  We construct and analyze BPS black hole solutions in gauged N=2, D=4
supergravity with charged hypermultiplets. A class of solutions can be found
through spontaneous symmetry breaking in vacua that preserve maximal
supersymmetry. The resulting black holes do not carry any hair for the scalars.
We demonstrate this with explicit examples of both asymptotically flat and
anti-de Sitter black holes. Next, we analyze the BPS conditions for
asymptotically flat black holes with scalar hair and spherical or axial
symmetry. We find solutions only in cases when the metric contains ripples and
the vector multiplet scalars become ghost-like. We give explicit examples that
can be analyzed numerically. Finally, we comment on a way to circumvent the
ghost-problem by introducing also fermionic hair.

High Energy Physics - Theory