Spectral properties and nodal solutions for second-order, m-point, boundary value problems

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Abstract

We consider the m-point boundary value problem consisting of the equation - u? = f (u), on (0, 1), where f : R ? R is C1, with f (0) = 0, together with the boundary conditions u (0) = 0, u (1) = underover(?, i = 1, m - 2) ai u (?i), where m = 3, ?i ? (0, 1) and ai > 0 for i = 1, ..., m - 2, with underover(?, i = 1, m - 2) ai < 1 . We first show that the spectral properties of the linearisation of this problem are similar to the well-known properties of the standard Sturm-Liouville problem with separated boundary conditions (with a minor modification to deal with the multi-point boundary condition). These spectral properties are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the above problem. Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a specified number of zeros) of the above problem, under various conditions on the asymptotic behaviour of f. © 2006 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)3318-3327
Number of pages10
JournalNonlinear Analysis: Theory, Methods and Applications
Volume67
Issue number12
DOIs
Publication statusPublished - 15 Dec 2007

Keywords

  • Eigenvalues
  • Multi-point boundary value problem
  • Nonlinear boundary value problem

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