Entire solutions to sublinear elliptic problems on harmonic NA groups and Euclidean spaces

We give necessary and sufficient conditions for the existence of entire solutions bounded or large of the equation ℒu - pψ (u) =0 , where ℒ is either the Laplace operator on ℝ ^d , d≥ 3 or the Laplace–Beltrami operator on the harmonic NA group and p is a function whose oscillation tends to zero at infinity at a specified rate. The results apply to noncompact rank one symmetric spaces.