Let Omega be a bounded smooth domain in R(N), N >= 3, and D(a)(1,2) (Omega) be the completion of C(0)(infinity) (Omega) with respect to the norm: ||u||(2)(a) = integral(Omega)|x|(-2a)|del u|(2)dx. The Caffarelli-Kohn-Nirenberg inequalities state that there is a constant C > 0 such that (integral(Omega)|x|(-bq)|u|(q)dx)(2/q) <= C integral(Omega)|x|(-2a)|del u|dx for u is an element of D(a)(1,2) (Omega) and [GRAPHICS] We prove the best constant for (0.1) [GRAPHICS] is always achieved in D(a)(1,2) (Omega) provided that 0 is an element of partial derivative Omega and the mean curvature H(0) < 0, where a, b satisfies (i) a < b < a + 1 and N >= 3, or (ii) b = a > 0 and N >= 4. If a = 0 and 1 > b > 0, then the result was proved by Ghoussoub and Robert [12].
