  We present a general and systematic treatment of multi-trace deformations in
the AdS/CFT correspondence in the large N limit, pointing out and clarifying
subtleties relating to the formulation of the boundary value problem on a
conformal boundary. We then apply this method to study multi-trace deformations
in the presence of a scalar VEV, which requires the coupling to gravity to be
taken into account. We show that supergravity solutions subject to `mixed'
boundary conditions are in one-to-one correspondence with critical points of
the holographic effective action of the dual theory in the presence of a
multi-trace deformation, and we find a number of new exact analytic solutions
involving a minimally or conformally coupled scalar field satisfying `mixed'
boundary conditions. These include the generalization to any dimension of the
instanton solution recently found in hep-th/0611315. Finally, we provide a
systematic method for computing the holographic effective action in the
presence of a multi-trace deformation in a derivative expansion away from the
conformal vacuum using Hamilton-Jacobi theory. Requiring that this effective
action exists and is bounded from below reproduces recent results on the
stability of the AdS vacuum in the presence of `mixed' boundary conditions.

High Energy Physics - Theory