From best constants to critical functions

Abstract.

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the $H_1^2$-Sobolev inequality. Then, we extend the notion of best constant 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 general question to know whether or not an extremal function always comes with a critical function are also given.