Sign-changing prescribed Gaussian curvature

I will consider the problem of prescribing the Gaussian curvature (under pointwise conformal change of the metric) on surfaces with conical singularities. This question has been first raised by Troyanov [TAMS,1991] and it is a generalization of the Kazdan-Warner problem for regular surfaces, known as the Nirenberg problem on the sphere. From the analytical point of view, this amounts to solve a singular Liouville-type equation on the surface. Initially, in the supercritical regime, only the case of positive prescribed Gaussian curvature has been attacked. In this talk I will present some new results (obtained in collaboration with T. D`Aprile, I. Ianni, S. Kallel, R. López-Soriano and D. Ruiz) concerning the sign-changing case.