  We consider asymptotically anti de Sitter gravity coupled to a scalar field
with mass slightly above the Breitenlohner-Freedman bound. This theory admits a
large class of consistent boundary conditions characterized by an arbitrary
function $W$. An important open question is to determine which $W$ admit stable
ground states. It has previously been shown that the total energy is bounded
from below if $W$ is bounded from below and the bulk scalar potential $V(\phi)$
admits a suitable superpotential. We extend this result and show that the
energy remains bounded even in some cases where $W$ can become arbitrarily
negative. As one application, this leads to the possibility that in
gauge/gravity duality, one can add a double trace operator with negative
coefficient to the dual field theory and still have a stable vacuum.
