Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Given $(M,g)$ a smooth compact Riemannian $n$-manifold, $n \ge 3$, we return in this article to the study of the sharp Sobolev-Poincaré type inequality

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 $n \ge 4$ and the sectional curvature of $g$ is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension $n$ is true and the sectional curvature of $g$ is nonpositive, but that $(0.1)$ is false if $n \ge 4$ 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

We assume in this article that $n \ge 4$, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality $(0.1)$. We prove that $(0.1)$ is true, and that $(0.2)$ possesses extremal functions 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 scalar curvature is not necessarily negative, but only nonpositive, is also given.