  We examine anti-de Sitter gravity minimally coupled to a self-interacting
scalar field in $D\geq 4$ dimensions when the mass of the scalar field is in
the range $m_{\ast}^{2}\leq m^{2}<m_{\ast} ^{2}+l^{-2}$. Here, $l$ is the AdS
radius, and $m_{\ast}^{2}$ is the Breitenlohner-Freedman mass. We show that
even though the scalar field generically has a slow fall-off at infinity which
back reacts on the metric so as to modify its standard asymptotic behavior, one
can still formulate asymptotic conditions (i) that are anti-de Sitter
invariant; and (ii) that allows the construction of well-defined and finite
Hamiltonian generators for all elements of the anti-de Sitter algebra. This
requires imposing a functional relationship on the coefficients $a$, $b$ that
control the two independent terms in the asymptotic expansion of the scalar
field. The anti-de Sitter charges are found to involve a scalar field
contribution. Subtleties associated with the self-interactions of the scalar
field as well as its gravitational back reaction, not discussed in previous
treatments, are explicitly analyzed. In particular, it is shown that the fields
develop extra logarithmic branches for specific values of the scalar field mass
(in addition to the known logarithmic branch at the B-F bound).
