• Blow-up analysis
• Geometric PDEs
• Min-max Schemes
• Variational Methods

We consider a class of elliptic PDEs on closed surfaces with exponential nonliearities and Dirac deltas on the right-hand side. The study arises from abelian Chern-Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities.

A general existence result is proved using global variational methods and some examples of application are studied in detail. The analytic problem is reduced to a topological problem concerning the contractibility of a model space associated to the equation, the so-called space of formal baricenters. Finally, a conjecture is presented in order to reduce such topological investigation to test a very simple algebraic relation involving the parameters that appear in the equation.