Optimal Hardy-Sobolev inequalities on compact Riemannian manifolds

Given a compact Riemannian manifold (M, g) of dimension n >= 3, a point x(0) is an element of M and s is an element of (0,2), the Hardy-Sobolev embedding yields the existence of A, B > 0 such that (integral(M)d(g)(x,x(0)/vertical bar u vertical bar(n-2)/(2(n-s))dv(g))(n-s)/(n-2) <= A integral(M) vertical bar del u vertical bar(2)(g)dv(g) + B integral(M) u(2) dv(g) for all u is an element of H-1(2) (M). It has been proved in Jaber [20] that A >= K (n, s) and that one can take any value A > K (n, s) in (1), where K (n, s) is the best possible constant in the Euclidean Hardy-Sobolev inequality. In the present manuscript, we prove that one can take A = K(n, s) in (1). (C) 2014 Published by Elsevier Inc.