  We explore the possibility of embedding regular compact objects with (anti)
de Sitter ((A)dS) core as solutions of Einstein's gravity minimally coupled to
a real scalar field. We consider, among others, solutions interpolating between
an inner, potential-dominated core and an outer, kinetic-term-dominated region.
Owing to their analogy with slow-roll inflation, we term them gravitational
vacuum inflative stars, or gravistars for short. We systematically discuss
approximate solutions of the theory describing either the core or the
asymptotically-flat region at spatial infinity. We extend nonexistence theorems
for smooth interpolating solutions, previously proved for black holes, to
compact objects without event horizons. This allows us to construct different
classes of exact (either smooth or non-smooth) singularity-free solutions of
the theory. We first find a smooth solution interpolating between an AdS
spacetime in the core and an asymptotically-flat spacetime (a Schwarzschild
solution with a subleading $1/r^2$ deformation). We proceed by constructing
non-smooth solutions describing gravistars. Finally, we derive a smooth scalar
lump solution interpolating between $\text{AdS}_4$ in the core and a Nariai
spacetime at spatial infinity.
