t| ≥ R, x ∈ Ω,
AR
Abstract and Applied Analysis
3
t
where Fx, t 0 fx, sds; one positive solutions for 1.7 was obtained. It is well known
that condition AR plays an important role for showing the boundedness of Palais-Smale
sequences. More recently, Corrêa and Nascimento in 13 studied a nonlocal elliptic system
of p-Kirchhoff type
p−1
Δp u fu, v ρ1 x, in Ω,
− M1
|∇u|p
Ω
p−1
Δp v gu, v ρ2 x, in Ω,
− M2
|∇v|p
P
Ω
∂u ∂v
0, on ∂Ω,
∂η ∂η
where η is the unit exterior vector on ∂Ω, and Mi , ρi i 1, 2, f, g satisfy suitable
assumptions, where a special and important condition: periodic condition on nonlinearity
was assumed. They obtained the existence of a weak solution for the nonlocal elliptic
system of p-Kirchhoff type P under Neumann boundary condition via Ekeland’s Variational
Principle.
In the present paper, our objective is to consider the nonlocal elliptic system of
p, q-Kirchhoff-type 1.1, instead of the nonlocal elliptic system of p-Kirchhoff type and
single Kirchhoff type equation. Under bounded condition on M and some novel conditions
without P S condition and periodic condition on F, we will prove the existence of two
solutions and three solutions of system 1.1 by means of one multiple critical points theorem
without the Palais-Smale condition of Bonanno in 14 and an equivalent formulation 15,
Theorem 2.3 of Ricceri’s three critical points theorem 16, Theorem 1, respectively.
In order to state our main results, we need the following preliminaries.
1,p
1,q
Let X W0 Ω × W0 Ω be the Cartesian product of two Sobolev spaces, which is a
reflexive real Banach space endowed with the norm
u, v
u
p v
q ,
1,p
1.8
1,q
where · p and · q denote the norms of W0 Ω and W0 Ω, respectively. That is,
u
p 1,p
Ω
|∇u|p
1/p
v
q ,
Ω
|∇v|q
1/q
1.9
1,q
for all u ∈ W0 Ω and v ∈ W0 Ω.
1,p
1,q
Since p > N and q > N, W0 Ω and W0 Ω are compactly embedded in C0 Ω. Let
C max
⎧
⎨
⎩
sup
1,p
u∈W0 Ω\{0}
maxx∈Ω |ux|p
p
u
p
,
sup
1,q
v∈W0 Ω\{0}
⎫
maxx∈Ω |vx|q ⎬
;
q
⎭
v
q
1.10
4
Abstract and Applied Analysis
then we have C < ∞. Furthermore, it is known from 17 that
1/N maxx∈Ω |ux|p
p − 1 1−1/P 1/N−1/P N
N −1/P
sup
Γ 1
≤ √
,
|Ω|
2
p−N
u
p
π
1,p
u∈W Ω\{0}
0
1.11
where Γ denotes the Gamma function and |Ω| is the Lebesgue measure of Ω. Additionally,
1.11 is an equality when Ω is a ball.
Recall that u, v ∈ X is called a weak solution of system 1.1 if
M1
Ω
|∇u|p
p−1 Ω
−λ
Ω
q−1 |∇u|p−2 ∇u∇ϕ M2
|∇v|q
|∇v|q−2 ∇v∇ψ
Ω
Fu x, u, vϕxdx − λ
Ω
Ω
1.12
Fx, u, vdx
1.13
Fv x, u, vψxdx 0,
for all ϕ, ψ ∈ X. Define the functional I : X → R given by
1
Iu, v M
1
p
Ω
|∇u|
p
1
M
2
q
Ω
|∇v|
q
−λ
Ω
for all u, v ∈ X, and where
1 t M
t
M1 sp−1 ds,
0
2 t M
t
M2 sq−1 ds,
∀t ≥ 0.
1.14
0
By the conditions M and F2, it is easy to see that I ∈ C1 X, R and a critical point of I
corresponds to a weak solution of the system 1.1.
Now, giving x0 ∈ Ω and choosing R2 > R1 > 0 such that Bx0 , R2 ⊆ Ω, where Bx, R N
{y ∈ R : |y − x| < R}. Next we give some notations.
