  Given a closed two dimensional manifold, we prove a general existence result
for a class of elliptic PDEs with exponential nonlinearities and negative Dirac
deltas on the right-hand side, extending a theory recently obtained for the
regular case. This is done by global methods: since the associated Euler
functional is in general unbounded from below, we need to define a new model
space, generalizing the so-called space of formal barycenters and
characterizing (up to homotopy equivalence) its very low sublevels. As a
result, the analytic problem is reduced to a topological one concerning the
contractibility of this model space. To this aim, we prove a new functional
inequality in the spirit of [16] and then we employ a min-max scheme based on a
cone-style construction, jointly with the blow-up analysis given in [5] (after
[6] and [8]). This study is motivated by abelian Chern- Simons theory in
self-dual regime, or from the problem of prescribing the Gaussian curvature in
presence of conical singularities (hence generalizing a problem raised by
Kazdan and Warner in [26]).
