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This paper deals with the question of prescribing the Gaussian curvature on adisk and the geodesic curvature of its boundary by means of a conformaldeformation of the metric. We restrict ourselves to a symmetric setting inwhich the Gaussian curvature is negative, and we are able to give generalexistence results. Our approach is variational, and solutions will be searchedas critical points of an associated functional. The proofs use a perturbationargument via the monotonicity trick of Struwe, together with a blow-up analysisand Morse index estimates. We also give a nonexistence result that shows that,to some extent, the assumptions required for existence are necessary.

Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature