  We study the transition of a scalar field in a fixed $AdS_{d+1}$ background
between an extremum and a minimum of a potential. We first prove that two
conditions must be met for the solution to exist. First, the potential involved
cannot be generic, i.e. a fine-tuning of their parameters is mandatory. Second,
at least in some region its second derivative must have a negative upper limit
which depends only on the dimensionality $d$. We then calculate the boundary
propagator for small momenta in two different ways: first in a WKB
approximation, and second with the usual matching method, generalizing the
known calculation to arbitrary order. Finally, we study a system with
spontaneously broken non-Abelian global symmetry, and show in the holographic
language why the Goldstone modes appear.
