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

Let $\text{L}$ be the Laplace operator on ${\mathbb{R}}^d$, $d\ge 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $\text{L}u-\phi (\cdot ,u)=0$ 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.