Spectral properties of p-Laplacian problems with Neumann and mixed-type multi-point boundary conditions

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Abstract

We consider the boundary value problem consisting of the p-Laplacian equation -fp(u')'=?fp(u),on (-1,1), where p>1, fp(s):=|s|p-1sgns for s?R, ??R, together with the multi-point boundary conditions fp(u'(±1))=?i=1ai±f p(u'(?i±)), or u(±1)=?i=1m±ai±u(?i±), or a mixed pair of these conditions (with one condition holding at each of x=-1 and x=1). In (2), (3), m± =1 are integers, ?i±?(-1,1), 1 =i=m±, and the coefficients ai± satisfy ? i=1|ai±|<1. We term the conditions (2) and (3), respectively, Neumann-type and Dirichlet-type boundary conditions, since they reduce to the standard Neumann and Dirichlet boundary conditions when a±=0. Given a suitable pair of boundary conditions, a number ? is an eigenvalue of the corresponding boundary value problem if there exists a non-trivial solution u (an eigenfunction). The spectrum of the problem is the set of eigenvalues. In this paper we obtain various spectral properties of these eigenvalue problems. We then use these properties to prove Rabinowitz-type, global bifurcation theorems for related bifurcation problems, and to obtain nonresonance conditions (in terms of the eigenvalues) for the solvability of related inhomogeneous problems. © 2010 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)1471-1484
Number of pages14
JournalNonlinear Analysis: Theory, Methods and Applications
Volume74
Issue number4
DOIs
Publication statusPublished - 15 Feb 2011

Keywords

  • Multi-point boundary conditions
  • Ordinary differential equations
  • Second order

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