  We study static, spherically symmetric solutions in a recently proposed
ghost-free model of non-linear massive gravity. We focus on a branch of
solutions where the helicity-0 mode can be strongly coupled within certain
radial regions, giving rise to the Vainshtein effect. We truncate the analysis
to scales below the gravitational Compton wavelength, and consider the weak
field limit for the gravitational potentials, while keeping all non-linearities
of the helicity-0 mode. We determine analytically the number and properties of
local solutions which exist asymptotically on large scales, and of local
(inner) solutions which exist on small scales. We find two kinds of asymptotic
solutions, one of which is asymptotically flat, while the other one is not, and
also two types of inner solutions, one of which displays the Vainshtein
mechanism, while the other exhibits a self-shielding behaviour of the
gravitational field. We analyse in detail in which cases the solutions match in
an intermediate region. The asymptotically flat solutions connect only to inner
configurations displaying the Vainshtein mechanism, while the non
asymptotically flat solutions can connect with both kinds of inner solutions.
We show furthermore that there are some regions in the parameter space where
global solutions do not exist, and characterise precisely in which regions of
the phase space the Vainshtein mechanism takes place.
