Optimal Sobolev inequalities and extremal functions. The three-dimensional case

The existence of extremal functions for sharp Sobolev inequalities on compact manifolds is fairly well understood when the dimension of the manifold is greater than or equal to four. We investigate in this paper the existence of extremal functions for such inequalities when the dimension is three, and, as a by-product, we investigate also low dimension phenomena that occur in the study of nonlinear elliptic PDEs involving the Laplace operator.