We study the existence and behavior of positive radial solutions of the equation Δu + f(u) = 0 in ℝn. This equation arises in various problems in applied mathematics, e.g. in the study of phase transitions, nuclear cores and more recently in population genetics and solitary waves. The important model case f(u) = - u + up, p>1, describes for instance the pressure distribution in a van der Waals fluid. In this case, we obtain fairly complete knowledge of all positive radial solutions.