Positive Solutions to Singular Semilinear Elliptic Problems

We obtain a characterization of all locally bounded functions p ≥ 0 for which the equation (E) Δu +p(x)ψ(u) = 0 has a positive solution in Ω vanishing on the boundary, where Ω is a domain of ℝN and ψ > 0 is a nonincreasing continuous function on ]0,∞[. In particular, for Ω = ℝN with N ≥ 3, it is shown that (E) has a (unique) positive solution in ℝN which decays to zero at infinity if and only if the set {p > 0} has positive Lebesgue measure and