From best constants to critical functions

The study of sharp Sobolev inequalities starts with the notion of best cons tant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this p aper to the H-1(2)-Sobolev inequality. Then, we extend the notion of best c onstant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp H-1(2)-Sobolev inequality. Partial answers to the more genera l question to know whether or not an extremal function always comes with a critical function are also given.