Multiplicity results for semipositone two-point boundary value problems

Abstract

In this paper we address two-point boundary value problems of the form

$$u^{\prime \prime }+f\left(u\right)=0,\text{ in }\left(0,1\right),u\left(0\right)=u\left(1\right)=0,$$

where the function $f$ resembles $f\left(u\right)=\lambda \left(exp\left(au∕\left(a+u\right)\right)-c\right)$ for some constants $c\ge 0$, $\lambda >0$, $a>4$. We prove the existence of positive solutions for the semipositone case where $f\left(0\right)<0$, and further prove multiplicity under certain conditions. In particular we extend theorems from Henderson and Thompson to the semipositone case.