### Abstract

We consider the boundary value problem consisting of the p-Laplacian equation -f_{p}(u')'=?f_{p}(u),on (-1,1), where p>1, f_{p}(s):=|s|^{p-1}sgns for s?R, ??R, together with the multi-point boundary conditions f_{p}(u'(±1))=?_{i=1}^{m±}a_{i}^{±}f _{p}(u'(?_{i}^{±})), or u(±1)=?_{i=1}^{m}±a_{i}^{±}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 a_{i}^{±} satisfy ? _{i=1}^{m±}|a_{i}^{±}|<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 language | English |
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Pages (from-to) | 1471-1484 |

Number of pages | 14 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 74 |

Issue number | 4 |

DOIs | |

Publication status | Published - 15 Feb 2011 |

### Keywords

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