1/P
N 1/P
C1/P RN
π N/2
2 − R1
,
R2 − R1
Γ1 N/2
1/q
N 1/q
C1/q RN
π N/2
2 − R1
.
R2 − R1
Γ1 N/2
α1 α1 N, p, R1 , R2
α2 α2 N, q, R1 , R2
1.15
Abstract and Applied Analysis
5
Moreover, let a, c be positive constants, denote
⎛
⎞
N 2 1/2
a ⎝
⎠,
R2 −
yx xi − x0i
R2 − R1
i1
Ac s, t ∈ R × R : |s|p |t|q ≤ c ,
ka Bx0 ,R2 \Bx0 ,R1 hc, a ka − gc,
∀x ∈ Bx0 , R2 \ Bx0 , R1 ,
gc F x, yx, yx dx ⎧
⎫
⎨ mp−1 mq−1 ⎬
1
, 1
,
M
max
⎩ p
q ⎭
sup Fx, s, tdx,
Ω s,t∈Ac
Bx0 ,R1 Fx, a, adx,
⎧
⎫
⎨ mp−1 mq−1 ⎬
0
M− min
, 0
.
⎩ p
q ⎭
1.16
Now we are ready to state our main results for the system 1.1
Theorem 1.1. Assume that F1-F2 hold and there are three positive constants a, c1 , c2 with
c1 < aα1 p aα2 q ≤ 1 < c2 such that
M
gc1 < M− hc1 , a,
M
gc2 < M− hc1 , a.
1.17
Then, for each
λ∈
!
"
M
aα1 p aα2 q M− c1 aα1 p aα2 q
1
1
,
,
min
,
Chc1 , a
C
gc1 gc2 1.18
there exists a positive real number ρ such that the system 1.1 has at least two weak solutions ui , vi ∈
X i 1, 2 whose norms in C0 Ω are less than some positive constant ρ.
Theorem 1.2. Assume that F1-F2 hold and there are two positive constants a, b, with aα1 p aα2 q > bM
/M− such that
F3 Fx, s, t ≥ 0 for a.e. x ∈ Ω \ Bx0 , R1 and all s, t ∈ 0,a × 0, a;
F4 aα1 p aα2 q |Ω|supx,s,t∈Ω×AbM
/M− Fx, s, t < b Bx0 ,R1 Fx, a, adx.
Then there exist an open interval Λ ⊆ 0, ∞ and a positive real number ρ such that, for each
λ ∈ Λ, the system 1.1 has at least three weak solutions wi ui , vi ∈ X i 1, 2, 3 whose norms
wi are less than ρ.
2. Proofs of Main Results
Before proving the results, we state one multiple critical points theorem without the PalaisSmale condition of Bonanno in 13 and an equivalent formulation 14, Theorem 2.3 of
Ricceri’s three critical points theorem 15, Theorem 1, which are our main tools.
6
Abstract and Applied Analysis
Theorem 2.1 see 14, Theorem 2.1. Let X be a reflexive real Banach space, and let Ψ, Φ : X → R
be two sequentially weakly lower semicontinuous functions. Assume that Ψ is (strongly) continuous
and satisfies lim
u
→ ∞ Ψu ∞. Assume also that there exist two constants r1 and r2 such that
j infX Ψ < r1 < r2 ;
jj ϕ1 r1 < ϕ2 r1 , r2 ;
jjj ϕ1 r2 < ϕ2 r1 , r2 ;
where
ϕ1 r Φu − infu∈Ψ−1 −∞,rw Φu
inf
r − Ψu
u∈Ψ−1 −∞,r
ϕ2 r1 , r2 inf
u∈Ψ−1 −∞,r1 ,
2.1
Φu − Φv
.
sup
Ψv
− Ψu
v∈Ψ−1 r1 ,r2 Then, for each
λ∈
!
"
1
1
1
, min
,
,
ϕ2 r1 , r2 ϕ1 r1 ϕ2 r2 2.2
the functional Ψ λΦ has two local minima which lie in Ψ−1 −∞, r1 and Ψ−1 r1 , r2 , respectively.
Theorem 2.2 see 15, Theorem 2.3. Let X be a separable and reflexive real Banach space.
