Critical aspects of Choquard type equations

We are concerned with a class of nonlocal Schroedinger equations which show up in many different applied contexts. In particular we consider the case in which the nonlinear interaction has critical features. In the classical Sobolev sense, when we are in presence of the Hardy-Littlewood-Sobolev upper-critical exponent, as well as new critical phenomena occur, the so-called 'bubbling at infinity', when the nonlinearity exhibits lower-critical growth. Assuming mild conditions on the nonlinearity and by using variational methods, we establish existence, non-existence and qualitative properties of finite energy solutions. Some partial results in the limiting case of dimension two will be also presented.
(Joint works in collaboration with: J. Zhang; J. Van Schaftingen and J. Zhang; C. Tarsi and M. Yang).