  In this paper we study the problem, posed by Troyanov, of prescribing the
Gaussian curvature under a conformal change of the metric on surfaces with
conical singularities. Such geometrical problem can be reduced to the
solvability of a nonlinear PDE with exponential type non-linearity admitting a
variational structure. In particular, we are concerned with the case where the
prescribed function $K$ changes sign. When the surface is the standard sphere,
namely for the singular Nirenberg problem, by a min-max approach and a new
compactness argument we give sufficient conditions on $K$, concerning mainly
the regularity of its nodal line and the topology of its positive nodal region,
to be the Gaussian curvature of a conformal metric with assigned conical
singularities.
  Besides, we find a class of functions on $\mathbb{S}^2$ which do not verify
our conditions and which can not be realized as the Gaussian curvature of any
conformal metric with one conical singularity. This shows that our result is
somehow sharp.