Ψ : X → R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous
functional whose Gâteaux derivative admits a continuous inverse on X ∗ ; Φ : X → R is a continuously
Gâteaux differentiable functional whose Gâteaux derivative is compact. Suppose that
i lim
u
→ ∞ Ψu λΦu ∞ for each λ > 0;
ii There are a real number r, and u0 , u1 ∈ X such that Ψu0 < r < Ψu1 ;
iii infu∈Ψ−1 −∞,r Φu > Ψu1 − rΦu0 r − Ψu0 Φu1 /Ψu1 − Ψu0 .
Then there exist an open interval Λ ⊆ 0, ∞ and a positive real number ρ such that, for each
λ ∈ Λ, the equation Ψ u λΦ u 0 has at least three weak solutions whose norms in X are less
than ρ.
First, we give one basic lemma.
Lemma 2.3. Assume that M and F2 hold; let
Ψu, v 1
M1
p
Ω
|∇u|p
1
M
2
q
Ω
|∇v|q ,
Φu, v −
Ω
Fx, u, vdx,
2.3
for all u, v ∈ X. Then Ψ and Φ are continuously Gâteaux differentiable and sequentially weakly
lower semicontinuous functionals. Moreover, the Gâteaux derivative of Ψ admits a continuous inverse
on X ∗ and the Gâteaux derivative of Φ is compact.
Abstract and Applied Analysis
7
Proof. By condition M, it is easy to see that Ψ is continuously Gâteaux differentiable.
Moreover, the Gâteaux derivative of Ψ admits a continuous inverse on X ∗ . Thanks to p >
N, q > N, and F2, Φ is continuously Gâteaux differentiable and sequentially weakly lower
semicontinuous functional whose Gâteaux derivative is compact. Next We will prove that Ψ
is a sequentially weakly lower semicontinuous functional. Indeed, for any un , vn ∈ X with
1,p
1,q
un , vn u, v in X, then un u in W0 Ω and vn v in W0 Ω. Therefore,
lim inf un p ≥ u
p ,
lim inf vn q ≥ v
q
n→∞
2.4
n→∞
due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and
1 and M
2 , we conclude that
monotonicity of M
1
1 lim inf
1
M
|∇u|p ≤ M
|∇un |p ≤ lim inf M
|∇un |p ,
n→∞
n→∞
Ω
Ω
Ω
q
q
2 lim inf
2
2
≤M
M
|∇v|
|∇vn | ≤ lim inf M
|∇vn |q ,
n→∞
Ω
n→∞
Ω
2.5
Ω
Consequently, Ψ is a sequentially weakly lower semicontinuous functional.
Proof of Theorem 1.1. Let
1
Ψu, v M
1
p
Ω
|∇u|
p
1
M
2
q
q
Ω
|∇v|
,
Φu, v −
Ω
Fx, u, vdx
2.6
for all u, v ∈ X. Under condition M, by a simple computation, we have
p
q
p
q
M− u
p v
q ≤ Ψu, v ≤ M
u
p v
q .
2.7
Therefore, 2.7 implies that
Ψu, v ∞.
lim
2.8
u,v
→ ∞
Put
r1 M− c 1 aα1 p aα2 q ,
C
r2 M− c 2 aα1 p aα2 q .
C
2.9
Denote
ϕ1 r ϕ2 r1 , r2 u,v∈
inf
Φu, v − infu,v∈
Ψ−1 −∞,r
inf
u,v∈ Ψ−1 −∞,r1 Ψ−1 −∞,rw
Φu, v
,
r − Ψu, v
Φu, v − Φu1 , v1 ,
sup
u1 ,v1 ∈ Ψ−1 r1 ,r2 Ψu1 , v1 − Ψu, v
and Ψ−1 −∞, rw is the closure of Ψ−1 −∞, r in the weak topology.
2.10
8
Abstract and Applied Analysis
Set
w0 x ⎧
⎪
0,
⎪
⎪
⎪
⎪
⎪
⎨
⎛
a ⎝
R2 −
⎪
⎪
R2 − R1
⎪
⎪
⎪
⎪
⎩
a,
x ∈ Ω \ Bx0 , R2 ,
1/2 ⎞
⎠,
xi − xi
N
x ∈ Bx0 , R2 \ Bx0 , R1 ,
0
2.11
i1
x ∈ Bx0 , R1 .
