This paper deals with the existence of classical positive solutions to the elliptic boundary value problem \[ \Delta u+f(x,u)=0,\quad x\in B=\{x\in {\mathbb{R}}^ N | \quad | x| <1\},\quad u=0\text{ on } \partial B, \] where \(\Delta\) is the N dimensional Laplacian, \(N\geq 2\), and f(x,u) is a positive, smooth function defined on \(B\times (0,\infty)\) and nonincreasing in \(u>0\). No smoothness is imposed on \(\partial (B\times (0,\infty))\) so that certain singularities are allowed. Additional assumptions on f are given which are sufficient to guarantee the existence of a positive solution \(u\in C^ 2(B)\cap C(\bar B)\).