Eigenvalue criteria for existence of positive solutions of second-order, multi-point, $p$-Laplacian boundary value problems

Abstract

In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem \begin{gather} - (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 , \quad\text{a.e. on $(-1,1)$}, \tag{1} \\ u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) , \tag{2} \end{gather} where $p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$, $m^\pm \ge 1$ are integers, and $$ \eta_i^\pm \in (-1,1),\quad \alpha_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad \sum^{m^\pm}_{i=1} \alpha_i^\pm < 1 . $$ Also, $a \in L^1(-1,1)$, and $g \colon [-1,1] \times \mathbb R^2 \to \mathbb R$ is Carathéodory, with \begin{equation} g(x,0,0) = 0, \quad x \in [-1,1]. \tag{3} \end{equation} Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of $g(x,s,t)$, as $s \to \infty$, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation \begin{equation} -\phi_p (u')' + a \phi_p (u) = \lambda \phi_p (u) , \quad \text{on $(-1,1)$}, \tag{4} \end{equation}