Open Access

Fourth order elliptic system with dirichlet boundary condition

Journal of Inequalities and Applications20112011:60

https://doi.org/10.1186/1029-242X-2011-60

Received: 13 February 2011

Accepted: 17 September 2011

Published: 17 September 2011

Abstract

We investigate the multiplicity of the solutions of the fourth order elliptic system with Dirichlet boundary condition. We get two theorems. One theorem is that the fourth order elliptic system has at least two nontrivial solutions when λ k < c < λk+1and λk+n(λk+n- c) < a + b < λk+n+1(λk+n+1- c). We prove this result by the critical point theory and the variation of linking method. The other theorem is that the system has a unique nontrivial solution when λ k < c < λk+1and λ k (λ k - c) < 0, a+b < λk+1(λk+1- c). We prove this result by the contraction mapping principle on the Banach space.

AMS Mathematics Subject Classification: 35J30, 35J48, 35J50

Keywords

Fourth order elliptic systemfourth order elliptic equationvariation linking theoremcontraction mapping principle

1. Introduction

Let Ω be a smooth bounded region in R n with smooth boundary ∂Ω. Let λ1< λ2 ≤ ... ≤ λ k ≤ ... be the eigenvalues of -Δ with Dirichlet boundary condition in Ω. In this paper we investigate the multiplicity of the solutions of the following fourth order elliptic system with Dirichlet boundary condition
Δ 2 u + c Δ u = a ( ( u + v + 1 ) + 1 ) in  Ω , Δ 2 v + c Δ v = b ( ( u + v + 1 ) + 1 ) in  Ω , u = 0, v = 0, Δ u = 0, Δ v = 0 on  Ω ,
(1.1)
where c R, u+ = max{u, 0} and a, b R are constant. The single fourth order elliptic equations with nonlinearities of this type arises in the study of travelling waves in a suspension bridge ([6]) or the study of the static deflection of an elastic plate in a fluid and have been studied in the context of the second order elliptic operators. In particular, Lazer and McKenna [6] studied the single fourth order elliptic equation with Dirichlet boundary condition
Δ 2 u + c Δ u = b ( ( u + 1 ) + 1 ) , in  Ω , u = 0 , Δ u = 0 on  Ω .
(1.2)
Tarantello [10] also studied problem (1.2) when c < λ1 and bλ1(λ1 - c). She show that (1.2) has at least two solutions, one of which is a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [8] proved that if c < λ1 and bλ2(λ2 - c), then (1.2) has at least four solutions by the Leray-Schauder degree theory. Micheletti, Pistoia and Sacon [9] also proved that if c < λ1 and bλ2(λ2 - c), then (1.2) has at least three solutions by variational methods. Choi and Jung [2] also considered the single fourth order elliptic problem
Δ 2 u + c Δ u = b u + + s in  Ω , u = 0 , Δ u = 0 on  Ω .
(1.3)

They show that (1.3) has at least two nontrivial solutions when c < λ1, λ1(λ1 - c) < b < λ2(λ2 - c) and s < 0 or when λ1< c < λ2, b < λ1(λ1 - c) and s > 0. They also obtained these results by using the variational reduction method. They [3] also proved that when c < λ1, λ1(λ1 - c) < b < λ2(λ2 - c) and s < 0, (1.3) has at least three solutions by using degree theory. In [79] the authors investigate the existence of solutions of jumping problems with Dirichlet boundary condition.

In this paper we improve the multiplicity results of the single fourth order elliptic problem to that of the fourth order elliptic system. Our main results are as follows:

THEOREM 1.1. Suppose that ab ≠ 0 and d e t 1 1 b - a 0 . Let λ k < c < λk+1and λk+n(λk+n- c) < a + b < λk+n+1(λk+n+1- c). Then system (1.1) has at least two nontrivial solutions.

THEOREM 1.2. Suppose that ab ≠ 0 and d e t 1 1 b - a 0 . Let λ k < c < λk+1and λ k (λ k - c) < 0, a + b < λk+1(λk+1- c). Then system (1.1) has a unique nontrivial solution.

