Spectral properties of [formula omitted]-Laplacian problems with Neumann and mixed-type multi-point boundary conditions

Abstract

We consider the boundary value problem consisting of the p -Laplacian equation (1) − ϕ p ( u ′ ) ′ = λ ϕ p ( u ) , on ( − 1 , 1 ) , where p > 1 , ϕ p ( s ) ≔ | s | p − 1 sgn s for s ∈ R , λ ∈ R , together with the multi-point boundary conditions (2) ϕ p ( u ′ ( ± 1 ) ) = ∑ i = 1 m ± α i ± ϕ p ( u ′ ( η i ± ) ) , or (3) u ( ± 1 ) = ∑ i = 1 m ± α i ± u ( η i ± ) , 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, η i ± ∈ ( − 1 , 1 ) , 1 ⩽ i ⩽ m ± , and the coefficients α i ± satisfy ∑ i = 1 m ± | 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.