Semipositone higher-order differential equations.
This paper deals with the existence of positive solutions of the following conjugate higher-order boundary value problem \[ \begin{aligned} (-1)^{(n-p)}y^{(n)}(t) = \mu f(t, y(t)), & \qquad 0 < t < 1,\\ y^{(i)}(0) = 0, & \qquad 0 \leq i \leq p-1,\\ y^{(i)}(1) = 0, & \qquad 0 \leq i \leq n-p-1, \end{aligned} \] where \(n \geq 2\), \(1\leq p\leq n-1\), and \(\mu > 0\) are constants. Problems of this type are referred to as semipositone problems in the literature and they arise naturally in chemical reactor theory. The constant \(\mu\) is usually called the Thiele modulus, and in applications, one is interested in showing the existence of positive solutions for \(\mu >0\) small enough. Most papers in the literature discuss the conjugate problem when \(f\) takes nonnegative values (i.e., positone problems) and only a handful of papers have appeared discussing the semipositone problem. This paper attempts to fill a part of this gap in the literature: the nonlinearity \(f\) may take negative values through the whole paper. Let us note that existence results are obtained using Krasnoselskii's fixed-point theorem in a cone. Moreover, the presented technique allows one to consider also other types of boundary data. Thus, the following \((n,p)\) problem is also discussed \[ \begin{aligned} y^{(n)}(t) + \mu f(t, y(t)) = 0, & \qquad 0 < t < 1,\\ y^{(i)}(0) = 0, & \qquad 0 \leq i \leq n-2,\\ y^{(p)}(1) = 0, \end{aligned} \] where \(n \geq 2\) and \(1\leq p\leq n-1\) are fixed.
scientific article; zbMATH DE number 2138572