Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

Let L be the Laplace operator on R-d, d >= 3, or the Laplace-Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation Lu - phi( . , u) = 0 we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of phi with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.