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

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-1sgns 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.