We explore the effects of a positive cosmological constant on astrophysical and cosmological configurations described by a polytropic equation of state. We derive the conditions for equilibrium and stability of such configurations and consider some astrophysical examples where our analysis may be relevant. We show that in the presence of the cosmological constant the isothermal sphere is not a viable astrophysical model since the density in this model does not go asymptotically to zero. The cosmological constant implies that, for a polytropic index smaller than 5, the central density has to exceed a certain minimal value in terms of the vacuum density in order to guarantee the existence of a finite-size object. We examine such configurations together with effects of Λ in other exotic possibilities, such as neutrino and boson stars, and we compare our results to N-body simulations. The astrophysical properties and configurations found in this paper are specific features resulting from the existence of a dark energy component. Hence, if found in nature they would be an independent probe of a cosmological constant, complementary to other observations.