Sala Riunioni (Dip. Matematica).
In dimension n elliptic PDE’s with an exponential non-linearity can have a critical behavior when the differential operator either involves derivatives of order >2 or is quasi-linear. Due to intrinsic invariances such PDE’s present a natural lack of compactness and in the quasi-linear case I aim to present some recent results concerning existence issues and the description of the blow-up mechanism. In the last part of the talk, I will report on an ongoing research project, in collaboration with A. Malchiodi, concerning a four-dimensional PDE arising in the theory of log-determinants in conformal geometry, where the differential operator is fourth-order and quasi-linear at the same time.