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