The author deals with the Sturm-Liouville boundary value problem associated to a second-order differential equation of the form $u''+g(t)f(u)=0$. Conditions are given for the existence of multiple positive solutions. The main features of the present paper are that it is assumed that $g$ is measurable (not necessarily integrable) and that no monotonicity assumption on $f$ is required. The proofs are performed by means of a fixed-point theorem in cones. Applications are provided to eigenvalue problems and to the search of radial solutions to elliptic equations. The results in this paper generalise various earlier contributions. See, among others, {\it R. W. Leggett} and {\it L. R. Williams} [Indiana Univ. Math. J. 28, 673-688 (1979;

Zbl 0421.47033)], {\it K. Lan} and {\it J. R. L. Webb} [J. Differ. Equations 148, 407-421 (1998;

Zbl 0909.34013)]; for related results obtained by the upper-lower solution method, we refer to {\it A. K. Ben-Naoum} and {\it C. De Coster} [Differ. Integral Equ. 10, 1093-1112 (1997;

Zbl 0940.35086)] and {\it M. Gaudenzi} and {\it P. Habets} [Topol. Methods Nonlinear Anal. 14, 131-150 (1999;

Zbl 0965.34011)].