  We consider the case with boundary of the classical Kazdan-Warner problem in
dimension greater or equal than three, i.e. the prescription of scalar and
boundary mean curvatures via conformal deformations of the metric. We deal in
particular with negative scalar curvature and boundary mean curvature of
arbitrary sign, which to our knowledge has not been treated in the literature.
We employ a variational approach to prove new existence results, especially in
three dimensions. One of the principal issues for this problem is to obtain
compactness properties, due to the fact that bubbling may occur with profiles
of hyperbolic balls or horospheres, and hence one may lose either pointwise
estimates on the conformal factor or the total conformal volume. We can
sometimes prevent them using integral estimates, Pohozaev identities and
domain-variations of different types.
