Elsevier

Applied Mathematics and Computation

Volume 168, Issue 2, 15 September 2005, Pages 1219-1231
Applied Mathematics and Computation

Existence of positive solutions of a fourth-order boundary value problem

https://doi.org/10.1016/j.amc.2004.10.014Get rights and content

Abstract

We consider the fourth-order boundary value problemu=f(t,u,u),0<t<1u(0)=u(1)=u(0)=u(1)=0where f(t, u, p) = au  b p + ∘(∣(u, p)∣) near (0, 0), and f(t, u, p) = cu  dp + ∘(∣(u, p)∣) near ∞. We give conditions on the constants a, b, c, d that guarantee the existence of positive solutions. The proof of our main result is based upon global bifurcation techniques.

Introduction

Let g:[0, 1] × [0, ∞)  [0, ∞) be continuous. Letlims0+g(t,s)sg0,lims0+g(t,s)sgexist. It is well-known that if eitherg0<π2<gorg<π2<g0,then the second-order boundary value problemu+g(t,u)=0,0<t<1,u(0)=u(1)=0has 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 problemu+λu=0,0<t<1,u(0)=u(1)=0.

In [5], Del Pino and Manásevich studied the two parameter linear eigenvalue problemu=νu-ηu,0<t<1,u(0)=u(1)=u(0)=u(1)=0.

They proved the following.

Theorem A [5, Proposition 2.1]

(ν, η) is an eigenvalue pair of (1.7), (1.8) if and only ifν(kπ)4+η(kπ)2=1forsomekN.

Motivated by the above results, we consider the fourth-order boundary value problemu=f(t,u,u),0<t<1,u(0)=u(1)=u(0)=u(1)=0.under 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 thatf(t,u,p)=au-bp+((u,p))as(u,p)(0,0)uniformly for t  [0, 1], andf(t,u,p)=cu-dp+((u,p))as(u,p)uniformly for t  [0, 1]. Here(u,p)u2+p2.

(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 a02+b02>0 andf(t,u,p)a0u-b0p,(t,u,p)[0,1]×[0,)×(-,0].

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 problemu=λ(αu-βu),u(0)=u(1)=u(0)=u(1)=0and 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. DefinecK(B)={λ[0,):thereexistsx

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 problemu=λ(αu-βu),0<t<1,u(0)=u(1)=u(0)=u(1)=0.if (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 byλ1(α,β)<λ2(α,β)<<λn(α,β)<,whereλk(α,β)=(kπ)4α+β(kπ)2,kN.The generalized eigenfunction corresponding to λk(α, β) isϕk(t)=sinkπt.

Proof

By Theorem A, λ is a generalized eigenvalue of (3.1), (3.2) if and only ifλα(kπ)4+λβ(kπ)2=1for

The main result

Theorem 4.1

Let (H1), (H2) and (H3) hold. Assume that eitherλ1(c,d)<1<λ1(a,b)orλ1(a,b)<1<λ1(c,d).Then (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 eithercπ4+dπ2>1andaπ4+bπ2<1oraπ4+bπ2>1andcπ4+dπ2<1.Then (1.10), (1.11) has at least one positive solution.

Remark 4.1

It is worth remarking that ifλ1(c,d)=1=λ1(a,b),then 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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