In section 2 we define a Banach space H spanned by eigenfunctions of Δ2 + c Δ with Dirichlet boundary condition and investigate some properties of system (1.1). In section 3, we prove Theorem 1.1 by using the critical point theory and variation of linking method. In section 4, we prove Theorem 1.2 by using the contraction mapping principle.

2. Fourth order elliptic system

The eigenvalue problem Δ2u + c Δu = μu in Ω with u = 0, Δu = 0 on ∂Ω has also infinitely many eigenvalues μ k = λ k (λ k - c), k ≥ 1 and corresponding eigenfunctions ϕ k , k ≥ 1. We note that λ1(λ1 - c) < λ2(λ2 - c) ≤ λ3(λ3 - c) < .

The system
Δ 2 u + c Δ u = a ( ( u + v + 1 ) + 1 ) Δ 2 v + c Δ v = b ( ( u + v + 1 ) + 1 ) u = 0, v = 0, Δ u = 0, Δ v = 0 in  Ω , in  Ω , on  Ω
can be transformed to the equation
Δ 2 ( u + v ) + c Δ ( u + v ) = ( a + b ) ( ( u + v + 1 ) + 1 ) in  Ω , u = 0, v = 0, Δ u = 0, Δ v = 0 on  Ω .
(2.1)
We also have
Δ 2 ( b u - a v ) + c Δ ( b u - a v ) = 0 in  Ω , u = 0 , v = 0 , Δ u = 0 , Δ v = 0 on  Ω .
It follows from the above equation that bu - av = 0. If u + v = w is a solution of (2.1), then the system
u + v = w , b u - a v = 0
has a unique solution of (1.1) since det 1 1 b - a 0 . Hence the number of the solutions w = u + v of (1.1) is equal to that of (2.1). To investigate the multiplicity of (1.1) it is enough to find the multiplicity of (2.1). Let us set w = u + v. Then (2.1) is equivalent to the equation
Δ 2 w + c Δ w = ( a + b ) ( ( w + 1 ) + 1 ) in  Ω , w = 0, Δ w = 0 , on Ω .
(2.2)
Any element u L2(Ω) can be expressed by
u = h k ϕ k  with h k 2 < .
Let H be a subspace of L2(Ω) defined by
H = { u L 2 ( Ω ) | | λ k ( λ k - c ) | h k 2 < } .
Then this is a complete normed space with a norm
u = [ | λ k ( λ k - c ) | h k 2 ] 1 2 .
Since λ k (λ k - c) → + ∞ and c is fixed, we have
  1. (i)

    Δ2 u + c Δu H implies u H.

     
  2. (ii)

    u C u L 2 ( Ω ) , for some C > 0.

     
  3. (iii)

    u L 2 ( Ω ) = 0 if and only if || u || = 0.

     

For the proof of the above results we refer [1].

LEMMA 2.1. Assume that c is not an eigenvalue of, a + bλ k (λ k - c) and bounded. Then all solutions in L2(Ω) of
Δ 2 w + c Δ w = ( a + b ) ( ( w + 1 ) + 1 ) i n L 2 ( Ω )

belong to H.

Proof. Let us write (a + b)((w + 1)+ - 1) = ∑h k ϕ k L2(Ω).
( Δ 2 + c Δ ) 1 ( a + b ) ( ( w + 1 ) + 1 ) = 1 λ k ( λ k c ) h k ϕ k L 2 ( Ω ) . ( Δ 2 + c Δ ) 1 ( a + b ) ( ( w + 1 ) + 1 ) = | λ k ( λ k c ) | 1 ( λ k ( λ k c ) ) 2 h k 2 C h k 2 = C w L 2 ( ω ) 2 <

for some C > 0. Thus (Δ2 + c Δ)-1((a + b)((w + 1)+ -1)) H.   ■

With the aid of Lemma 2.1 it is enough that we investigate the existence of the solutions of (1.1) in the subspace H of L2(Ω).

Let us define the functional
F ( w ) = Ω 1 2 | Δ w | 2 - c 2 | w | 2 - a + b 2 | w + 1 | + - ( a + b ) w .
(2.3)

If we assume that λ k < c < λk+1and a + b is bounded, F (u) is well defined. By the following lemma, F(w) C1. Thus the critical points of the functional F(w) coincide with the weak solutions of (2.2).

