Second order, multi-point problems with variable coefficients

TL;DR

This paper investigates the spectral properties of second order multi-point eigenvalue problems with variable coefficients, establishing conditions under which classical spectral results hold and exploring nonlinear problem bifurcations.

Contribution

It extends spectral analysis of multi-point boundary value problems to variable coefficient cases, previously only studied for constant coefficients, and analyzes nonlinear bifurcation phenomena.

Findings

Spectral properties similar to separated problems hold for small boundary coefficients.

Standard spectral properties may fail when boundary coefficients are large.

Established a Rabinowitz-type bifurcation result for nonlinear multi-point problems.

Abstract

In this paper we consider the eigenvalue problem consisting of the equation -u" = \la r u, \quad \text{on $(-1,1)$}, where $r \in C^1[-1,1], \ r>0$ and $\la \in \R$, together with the multi-point boundary conditions u(\pm 1) = \sum^{m^\pm}_{i=1} \al^\pm_i u(\eta^\pm_i), where $m^\pm \ge 1$ are integers, and, for $i = 1,...,m^\pm$, $\al_i^\pm \in \R$, $\eta_i^\pm \in [-1,1]$, with $\eta_i^+ \ne 1$, $\eta_i^- \ne -1$. We show that if the coefficients $\al_i^\pm \in \R$ are sufficiently small (depending on $r$) then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients $\al_i^\pm$ are not sufficiently small then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ($r \equiv 1$), but the variable coefficient case has not been considered previously (apart from the existence of `principal' eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for existence of general solutions and also of nodal solutions --- these results rely on the spectral properties of the linear problem.