Given (M;g) a smooth compact Riemannian n-manifold, n greater than or equal
to 3, we return in this article to the study of the sharp Sobolev-Poincare
type-inequality
(0.1) parallel tou parallel to (2)(2*) less than or equal to K(n)(2)paralle
l to delu parallel to (2)(2) + B parallel tou parallel to (2)(1)
where 2(star) = 2n/(n - 2) is the critical Sobolev exponent, and K-n is the
sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that (0.1
) is true if n = 3, that(0.1) is true if n greater than or equal to 4 and t
he sectional curvature of g is a nonpositive constant, or the Cartan-Hadama
rd conjecture in dimension n is true and the sectional curvature of g is no
npositive, but that (0.1) is false if n greater than or equal to 4 and the
scalar curvature of g is positive somewhere. When (0.1) is true, we define
B(g) as the smallest B in (0.1). The saturated form of (0.1) reads as
(0.2) parallel tou parallel to (2)(2 star) less than or equal to K(n)(2)par
allel to delu parallel to (2)(2) + B(g)parallel tou parallel to (2)(1).
We assume in this article that n greater than or equal to 4, and complete t
he study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincare inequalit
y (0.1). We prove that (0.1) is true, and that (0.2) possesses extremal fun
ctions when the scalar curvature of g is negative. A fairly complete answer
to the question of the validity of (0.1) under the assumption that the sca
lar curvature is not necessarily negative, but only nonpositive, is also gi
ven.