  To study the problem of the assigned Gauss curvature with conical
singularities on Riemanian manifolds, we consider the Liouville equation with a
single Dirac measure on the two-dimensional sphere. By a stereographic
projection, we reduce the problem to a Liouville equation on the euclidean
plane. We prove new multiplicity results for bounded radial solutions, which
improve on earlier results of C.-S. Lin and his collaborators. Based on
numerical computations, we also present various conjectures on the number of
unbounded solutions. Using symmetries, some multiplicity results for non radial
solutions are also stated.