Then u0 , v0 ∈ X, where u0 x v0 x w0 x and
p
p
u0 p w0 p aα1 p
,
C
q
q
v0 q w0 q aα1 q
.
C
2.12
Consequently, 2.7 and 2.12 imply that
r1 < Ψu0 , v0 < r2 .
2.13
Furthermore, 2.13 implies that
ϕ2 r1 , r2 inf
sup
Φu, v − Φu1 , v1 − Ψu, v
u,v∈ Ψ−1 −∞,r1 u ,v ∈ Ψ−1 r ,r Ψu1 , v1 1 1
1 2
Φu, v − Φu0 , v0 .
≥
inf
−1
u,v∈Ψ −∞,r1 Ψu0 , v0 − Ψu, v
2.14
On the other hand, by F1, 1.17, and, 2.11, one has
M
Fx, u0 , v0 dx ka > hc1 , a >
gc1 > gc1 M−
Ω
sup Fx, s, tdx.
Ω s,t∈Ac1 2.15
For each u, v ∈ X with Ψu, v ≤ r1 , and x ∈ Ω, by 2.7, we conclude
Cr
p
q
1
c1 aα1 p aα2 q ≤ c1 .
|ux|p |vx|q ≤ C u
p v
q ≤
M−
2.16
Abstract and Applied Analysis
9
Therefore, the combination of 2.15 and 2.16 implies
Φu, v − Φu0 , v0 Ψu0 , v0 − Ψu, v
≥
≥
Fx, u0 , v0 dx − Ω Fx, u, vdx
Ψu0 , v0 − Ψu, v
Fx, u0 , v0 dx − Ω sup|ux|p |vx|q ≤c1 Fx, u, vdx
Ω
Ω
Ψu0 , v0 − Ψu, v
Fx, u0 , v0 dx − Ω sup|ux|p |vx|q ≤c1 Fx, u, vdx
Ω
Ω
≥
Fx, u0 , v0 dx −
M
Ψu0 , v0 2.17
sup|ux|p |vx|q ≤c1 Fx, u, v dx
p
q
u0 p v0 q
Ω
C
hc1 , a.
M
aα1 p aα2 q
By 2.14 and 2.17, we have
ϕ2 r1 , r2 ≥
M
C
hc1 , a.
aα1 p aα2 q
2.18
Similarly, for every u, v ∈ X such that Ψu, v ≤ r, where r is a positive real number, and
x ∈ Ω, one has
Cr
p
q
.
|ux|p |vx|q ≤ C u
p v
q ≤
M−
2.19
By virtue of Ψ being sequentially weakly lower semicontinuous, then Ψ−1 −∞, rw Ψ−1 −∞, r. Consequently,
ϕ1 r ≤
≤
inf
Φ0, 0 − infu,v∈
Ψ−1 −∞,rw
Ψ−1 −∞,rw
Φu, v
r − Ψ0, 0
−infu,v∈
Ω
Φu, v
r − Ψu, v
u,v∈ Ψ−1 −∞,r
≤
Φu, v − infu,v∈
Ψ−1 −∞,rw
2.20
Φu, v
r
sup|ux|p |vx|q ≤Cr/M− Fx, u, vdx
r
.
10
Abstract and Applied Analysis
It implies that
ϕ1 r1 ≤
gc1 C
C
hc1 , a,
p
q gc1 <
r1
M− c1 aα1 aα2 M aα1 p aα2 q
2.21
ϕ1 r2 ≤
gc2 C
C
hc1 , a.
p
q gc2 <
r2
M− c2 aα1 aα2 M aα1 p aα2 q
2.22
By 2.18–2.22, we conclude
ϕ1 r1 ≤ ϕ2 r1 , r2 ,
ϕ1 r2 ≤ ϕ2 r1 , r2 .
2.23
Therefore, the conditions j, jj, and jjj in Theorem 2.1 are satisfied. Consequently, by
Lemma 2.3 and above facts, the functional Ψ λΦ has two local minima u1 , v1 , u2 , v2 ∈ X,
which lie in Ψ−1 −∞, r1 and Ψ−1 r1 , r2 , respectively. Since I Ψ λΦ ∈ C1 , u1 , v1 , u2 , v2 ∈
X are the solutions of the equation
Ψ u, v λΦ u, v 0.
