Given a smooth function on a compact surface, does there exist a conformal metric realizing that function as its curvature? This is a classical problem in Geometric Analysis that goes back to the works of Berger, Kazdan & Warner and Moser in the 70's.
This question leads to a semilinear elliptic problem with a exponential nonlinearity, which has a variational structure. This kind of problems appear also in very different contexts: for instance, in the abelian Chern-Simons theory or in the Electroweak theory. In this talk I will try to give an overview on the topic, that still yields intricate open problems. In particular, the case of the sphere (the so-called Nirenberg problem) is not yet completely understood.
After that we will adress the same question on a surface with conical points, or with boundaries. We will show that, in some cases, the corresponding Euler-Lagrange functional is not bounded from below . Therefore we shall look for saddle-points of the functional, with the help of min-max theory.