  A step-by-step approach is followed to study cosmic structures in the context
of Brans-Dicke theory with positive cosmological constant $\Lambda$ and
parameter $\omega$. First, it is shown that regular stationary black-hole
solutions not only have constant Brans-Dicke field $\phi$, but can exist only
for $\omega=\infty$, which forces the theory to coincide with the General
Relativity. Generalizations of the theory in order to evade this black-hole
no-hair theorem are presented. It is also shown that in the absence of a
stationary cosmological event horizon in the asymptotic region, a stationary
black hole horizon can support a non-trivial Brans-Dicke hair. Even more
importantly, it is shown next, that the presence of a stationary cosmological
event horizon rules out any regular stationary solution, appropriate for the
description of a star. Thus, to describe a star one has to assume that there is
no such stationary horizon in the faraway asymptotic region. Under this
implicit assumption generic spherical cosmic structures are studied
perturbatively and is shown that only for $\omega>0$ or $\omega\lesssim -5$
their predicted maximum sizes are consistent with observations. We also point
out how, many of the conclusions of this work differ qualitatively from the
$\Lambda=0$ spacetimes.
