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

We consider the boundary value problem consisting of the pp-Laplacian equation equation(1)−ϕp(u′)′=λϕp(u),on (−1,1), where p>1p>1, ϕp(s)≔|s|p−1sgns for s∈Rs∈R, λ∈Rλ∈R, together with the multi-point boundary conditions equation(2)ϕp(u′(±1))=∑i=1m±αi±ϕp(u′(ηi±)), or equation(3)u(±1)=∑i=1m±αi±u(ηi±), or a mixed pair of these conditions (with one condition holding at each of x=−1x=−1 and x=1x=1). In (2), (3), m±⩾1m±⩾1 are integers, ηi±∈(−1,1), 1⩽i⩽m±1⩽i⩽m±, and the coefficients αi± satisfy ∑i=1m±|αi±|<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 α±=0α±=0.Given a suitable pair of boundary conditions, a number λλ is an eigenvalue   of the corresponding boundary value problem if there exists a non-trivial solution uu (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.