Singular mean field equations on compact Riemann surfaces

For a general class of elliptic PDE’s in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack.