Difficulties in reconciling non-negligible differences between the local
  and cosmological values of the gravitational coupling in extended Brans-Dicke
  theories
  Recent studies by Sol\`a Peracaula, G\'omez-Valent, de Cruz P\'erez and
Moreno-Pulido (2019,2020) have pointed out the intriguing possibility that
Brans-Dicke cosmology with constant vacuum energy density (BD-$\Lambda$CDM) may
be able to alleviate the $H_0$ and $\sigma_8$ tensions that are found in the
framework of the concordance cosmological model (GR-$\Lambda$CDM). The fitting
analyses presented in these works indicate a preference for values of the
effective gravitational coupling appearing in the Friedmann equation, $G$,
about $4-9\%$ larger than Newton's constant (as measured on Earth), and mildy
evolving with the expansion of the universe. The signal reaches the $\sim
3.5\sigma$ c.l. when the prior on $H_0$ from SH0ES and the angular diameter
distances to strong gravitationally lensed quasars measured by H0LICOW are
considered, and the $\sim 3\sigma$ c.l. when only the former is included. Thus,
the improvement in the description of the cosmological datasets relies on the
existence of a mechanism capable of screening the modified gravity effects at
those scales where deviations from standard General Relativity (GR) are highly
constrained, as in the Solar System. In this paper we explore several
extensions of BD-$\Lambda$CDM that can leave the cosmological evolution
basically unaltered at the background and linear perturbations level, while
being able to screen the Brans-Dicke effects inside the regions of interest,
leading to standard GR. We search for weak-field solutions around spherical
static massive objects with no internal pressure and show that, unfortunately,
these mechanisms can only explain very tiny departures of the effective
cosmological gravitational coupling from the one measured locally. This might
hinder the ability of BD-$\Lambda$CDM to alleviate the cosmological tensions.
