Spectral properties of second-order, multi-point, [formula omitted]-Laplacian boundary value problems

Abstract

We consider the multi-point boundary value problem − ϕ p ( u ′ ) ′ = λ ϕ p ( u ) , on ( − 1 , 1 ) , u ( ± 1 ) = ∑ i = 1 m ± α i ± u ( η i ± ) , where p > 1 , ϕ p ( s ) ≔ | s | p − 1 sgn s for s ∈ R , λ ∈ R , m ± ⩾ 1 are integers, η i ± ∈ ( − 1 , 1 ) , 1 ⩽ i ⩽ m ± , and the coefficients α i ± satisfy ∑ i = 1 m ± | α i ± | < 1 . A number λ ∈ R is said to be an eigenvalue of the above problem if there exists a non-trivial solution u . 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.