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

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Abstract

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=1ai± 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.

Original languageEnglish
Pages (from-to)311-326
Number of pages16
JournalTopological Methods in Nonlinear Analysis
Volume36
Issue number2
Publication statusPublished - 2010

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Multi-point Boundary Value Problem
Existence of Positive Solutions
P-Laplacian
Boundary Value Problem
Eigenvalue
Principal Eigenvalue
Positive Solution
Existence and Uniqueness
Asymptotic Behavior
Integer

Keywords

  • Positive solutions of nonlinear boundary value problems
  • Principal eigenvalues

Cite this

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title = "Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems",
abstract = "In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - ({\O}p(u?)?+(a+g(x,u,u?){\O}p(u) =0, a.e. on (-1,1), (2) u(±1)=Si=1 ±ai±u(? i±) where p>1, {\O}p(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) -{\O}(u?)?+a{\O}p(u)= ?{\O}p(u), on (-1,1) Copyright {\circledC} 2010 Juliusz Schauder Center for Nonlinear Studies.",
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author = "Rynne, {Bryan P.}",
year = "2010",
language = "English",
volume = "36",
pages = "311--326",
journal = "Topological Methods in Nonlinear Analysis",
issn = "1230-3429",
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T1 - Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems

AU - Rynne, Bryan P.

PY - 2010

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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.

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KW - Principal eigenvalues

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