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

Abstract

We consider the $m$-point boundary value problem consisting of the equation \begin{equation} -\phi_p (u')'=f(u), \quad \text{on $(0,1)$},\tag{$1$} \end{equation} together with the boundary conditions \begin{equation} u(0) = 0,\quad u(1) = \sum^{m-2}_{i=1}\alpha_i u(\eta_i) ,\tag{$2$} \end{equation} where $p> 1$, $\phi_p(s) := |s|^{p-1} \text{\rm sgn} s$, $s \in {\mathbb R}$, $m \ge 3$, $\alpha_i , \eta_i \in (0,1)$, for $i=1,\dots,m-2$, and $\sum^{m-2}_{i=1} \alpha_i < 1$. We assume that the function $f \colon {\mathbb R} \to{\mathbb R}$ is continuous, satisfies $sf(s) > 0$ for $s \in {\mathbb R} \setminus \{0\}$, and that $f_0 := \lim_{\xi \rightarrow 0}{f(\xi)}/{\phi_p(\xi)} > 0$.

Closely related to the problem (1), (2), is the spectral problem consisting of the equation \begin{equation} -\phi_p (u')' = \lambda \phi_p(u) , \tag{$3$} \end{equation} together with the boundary conditions (2). It will be shown that the spectral properties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modification to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions of (1), (2), under various conditions on the asymptotic behaviour of $f$.