### Abstract

In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - (Ø_{p}(u?)?+(a+g(x,u,u?)Ø_{p}(u) =0, a.e. on (-1,1), (2) u(±1)=S_{i}=1 ^{±}a_{i}^{±}u(? _{i}^{±}) where p>1, Ø_{p}(s) ^{p-2}s, se R,m± = 1 are integer, and ?_{i}^{±}?(-1,1), a_{i}^{±} > 0, i=1 ,?,m^{±}, S_{i}=1^{m±}a_{i}^{±} Also a ? L^{1}(-1,1) and g: [-1,1]× R ^{2}?R is Carath´eodory, with (3) g(x,0,0)=0, x ?[-1,1]. Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of g(x, s, t), as s ? 8, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation (4) -Ø(u?)?+aØp(u)= ?Øp(u), on (-1,1) Copyright © 2010 Juliusz Schauder Center for Nonlinear Studies.

Original language | English |
---|---|

Pages (from-to) | 311-326 |

Number of pages | 16 |

Journal | Topological Methods in Nonlinear Analysis |

Volume | 36 |

Issue number | 2 |

Publication status | Published - 2010 |

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### Keywords

- Positive solutions of nonlinear boundary value problems
- Principal eigenvalues

### Cite this

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**Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems.** / Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems

AU - Rynne, Bryan P.

PY - 2010

Y1 - 2010

N2 - In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - (Øp(u?)?+(a+g(x,u,u?)Øp(u) =0, a.e. on (-1,1), (2) u(±1)=Si=1 ±ai±u(? i±) where p>1, Øp(s) p-2s, se R,m± = 1 are integer, and ?i±?(-1,1), ai± > 0, i=1 ,?,m±, Si=1m±ai± Also a ? L1(-1,1) and g: [-1,1]× R 2?R is Carath´eodory, with (3) g(x,0,0)=0, x ?[-1,1]. Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of g(x, s, t), as s ? 8, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation (4) -Ø(u?)?+aØp(u)= ?Øp(u), on (-1,1) Copyright © 2010 Juliusz Schauder Center for Nonlinear Studies.

AB - In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - (Øp(u?)?+(a+g(x,u,u?)Øp(u) =0, a.e. on (-1,1), (2) u(±1)=Si=1 ±ai±u(? i±) where p>1, Øp(s) p-2s, se R,m± = 1 are integer, and ?i±?(-1,1), ai± > 0, i=1 ,?,m±, Si=1m±ai± Also a ? L1(-1,1) and g: [-1,1]× R 2?R is Carath´eodory, with (3) g(x,0,0)=0, x ?[-1,1]. Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of g(x, s, t), as s ? 8, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation (4) -Ø(u?)?+aØp(u)= ?Øp(u), on (-1,1) Copyright © 2010 Juliusz Schauder Center for Nonlinear Studies.

KW - Positive solutions of nonlinear boundary value problems

KW - Principal eigenvalues

UR - http://www.scopus.com/inward/record.url?scp=78751672999&partnerID=8YFLogxK

M3 - Article

VL - 36

SP - 311

EP - 326

JO - Topological Methods in Nonlinear Analysis

JF - Topological Methods in Nonlinear Analysis

SN - 1230-3429

IS - 2

ER -