  We demonstrate the existence of static, spherically symmetric globally
regular, i.e. solitonic solutions of a shift-symmetric scalar-tensor gravity
model with negative cosmological constant. The norm of the Noether current
associated to the shift symmetry is finite in the full space-time. We also
discuss the corresponding black hole solutions and demonstrate that the
interplay between the scalar-tensor coupling and the cosmological constant
leads to the existence of new branches of solutions. To linear order in the
scalar-tensor coupling, the asymptotic space-time corresponds to an Anti-de
Sitter space-time with a non-trivial scalar field on its conformal boundary.
This allows the interpretation of our solutions in the context of the AdS/CFT
correspondence. Finally, we demonstrate that - for physically relevant, small
values of the scalar-tensor coupling - solutions with positive cosmological
constant do not exist in our model.
