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 < 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
Keywords:
Fourth order elliptic system; fourth order elliptic equation; variation linking theorem; contraction mapping principle1. Introduction
Let Ω be a smooth bounded region in R^{n }with smooth boundary ∂Ω. Let λ_{1 }< λ_{2 }≤ ... ≤ 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 < λ_{1 }and b ≥ λ_{1}(λ_{1 } 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 < λ_{1 }and b ≥ λ_{2}(λ_{2 } c), then (1.2) has at least four solutions by the LeraySchauder degree theory. Micheletti, Pistoia and Sacon [9] also proved that if c < λ_{1 }and b ≥ λ_{2}(λ_{2 } 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 < λ_{1}, λ_{1}(λ_{1 } c) < b < λ_{2}(λ_{2 } c) and s < 0 or when λ_{1 }< c < λ_{2}, b < λ_{1}(λ_{1 } c) and s > 0. They also obtained these results by using the variational reduction method. They [3] also proved that when c < λ_{1}, λ_{1}(λ_{1 } c) < b < λ_{2}(λ_{2 } 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
THEOREM 1.2. Suppose that ab ≠ 0 and
In section 2 we define a Banach space H spanned by eigenfunctions of Δ^{2 }+ 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.
2. Fourth order elliptic system
The eigenvalue problem Δ^{2}u + 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 λ_{1}(λ_{1 } c) < λ_{2}(λ_{2 } c) ≤ λ_{3}(λ_{3 } 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
Any element u ∈ L^{2}(Ω) can be expressed by
Let H be a subspace of L^{2}(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) Δ^{2}u + cΔu ∈ H implies u ∈ H.
(ii)
(iii)
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 L^{2}(Ω) of
belong to H.
Proof. Let us write (a + b)((w + 1)^{+ } 1) = ∑h_{k}ϕ_{k }∈ L^{2}(Ω).
for some C > 0. Thus (Δ^{2 }+ 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 L^{2}(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) ∈ C^{1}. 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_{k}ϕ_{k},
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. ■
3. Proof of Theorem 1.1
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 X_{1 }and X_{2}. Let F ∈ C^{1}(X, R). Moreover assume:
(a) dim X_{1 }< +∞;
(b) there exist ρ > 0, R > 0 and e ∈ X_{1}, e ≠ 0 such that ρ < R and
(c)
(d) (P.S.)_{c }holds for any c ∈ [a, b], where
Then there exist at least two critical levels c_{1 }and c_{2 }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 ϕ_{1}, ⋯, ϕ_{k }whose eigenvalues are λ_{1}(λ_{1 } c), ⋯ , λ_{k}(λ_{k } c). Let
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},
We can write
where
Since H is compactly embedded in L^{2}, 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
Let
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
(ii) there exists ρ_{k+n }> 0 such that
Proof. (i) Let
since
Thus we have
(ii) Let w ∈ H_{k+n}. Then
so that
Since
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 e_{1 }∈ H^{+ }∩ H_{k+n }with e_{1} = 1. Then there exists
Proof. Let us chose
where λ_{k+1 }(λ_{k+1 } c) ≤ Λ ≤ λ_{k+1 }(λ_{k+1 } c). Choose σ > 0 so mall that
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
By Lemma 3.3, we also have
4. Proof of Theorem 1.2
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
or
We note that the operator (Δ^{2 }+cΔ  r)^{1 }is a compact, selfadjoint and linear map from L^{2}(Ω) into L^{2}(Ω) with norm
Thus the right hand side of (4.2) defines a Lipschitz mapping from L^{2}(Ω) into L^{2}(Ω) with Lipschitz constant < 1. By the contraction mapping principle, there exists a unique solution w ∈ L^{2}(Ω) of (4.2). By Lemma 2.1, the solution u ∈ H. We complete the proof. ■
Abbreviations
(FESDBC): fourthorder elliptic system with Dirichlet boundary condition.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
TJ carried out (FESDBC) studies, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment. All authors read and approved the final manuscript.
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 (KRF20100023985).
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