Fourth order elliptic system with dirichlet boundary condition

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 < λ_k+1 and λ_k+n(λ_k+n - c) < a + b < λ_k+n+1(λ_k+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 < λ_k+1 and λ_k(λ_k - c) < 0, a+b < λ_k+1(λ_k+1 - c). We prove this result by the contraction mapping principle on the Banach space.

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

Let Ω be a smooth bounded region in Rⁿ 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

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

Tarantello [10] also studied problem (1.2) when c < λ₁ and b ≥ λ₁(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 b ≥ λ₂(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 b ≥ λ₂(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

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 [7-9] 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 < λ_k+1 and λ_k+n(λ_k+n - c) < a + b < λ_k+n+1(λ_k+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 < λ_k+1 and λ_k(λ_k - c) < 0, a + b < λ_k+1(λ_k+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 + cΔ 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.

The eigenvalue problem Δ²u + cΔu = 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

can be transformed to the equation

We also have

It follows from the above equation that bu - av = 0. If u + v = w is a solution of (2.1), then the system

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

Any element u ∈ L²(Ω) can be expressed by

Let H be a subspace of L²(Ω) defined by

Then this is a complete normed space with a norm

Since λ_k(λ_k - c) → + ∞ and c is fixed, we have

(i) Δ²u + cΔu ∈ H implies u ∈ H.

(ii) ∥ u ∥ ≥ C ∥ u ∥ L 2 ( Ω ) , for some C > 0.

(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 -Δ, a + b ≠ λ_k(λ_k - c) and bounded. Then all solutions in L²(Ω) of

belong to H.

Proof. Let us write (a + b)((w + 1)⁺ - 1) = ∑h_kϕ_k ∈ L²(Ω).

for some C > 0. Thus (Δ² + cΔ)^-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

If we assume that λ_k < c < λ_k+1 and 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 < λ_k+1 and a + b is bounded. Then the functional F(w) is continuous and Frechét differentiable in H and

for h ∈ H.

Proof. First we shall prove that F(w) is continuous at w. Let w, z ∈ H.

Let w = ∑h_kϕ_k, z = ∑ h ̃ k ϕ k . Then we have

On the other hand, by Mean Value Theorem, we have

Thus we have

Thus F(w) is continuous at w. Next we shall prove that F(w) is Fréchet differentiable at w ∈ H. We consider

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

Throughout this section we assume that λ_k < c < λ_k+1 and λ_k+n(λ_k+n - c) < a + b < λ_k+n+1(λ_k+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:

THEOREM 3.1. ("A Variation of Linking") Let × be an Hilbert space, which is topological direct sum of the subspaces X₁ and 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

(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 c₁ and c₂ for the functional F such that :

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

Let e ∈ H⁺ ∩ H_k+n, e ≠ 0 and ρ > 0. Let us set

Let L : H → H be the linear continuous operator such that

Then L is an isomorphism and H_k+n, H k + n ⊥ are the negative space and the positive space of L. Thus we have

We can write

where

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

LEMMA 3.1. Let λ_k < c < λ_k+1 and λ_k+n(λ_k+n - c) < a + b < λ_k+n+1(λ_k+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 Dψ 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

Let P + : H → H k + n ⊥ and P^- : H → H_k+n denote the orthogonal projections. Since P⁺ z_n -P^- z_n is bounded in H, we have

Since P⁺ z_n - P^- z_n → P⁺ z - P^- z in H, from (3.2) and (3.3) we get

Hence z ≠ 0. On the other hand, from (3.5), we get

The last line of (3.6) is positive or equal to 0 since λ_k+n+1(λ_k+n+1 - c) - (a + b) > 0 and - (λ_k+n(λ_k+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 < λ_k+1 and λ_k+n(λ_k+n - c) < b < λ_k+n+1(λ_k+n+1 - c).

Then

(i) there exists R_k+n > 0 such that the functional F(w) is bounded from below on H k + n ⊥ ;

(ii) there exists ρ_k+n > 0 such that

Proof. (i) Let w ∈ H k + n ⊥ . Then we have

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 R_k+n > 0 such that inf   w ∈ H k + n ⊥ | | w | | = R k + n F ( w ) > 0 . Moreover if w ∈ H k + n ⊥ with ||w|| < R_k+n, then we have

Thus we have inf   w ∈ H k + n ⊥ | | w | | < R k + n F ( w ) > - - ∞ .

(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

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 < λ_k+1, λ_k+n (λ_k+n - c) < a + b < λ_k+n+1 (λ_k+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 ,

Proof. Let us chose w ∈ H k + n ⊥ and σ ≥ 0 and e₁ ∈ H⁺ ∩ H_k+n with ||e₁|| = 1. Then we get

where λ_k+1 (λ_k+1 - c) ≤ Λ ≤ λ_k+1 (λ_k+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

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.

Proof of Theorem 1.2

Assume that λ_k < c < λ_k+1 and λ_k(λ_k - c) < 0, b < λ_k+1(λ_k+1 - c). Let r = 1 2 { λ k ( λ k - c ) + λ k + 1 ( λ k + 1 - c ) } . We can rewrite (2.2) as

or

We note that the operator (Δ² +cΔ - 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

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.   ■