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

Abstract

We consider the nonlinear boundary value problem consisting of the equation −u″=f(u),on(−1,1), where f:R→R is continuous, together with general Sturm–Liouville type, multi-point boundary conditions at ±1. We obtain Rabinowitz-type global bifurcation results, and then use these to obtain ‘nodal’ solutions of the problem. We conclude with a nonresonance result for an inhomogeneous form 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 Prüfer angle construction.Unfortunately, it seems to be difficult to apply the Prüfer 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.