LEMMA 2.2. Assume that λ k < c < λk+1and a + b is bounded. Then the functional F(w) is continuous and Frechét differentiable in H and
D F ( w ) ( h ) = Ω [ Δ w Δ h c w h ( a + b ) ( w + 1 ) + h ( a + b ) h ] d x
(2.4)

for h H.

Proof. First we shall prove that F(w) is continuous at w. Let w, z H.
F ( w + z ) F ( w ) = Ω [ 1 2 | Δ ( w + z ) | 2 c 2 | ( w + z ) | 2 a + b 2 | ( w + z + 1 ) + | 2 ( a + b ) ( w + z ) ] d x Ω [ 1 2 | Δ w | 2 c 2 | w | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) w ] d x = Ω [ w ( Δ 2 z + c Δ z ) + 1 2 z ( Δ 2 z + c Δ z ) ( a + b 2 | ( w + z + 1 ) + | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) z ) ] d x .
Let w = ∑h k ϕ k , z = h ̃ k ϕ k . Then we have
| Ω w ( Δ 2 z + c Δ z ) d x | = | Ω λ k ( λ k - c ) h k h ̃ k | w z , | Ω z ( Δ 2 z + c Δ z ) d x | = | λ k ( λ k - c ) h ̃ k 2 | z 2 .
On the other hand, by Mean Value Theorem, we have
a + b 2 | ( w + z + 1 ) + | 2 a + b 2 | ( w + 1 ) + | 2 ( a + b ) z .
Thus we have
a + b 2 | ( w + z + 1 ) + | 2 - a + b 2 | ( w + 1 ) + | 2 - ( a + b ) z 2 ( a + b ) z = O ( z ) .
Thus F(w) is continuous at w. Next we shall prove that F(w) is Fréchet differentiable at w H. We consider
| F ( w + z ) - F ( w ) - D F ( w ) z | = | Ω 1 2 z ( Δ 2 z + c Δ z ) - ( a + b 2 | ( w + z + 1 ) + | 2 - a + b 2 | ( w + 1 ) + | 2 + ( a + b ) ( w + 1 ) + z ) | 1 2 z 2 + ( a + b ) z + ( a + b ) ( w + 1 ) z = z ( 1 2 z + ( a + b ) + ( a + b ) ( w + 1 ) ) = O ( z ) .

Thus F(w) is Fréchet differentiable at w H.   ■

3. Proof of Theorem 1.1

Throughout this section we assume that λ k < c < λk+1and λk+n(λk+n- c) < a + b < λk+n+1(λk+n+1- c). We shall prove Theorem 1.1 by applying the variation of linking method (cf. Theorem 4.2 of [8]). Now, we recall a variation of linking theorem of the existence of critical levels for a functional.

Let X be an Hilbert space, Y X, ρ > 0 and e X\Y , e ≠ 0. Set:
B ρ ( Y ) = { x Y : x X ρ } , S ρ ( Y ) = { x Y : x X = ρ } , Δ ρ ( e , Y ) = { σ e + v : σ 0 , v Y , σ e + v X ρ } , Σ ρ ( e , Y ) = { σ e + v : σ 0 , v Y , σ e + v X = ρ } { v : v Y , v X ρ } .

THEOREM 3.1. ("A Variation of Linking") Let × be an Hilbert space, which is topological direct sum of the subspaces X1and X2. Let F C1(X, R). Moreover assume:

(a) dim X1< +∞;

(b) there exist ρ > 0, R > 0 and e X1, e ≠ 0 such that ρ < R and
sup S ρ ( X 1 ) F < inf Σ R ( e , X 2 ) F ;

(c) - < a = in f Δ R ( e , X 2 ) F ;

(d) (P.S.) c holds for any c [a, b], where b = sup B ρ ( X 1 ) F .

