Sharp Sobolev inequalities involving boundary terms

Let (M, g) be a compact Riemannian manifold of dimension n (n ≥ 3) with smooth boundary. In [LZ], we established some sharp trace inequality on (M, g). In this paper we establish some sharp Sobolev inequalities using the method in [LZ]. For n ≥ 3, it was shown by Aubin [Au1] and Talenti [T] that, for p = 2n/(n − 2), 1 S 1 = inf R n |∇u| 2 R n |u| p 2/p u ∈ L p (R n) \ {0}, ∇u ∈ L 2 (R n) , (0.1) is achieved and the extremal functions are found. In particular, 1 S 1 = πn(n − 2) Γ(n/2)/Γ(n) 2/n .