Spectral properties of second-order, multi-point, image-Laplacian boundary value problems Original Research Article

We consider the multi-point boundary value problem View the MathML source−ϕp(u′)′=λϕp(u),on (−1,1), Turn MathJax on View the MathML sourceu(±1)=∑i=1m±αi±u(ηi±), Turn MathJax on where p>1p>1, View the MathML sourceϕp(s)≔|s|p−1sgns for s∈Rs∈R, λ∈Rλ∈R, m±⩾1m±⩾1 are integers, View the MathML sourceηi±∈(−1,1), 1⩽i⩽m±1⩽i⩽m±, and the coefficients View the MathML sourceαi± satisfy View the MathML source∑i=1m±|αi±|<1. Turn MathJax on 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.