Then there exist at least two critical levels c 1 and c 2 for the functional F such that :
inf Δ R ( e , X 2 ) F c 1 sup S ρ ( X 1 ) F < inf Σ R ( e , X 2 ) F c 2 sup B ρ ( X 1 ) F .
Let H+be the subspace of H spanned by the eigenfunctions corresponding to the eigenvalues λ k (λ k - c) > 0 and H - the subspace of H spanned by the eigenfunctions corresponding to the eigenvalues λ k (λ k - c) < 0. Then H = H+ H-. Let H k be the subspace of H spanned by ϕ1, , ϕ k whose eigenvalues are λ1(λ1 - c), , λ k (λ k - c). Let H k be the orthogonal complement of H k in H. Then
H = H k H k .
Let e H+Hk+n, e ≠ 0 and ρ > 0. Let us set
B ρ ( H k + n ) = { w H k + n | w ρ } , S ρ ( H k + n ) = { w H k + n | w = ρ } , Δ ρ ( e , H k + n ) = { σ e + w | σ 0 , w H k + n , σ e + w ρ } , Σ ρ ( e , H k + n ) = { σ e + w | σ 0 , w H k + n , σ e + w = ρ } { w | w H k + n , w ρ } .
Let L : HH be the linear continuous operator such that
( L w ) z = Ω ( Δ 2 w + c Δ w ) z d x - ( a + b ) Ω w z d x .
(3.1)
Then L is an isomorphism and Hk+n, H k + n are the negative space and the positive space of L. Thus we have
( L w ) w - ( ( a + b ) - λ k + n ( λ k + n - c ) ) w 2 , w H k + n ,
(3.2)
( L w ) w ( λ k + n + 1 ( λ k + n + 1 - c ) - ( a + b ) ) w 2 , w H k + n .
(3.3)
We can write
F ( w ) = 1 2 ( L w ) w - ψ ( w ) ,
where
ψ ( w ) = Ω a + b 2 | ( w + 1 ) - | 2 d x .

Since H is compactly embedded in L2, the map Dψ : HH is compact.

LEMMA 3.1. Let λ k < c < λk+1and λk+n(λk+n- c) < a + b < λk+n+1(λk+n+1- c). Then F(w) satisfies the (P.S.) γ condition for any γ R.

Proof. Let (w n ) be a sequence in H with DF(w n ) → 0 and F(w n ) → γ. Since L is an isomorphism and is compact, it is sufficent to show that (w n ) is bounded in H. We argue by contradiction. we suppose that ||w n || → +∞. Let z n = w n w n . Up to a subsequence, we have z n z in H. Since DF (w n ) → 0, we get
D F ( w n ) w n w n 2 = Ω ( Δ 2 + c Δ ) z n 2 - Ω [ ( a + b ) ( z n + 1 w n ) + z n - ( a + b ) z n w n ] 0 .
(3.4)
Let P + : H H k + n and P- : HH k+n denote the orthogonal projections. Since P+z n -P-z n is bounded in H, we have
Ω ( Δ 2 + c Δ ) ( P + z n + P - z n ) ( P + z n - P - z n ) - Ω [ ( a + b ) ( P + z n + P - z n + 1 w n ) + ( P + z n - P - z n ) ] 0 .
(3.5)
Since P+z n - P-z n P+z - P-z in H, from (3.2) and (3.3) we get
0 Ω [ ( ( a + b ) z + ) ( P + z - P - z ) ] d x .
Hence z ≠ 0. On the other hand, from (3.5), we get
0 = Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω [ ( a + b ) z + ( P + z P z ) ] Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω [ ( a + b ) z ( P + z ) P z ) ] = Ω ( Δ 2 + c Δ ) ( P + z + P z ) ( P + z P z ) Ω ( a + b ) ( P + z ) + P z ) ( P + z ) P z ) = Ω ( Δ 2 + c Δ ( a + b ) ) ( P + z ) 2 d x Ω ( Δ 2 + c Δ ( a + b ) ) ( P z ) 2 ( λ k + n + 1 ( λ k + n + 1 c ) ( a + b ) ) P + z L Ω 2 ( λ k + n ( λ k + n c ) ( a + b ) ) P z L 2 ( Ω ) 2 .
(3.6)

The last line of (3.6) is positive or equal to 0 since λk+n+1(λk+n+1- c) - (a + b) > 0 and - (λk+n(λk+n- c) - (a + b)) > 0. Thus the only possibility to hold (3.6) is that P+z = 0 and P-z = 0. Thus z = 0, which gives a contradiction.

LEMMA 3.2. Let λ k < c < λk+1and λk+n(λk+n- c) < b < λk+n+1(λk+n+1- c).

