Existence of positive solutions of a fourth-order boundary value problem
Introduction
Let g:[0, 1] × [0, ∞) → [0, ∞) be continuous. Letexist. It is well-known that if eitherorthen the second-order boundary value problemhas at least one positive solution, see Liu and Li [9]. We note that the constant π2 in (1.1), (1.2) is the first eigenvalue of the linear eigenvalue problem
In [5], Del Pino and Manásevich studied the two parameter linear eigenvalue problem
They proved the following. Theorem A [5, Proposition 2.1] (ν, η) is an eigenvalue pair of (1.7), (1.8) if and only if
Motivated by the above results, we consider the fourth-order boundary value problemunder the assumptions.
(H1) f : [0, 1] × [0, ∞) × (−∞, 0] → [0, ∞) is continuous and there exist constants a, b, c, d ∈ [0, ∞) with a + b > 0 and c + d > 0 such thatuniformly for t ∈ [0, 1], anduniformly for t ∈ [0, 1]. Here
(H2) f(t, u, p) > 0 for t ∈ [0, 1] and (u, p) ∈ ([0, ∞) × (−∞, 0])⧹{(0, 0)}.
(H3) There exist constants a0, b0 ∈ [0, ∞) satisfy and
We give conditions on a, b, c, d which are related to the eigenvalue pair of (1.7), (1.8) and guarantee the existence of positive solutions of (1.10), (1.11).
Finally we note that Problem (1.10), (1.11) describes the deformation of an elastic beam whose both ends simply supported, see Gupta [7]. For early results on the existence of positive solutions of fourth-order problems, see Bai and Wang [3], Liu [8] and Ma and Wang [10]. For other related results on the existence of positive solutions of second-order ordinary differential equations and second-order elliptic problem, see Amann [1], Ambrosetti and Hess [2], Erbe and Wang [6] and references therein.
The rest of the paper is organized as follows: in Section 2, we state a version of Dancer’s results on the global solution branches for positive mapping. Section 3 is devoted to study the generalized eigenvalues λn(α, β) of linear fourth-order problemand prove some elementary properties of λn(α, β). Finally in Section 4, we state and prove our main result.
Section snippets
Global solutions branches for positive mappings
Suppose that E is a real Banach space with norm ∥ · ∥. Let K be a cone in E. A nonlinear mapping A : [0, ∞) × K → E is said to be positive if A([0, ∞) × K) ⊆ K. It is said to be K-completely continuous if A is continuous and maps bounded subsets of [0, ∞) × K to precompact subset of E. Finally, a positive linear operator V on E is said to be a linear minorant for A if A(λ, u) ⩾ λV(u) for (λ, u) ∈ [0, ∞) × K. If B is a continuous linear operator on E, denote r(B) the spectrum radius of B. Define
Generalized eigenvalues
Let
(C1) (α, β) ∈ [0, ∞) × [0, ∞) be given constants with α + β > 0. Definition 3.1 We say λ is a generalized eigenvalue of linear problemif (3.1), (3.2) has nontrivial solutions. Theorem 3.1 Let (C1) hold. Then the generalized eigenvalues of (3.1) and (3.2) are given bywhereThe generalized eigenfunction corresponding to λk(α, β) is Proof By Theorem A, λ is a generalized eigenvalue of (3.1), (3.2) if and only iffor
The main result
Theorem 4.1 Let (H1), (H2) and (H3) hold. Assume that eitherorThen (1.10), (1.11) has at least one positive solution.
As an immediate consequence of Theorem 3.2, Theorem 3.3, we have the following. Corollary 4.1 Let (H1) and (H2) and (H3) hold. Assume that eitherorThen (1.10), (1.11) has at least one positive solution. Remark 4.1 It is worth remarking that ifthen the existence of positive solutions of (1.10), (1.11) can not guarantee. Let
Acknowledgment
Supported by the NSFC (No. 10271095), GG-110-10736-1003, NWNU-KJCXGC-212, the Foundation of Excellent Young Teacher of the Chinese Education Ministry.
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