Heriot-Watt Mathematics Report Series
HWM06-45, 29 Nov 2006

Spectral properties and nodal solutions for second-order, $m$-point, boundary value problems

B P Rynne

Abstract

We consider the $m$-point boundary value problem consisting of the equation $$ -u'' = f(u) , \quad \text{on $(0,1)$,} $$ where $f : R \to R$ is $C^1$, with $f(0) = 0$, together with the boundary conditions $$ u(0)=0,\quad u(1) = \sum^{m-2}_{i=1}\alpha_i u(\eta_i) , $$ where $m \ge 3$, $\eta_i \in (0,1)$ and $\alpha_i > 0$ for $i=1,\dots,m-2$, with $$ \sum^{m-2}_{i=1} \alpha_i < 1 . $$

We first show that the spectral properties of the linearisation of this problem are similar to the well-known properties of the standard Sturm-Liouville problem with separated boundary conditions (with a minor modification to deal with the multi-point boundary condition). These spectral properties are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the above problem. Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a specified number of zeros) of the above problem, under various conditions on the asymptotic behaviour of $f$.

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