2.24
Then u1 , v1 , u2 , v2 ∈ X are the weak solutions of system 1.1.
Since Ψui , vi < r2 , i 1, 2, by 1.10 and 2.7,
|ui x|p |vi x|q ≤
Cr2
≤ c2 ,
M−
2.25
i 1, 2;
which implies there exists a positive real number ρ such that the norms of ui , vi ∈ X i 1, 2 in C0 Ω are less than some positive constant ρ. This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. Let
Ψu, v 1
M1
p
Ω
|∇u|p
1
M
2
q
Ω
|∇v|q ,
Φu, v −
Ω
Fx, u, vdx
2.26
for all u, v ∈ X. By F2 and 2.7, we have
1
q
Fx, u, vdx
|∇u| M2
|∇v| − λ
q
Ω
Ω
Ω
p
q
αx 1 |ux|γ |vx|β dx
≥ M− u
p v
q − λ
1
Ψu, v λΦu, v M
1
p
p
Ω
p
q
γ
β
≥ M− u
p v
q − λ
α
∞ |Ω| k1 u
p k2 v
q ,
2.27
Abstract and Applied Analysis
11
where k1 , k1 are positive constants. Since γ < p, β < q, 2.27 implies that
lim
Ψu, v λΦu, v ∞.
2.28
u,v
→ ∞
The same as in 2.11, defining a function w0 x, and letting u0 x v0 x w0 x, then
2.12 is also satisfied. Choosing r bM
/C, by 2.7, 2.12, and aα1 p aα2 q > bM
/M− ,
we conclude
M M− bM
p
q
−
Ψu0 , v0 ≥ M− u0 p v0 q r.
aα1 p aα2 q >
C
C M−
2.29
By F3 and the definitions of u0 and v0 , one has
|Ω|
sup
x,s,t∈Ω×AbM
/M− b
Fx, a, adx
aα1 p aα2 q Bx0 ,R1 Fx, a, adx
bM
Bx0 ,R1 C M
aα1 p aα2 q /C
bM
Ω\Bx0 ,R1 Fx, u0 , v0 dx Bx0 ,R1 Fx, u0 , v0 dx
≤
p
q
C
M
u v Fx, s, t <
0 p
bM
≤
C
Ω
0 q
Fx, u0 , v0 dx
.
Ψu0 , v0 2.30
For every u, v ∈ X such that Ψu, v ≤ r, and x ∈ Ω, one has
Cr
C bM
bM
p
q
.
|ux|p |vx|q ≤ C u
p v
q ≤
M− M− C
M−
By the combination of 2.30 and 2.31, we have
sup
u,v∈Ψ−1 −∞,r
−Φu, v ≤
sup
{u,v|Ψu,v≤r}
sup
Ω
Fx, u, vdx
{u,v||ux|p |vx|q ≤bM
/M− }
Ω
Fx, u, vdx
2.31
12
Abstract and Applied Analysis
≤
sup
Ω s,t∈AbM
/M− ≤ |Ω|
sup
Fx, s, tdx
x,s,t∈Ω×AbM
/M− bM
≤
C
r
Ω
Fx, s, t
Fx, u0 , v0 dx
Ψu0 , v0 −Φu0 , v0 .
Ψu0 , v0 2.32
Therefore,
Φu0 , v0 .
Ψu0 , v0 2.33
Ψu0 , v0 − rΦ0, 0 r − Ψ0, 0Φu0 , v0 .
Ψu0 , v0 − Ψ0, 0
2.34
inf
Φu, v > r
u,v∈Ψ−1 −∞,r
Note that Φ0, 0 Ψ0, 0 0, we conclude that
inf
u,v∈Ψ−1 −∞,r
Φu, v >
Hence, by Lemma 2.3 and above facts, Ψ and Φ satisfy all conditions of Theorem 2.2; then the
conclusion directly follows from Theorem 2.2.
Acknowledgments
The authors would like to thank the referee for the useful suggestions. This work is supported
in part by the National Natural Science Foundation of China 10961028, Yunnan NSF
Grant no. 2010CD086, and the Foundation of young teachers of Qujing Normal University
2009QN018.
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