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

摘要

We consider the boundary value problem consisting of the p-Laplacian equation -φp(u′)′=λφp(u),on (-1,1), where p>1, φ_p(s):=|s|p-1sgns for s∈?, λ∈?, together with the multi-point boundary conditions φp(u′(±1))=∑i=1m±αi±φ p(u′(ηi±)), or u(±1)=∑i=1m±αi±u(ηi±), 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, ηi±∈(-1,1), 1 ≤i≤m±, and the coefficients αi± satisfy ∑ i=1m±|α_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 α±=0. Given a suitable pair of boundary conditions, a number λ 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.