  We consider black hole solutions with a dilaton field possessing a nontrivial
potential approaching a constant negative value at infinity. The asymptotic
behaviour of the dilaton field is assumed to be slower than that of a localized
distribution of matter. A nonabelian SU(2) gauge field is also included in the
total action. The mass of the solutions admitting a power series expansion in
$1/r$ at infinity and preserving the asymptotic anti-de Sitter geometry is
computed by using a counterterm subtraction method. Numerical arguments are
presented for the existence of hairy black hole solutions for a dilaton
potential of the form $V(\phi)=C_1 \exp(2\alpha_1 \phi)+C_2 \exp(2\alpha_2
\phi)+C_3$, special attention being paid to the case of ${\cal N}=4, D=4$
gauged supergravity model of Gates and Zwiebach.
