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A step-by-step approach is followed to study cosmic structures in the contextof Brans-Dicke theory with positive cosmological constant $\Lambda$ andparameter $\omega$. First, it is shown that regular stationary black-holesolutions not only have constant Brans-Dicke field $\phi$, but can exist onlyfor $\omega=\infty$, which forces the theory to coincide with the GeneralRelativity. Generalizations of the theory in order to evade this black-holeno-hair theorem are presented. It is also shown that in the absence of astationary cosmological event horizon in the asymptotic region, a stationaryblack hole horizon can support a non-trivial Brans-Dicke hair. Even moreimportantly, it is shown next, that the presence of a stationary cosmologicalevent horizon rules out any regular stationary solution, appropriate for thedescription of a star. Thus, to describe a star one has to assume that there isno such stationary horizon in the faraway asymptotic region. Under thisimplicit assumption generic spherical cosmic structures are studiedperturbatively and is shown that only for $\omega>0$ or $\omega\lesssim -5$their predicted maximum sizes are consistent with observations. We also pointout how, many of the conclusions of this work differ qualitatively from the$\Lambda=0$ spacetimes.

Can Brans-Dicke theory with $\Lambda>0$ describe stars?