PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 6, Pages 1685–1691 S 0002-9939(04)07245-4 Article electronically published on January 16, 2004 SYMMETRY OF EXTREMAL FUNCTIONS FOR THE CAFFARELLI-KOHN-NIRENBERG INEQUALITIES CHANG-SHOU LIN AND ZHI-QIANG WANG (Communicated by David S. Tartakoff) Abstract. We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin. 1. Introduction This paper is concerned with symmetry properties of extremal functions for the following weighted Sobolev inequalities due to Caffarelli, Kohn and Nirenberg ([3]): for all u ∈ C ∞ 0 (R N ), (1)   N R |x| −bp |u| p dx  2/p ≤ C a,b  N R |x| −2a |∇u| 2 dx where for N ≥ 3: (2) −∞ <a< N − 2 2 ,a ≤ b ≤ a +1, and p = 2N N − 2 + 2(b − a) , and for N = 2: (3) −∞ <a< 0, a<b ≤ a +1, and p = 2 b − a . Let D 1,2 a (R N ) be the completion of C ∞ 0 (R N ), with respect to the inner product (4) (u,v) a =  N R |x| −2a ∇u ·∇v dx. Then inequalities (1) are extended to all u ∈D 1,2 a (R N ). Define (5) S(a,b)= inf u∈D 1,2 a (R N )\{0} E a,b (u) