Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions

TL;DR

This paper studies nonlinear second-order differential equations with multi-point Sturm-Liouville boundary conditions, establishing existence, bifurcation, and spectral properties using alternative oscillation methods due to the complexity of traditional approaches.

Contribution

It introduces new oscillation counting techniques to analyze spectral properties of linear problems, enabling the study of nonlinear problems with complex boundary conditions.

Findings

Existence of solutions under nonresonance conditions

Global bifurcation results for nodal solutions

Spectral properties similar to classical Sturm-Liouville problems

Abstract

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary conditions at $\pm 1$. We will obtain existence of solutions of this boundary value problem under certain `nonresonance' conditions, and also Rabinowitz-type global bifurcation results, which yield nodal solutions of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation $$-u'' = \lambda u, \quad \text{on} \quad (-1,1),$$ together with the multi-point boundary conditions. In a previous paper it was shown that, under certain `optimal' conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm-Liouville problem with single-point boundary conditions. In particular, for each integer $k \geq 0$ there exists a unique, simple eigenvalue $\lambda_k$, whose eigenfunctions have `oscillation count' equal to $k$, where the `oscillation count' was defined in terms of a complicated Pr\"ufer angle construction. Unfortunately, it seems to be difficult to apply the Pr\"ufer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above.