  We investigate the non-perturbative stability of asymptotically anti-de
Sitter gravity coupled to tachyonic scalar fields with mass saturating the
Breitenlohner-Freedman bound. Such "designer gravity" theories admit a large
class of boundary conditions at asymptotic infinity. At this mass, the
asymptotic behavior of the scalar field develops a logarithmic branch, and
previous attempts at proving a minimum energy theorem failed due to a large
radius divergence in the spinor charge. In this paper, we finally resolve this
issue and derive a lower bound on the conserved energy. Just as for masses
slightly above the BF bound, a given scalar potential can admit two possible
branches of the corresponding superpotential, one analytic and one
non-analytic. The key point again is that existence of the non-analytic branch
is necessary for the energy bound to hold. We discuss several AdS/CFT
applications of this result, including the use of double-trace deformations to
induce spontaneous symmetry breaking.
