  This paper deals with the question of prescribing the Gaussian curvature on a
disk and the geodesic curvature of its boundary by means of a conformal
deformation of the metric. We restrict ourselves to a symmetric setting in
which the Gaussian curvature is negative, and we are able to give general
existence results. Our approach is variational, and solutions will be searched
as critical points of an associated functional. The proofs use a perturbation
argument via the monotonicity trick of Struwe, together with a blow-up analysis
and Morse index estimates. We also give a nonexistence result that shows that,
to some extent, the assumptions required for existence are necessary.
