TY - JOUR AU - Rynne, Bryan P. PY - 2010/04/23 Y2 - 2024/03/29 TI - Eigenvalue criteria for existence of positive solutions of second-order, multi-point, $p$-Laplacian boundary value problems JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 36 IS - 2 SE - DO - UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2010.038 SP - 311 - 326 AB - In this paper we consider the existence and uniqueness of positivesolutions of the multi-point boundary value problem$$\gather- (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 ,\quad\text{a.e. on $(-1,1)$},\tag{1}\\u(\pm 1) = \sum^{m^\pm}_{i=1}\al^\pm_i u(\eta^\pm_i) ,\tag{2}\endgather$$where$p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$,$m^\pm \ge 1$ are integers,and$$\eta_i^\pm \in (-1,1),\quad\al_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad\sum^{m^\pm}_{i=1} \al_i^\pm < 1 .$$Also,$a \in L^1(-1,1),$and$g \colon [-1,1] \X \mathbb R^2 \to \mathbb R$ is Carathéodory,with$$g(x,0,0) = 0, \quad x \in [-1,1].\tag{3}$$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 themulti-point boundary value problem consisting of the equation$$ -\phi_p (u')' + a \phi_p (u) = \la \phi_p (u) , \quad \text{on $(-1,1)$},\tag{4}$$ ER -