Spectral properties of second-order, multi-point, pp-Laplacian boundary value problems

We consider the multi-point boundary value problem −ϕp(u′)′=λϕp(u),on (−1,1),u(±1)=∑i=1m±αi±u(ηi±), where p>1p>1, ϕp(s)≔|s|p−1sgns for s∈Rs∈R, λ∈Rλ∈R, m±⩾1m±⩾1 are integers, ηi±∈(−1,1), 1⩽i⩽m±1⩽i⩽m±, and the coefficients αi± satisfy ∑i=1m±|αi±|<1. A number λ∈Rλ∈R is said to be an eigenvalue   of the above problem if there exists a non-trivial solution uu. The spectrum is the set of eigenvalues. In this paper we obtain some basic spectral and degree-theoretic properties of this eigenvalue problem. These results have numerous applications to more general problems. As an example, a Rabinowitz-type, global bifurcation theorem is briefly described.