Then

(i) there exists Rk+n> 0 such that the functional F(w) is bounded from below on H k + n ;
inf w H k + n | | w | | = R k + n F ( w ) > 0 a n d inf w H k + n | | w | | < R k + n F ( w ) > - .
(3.7)
(ii) there exists ρk+n> 0 such that
sup w H k + n | | w | | = ρ k + n F ( w ) < 0 a n d sup w H k + n | | w | | ρ k + n F ( w ) < .
(3.8)
Proof. (i) Let w H k + n . Then we have
lim w H k + n | | w | | + F ( w ) lim w H k + n | | w | | 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + Ω [ a + b 2 | ( w + 1 ) + | 2 - ( a + b ) w - r 2 w 2 ] d x lim w H k + n | | w | | 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + Ω [ a + b 2 ( w 2 + 1 ) - r 2 w 2 ] d x lim w H k + n | | w | | + 1 2 ( 1 - r λ k + n + 1 ( λ k + n + 1 - c ) ) w 2 - lim w H k + n | | w | | + 1 2 ( ( a + b ) - r ) Ω w 2 - a + b 2 | Ω | + ,
since a + b - r < λ k + n + 1 ( λ k + n + 1 - c ) - r = λ k + n + 1 ( λ k + n + 1 - c ) - λ k + n ( λ k + n - c ) 2 . Thus there exists Rk+n> 0 such that inf w H k + n | | w | | = R k + n F ( w ) > 0 . Moreover if w H k + n with ||w|| < Rk+n,then we have
F ( w ) 1 2 ( λ k + n + 1 ( λ k + n + 1 - c ) ) | | w | | L 2 ( Ω ) 2 - Ω [ a + b 2 ( w + 1 ) 2 - - ( a + b ) w ] d x (1) > 1 2 { ( λ k + n + 1 ( λ k + n + 1 - c ) ) - ( a + b ) } w L 2 ( Ω ) 2 - Ω a + b 2 d x > - . (2) (3) 
Thus we have inf w H k + n | | w | | < R k + n F ( w ) > - - .
  1. (ii)

    Let w H k+n. Then

     

( L w ) w ( λ k + n ( λ k + n - c ) - r ) Ω w 2 d x λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 Ω w + 2 ,

Ω [ 1 2 ( a + b ) | ( w + 1 ) + | 2 - ( a + b ) w - r w 2 ] d x Ω [ 1 2 ( a + b ) | w + | 2 - ( a + b ) w - r w + 2 ] d x , ,

so that
F ( w ) 1 2 λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 Ω w + 2 (1)  - a + b - r 2 Ω w + 2 + Ω ( a + b ) w d x (2)  1 2 { λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) } w + L 2 ( Ω ) 2 (3)  + ( a + b ) w L 2 ( Ω ) . (4)  (5) 

Since λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) < 0 , there exists ρ k+n > 0 such that if w H k+n and ||w|| = ρ k+n , then sup F(w) < 0. Moreover, if w H k+n and ||w|| ≤ ρ k+n , then we have F ( w ) 1 2 { λ k + n ( λ k + n - c ) - λ k + n + 1 ( λ k + n + 1 - c ) 2 - ( a + b - r ) } w + L 2 ( Ω ) 2 + ( a + b ) w L 2 ( Ω ) ( a + b ) w L 2 ( Ω ) < .   ■

LEMMA 3.3. Let λ k < c < λk+1, λ k+n (λ k+n - c) < a + b < λk+n+1(λk+n+1- c) and let e1 H+H k+n with ||e1|| = 1. Then there exists R k + n - such that R k + n - > ρ k + n ,
inf w Σ R k + n - ( e 1 , H k + n ) F ( w ) 0 a n d inf w Δ R k + n - ( e 1 , H k + n ) F ( w ) - .
Proof. Let us chose w H k + n and σ ≥ 0 and e1 H+H k+n with ||e1|| = 1. Then we get
F ( w + σ e 1 ) 1 2 λ k + n + 1 ( λ k + n + 1 - c ) w L 2 ( Ω ) 2 + σ 2 2 e 1 2 (1) - Ω [ a + b 2 ( w + σ e 1 + 1 ) 2 - ( a + b ) ( w + σ e 1 ) ] d x (2) = 1 2 { λ k + n + 1 ( λ k + n + 1 - c ) - ( a + b ) } w L 2 ( Ω ) 2 + σ 2 2 ( Λ - ( a + b ) ) e 1 L 2 ( Ω ) 2 (3) - ( a + b ) σ 2 w L 2 ( Ω ) e 1 L 2 ( Ω ) - a + b 2 | Ω | , (4) (5) 

