Several results on existence, nonexistence and large-time behavior of small positive solutions were obtained before for the equation , , , with a linear dynamical boundary condition. Here is the N-dimensional Laplacian (in x). We study the effects of the change of the domain from the half-space to the exterior of the unit ball when . We show that the critical exponent for the existence of positive solutions and the decay rate of small solutions are different. More precisely, for the half-space problem the critical exponent is and the decay rate is , while for the exterior problem we obtain the exponent and the exponential rate e(-(N-2)t)..