  We consider a massive scalar field theory in anti-de Sitter space, in both
minimally and non-minimally coupled cases. We introduce a relevant double-trace
perturbation at the boundary, by carefully identifying the correct source and
generating functional for the corresponding conformal operator. We show that
such relevant double-trace perturbation introduces changes in the coefficients
in the boundary terms of the action, which in turn govern the existence of a
bound state in the bulk. For instance, we show that the usual action,
containing no additional boundary terms, gives rise to a bound state, which can
be avoided only through the addition of a proper boundary term. Another
notorious example is that of a conformally coupled scalar field, supplemented
by a Gibbons-Hawking term, for which there is no associated bound state. In
general, in both minimally and non-minimally coupled cases, we explicitly
compute the boundary terms which give rise to a bound state, and which ones do
not. In the non-minimally coupled case, and when the action is supplemented by
a Gibbons-Hawking term, this also fixes allowed values of the coupling
coefficient to the metric. We interpret our results as the fact that the
requirement to satisfy the Breitenlohner-Freedman bound does not suffice to
prevent tachyonic behavior from existing in the bulk, as it must be
supplemented by additional conditions on the coefficients in the boundary terms
of the action.