where λk+1(λk+1- c) ≤ Λ ≤ λk+1(λk+1- c). Choose σ > 0 so mall that σ 2 e 1 2 is small. We can choose a number R k + n - > 0 such that R k + n - > σ , R k + n - > ρ k + n , and inf w H k + n , σ 0 | | w + σ e 1 | | = R k + n F ( w + σ e 1 ) 0 : Moreover if w H k + n , σ 0 w + σ e 1 R k + n - , then F ( w ) σ 2 2 ( Λ - b ) e 1 L 2 ( Ω ) 2 - ( a + b ) σ w L 2 ( Ω ) e 1 L 2 ( Ω ) - a + b 2 | Ω | - . Thus we prove the lemma.   ■

Proof of Theorem 1.1

By Lemma 2.2, F(w) is continuous and Frechét differentiable in H. By Lemma 3.1. F(w) satisfies the (P.S.) γ condition for any γ R. We note that the subspace S ρ k + n H k + n and the subspace Σ R k + n - ( e 1 , H k + n ) link at the subspace {e1}. By Lemma 3.2 and Lemma 3.3, we have
sup w S ρ k + n H k + n F ( w ) < inf w Σ R k + n - ( e 1 , H k + n ) F ( w ) .

By Lemma 3.3, we also have inf w Δ R k + n - ( e 1 , H k + n ) F ( w ) > - Thus by the variation of linking theorem, there exists at least two nontrivial solutions of (2.2). Thus we complete the Theorem 1.1.

4. Proof of Theorem 1.2

Proof of Theorem 1.2

Assume that λ k < c < λk+1and λ k (λ k - c) < 0, b < λk+1(λk+1- c). Let r = 1 2 { λ k ( λ k - c ) + λ k + 1 ( λ k + 1 - c ) } . We can rewrite (2.2) as
( Δ 2 + c Δ - r ) w = ( a + b ) ( w + 1 ) + - r ( w + 1 ) + + r ( w + 1 ) + - r w - ( a + b ) in  L 2 ( Ω ) , w = 0 , Δ w = 0 on  Ω .
(4.1)
or
w = ( Δ 2 + c Δ - r ) - 1 [ ( a + b ) ( w + 1 ) + - r ( w + 1 ) + + r ( w + 1 ) + - r w - ( a + b ) ] in  L 2 ( Ω ) , w = 0 , Δ w = 0 on  Ω .
(4.2)
We note that the operator (Δ2 +c Δ - r)-1 is a compact, self-adjoint and linear map from L2(Ω) into L2(Ω) with norm 2 λ k + 1 ( λ k + 1 - c ) - λ k ( λ k - c ) , and
( ( a + b ) - r ) { ( w 2 + 1 ) + - ( w 1 + 1 ) + } + r { ( w 2 + 1 ) + - ( w 1 + 1 ) + } - r ( w 2 - w 1 ) max { ( a + b ) - r , r } | | w 2 - w 1 | | < 1 2 { λ k + 1 ( λ k + 1 - c ) - λ k ( λ k - c ) } | | w 2 - w 1 | | .

Thus the right hand side of (4.2) defines a Lipschitz mapping from L2(Ω) into L2(Ω) with Lipschitz constant < 1. By the contraction mapping principle, there exists a unique solution w L2(Ω) of (4.2). By Lemma 2.1, the solution u H. We complete the proof.   ■

Abbreviations

(FESDBC)

fourth-order elliptic system with Dirichlet boundary condition.

Declarations

Acknowledgements

This work(Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).

Authors’ Affiliations

(1)
Department of Mathematics, Kunsan National University
(2)
Department of Mathematics Education, Inha University

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Copyright

© Jung and Choi; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.