This is joint work with Francisco Martin (Granada) and Andreas Savas-Halilaj (Hannover). We obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

We consider Liouville equations arising from curvature prescription problems and from models in Electroweak and Chern-Simons theory. We show how improved versions of the Moser-Trudinger inequality, combined with min-max theory, may reduce these PDEs to the study of finite-dimensional objects consisting of measures supported at finitely-many points. These are joint works with D. Bartolucci, A. Carlotto, F. De Marchis and D. Ruiz.