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

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Abstract

We consider the nonlinear equation−u′′=f(u)+h,on(−1,1),where f:R→R and h:[−1,1]→R are continuous, together with general Sturm-Liouville type, multi-point boundary conditions at ±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′′=λu,on(−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≥0 there exists a unique, simple eigenvalue λk, 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.
Original languageEnglish
Pages (from-to)195–214
Number of pages20
JournalNonlinear Analysis: Theory, Methods and Applications
Volume136
Early online date9 Mar 2016
DOIs
Publication statusPublished - May 2016

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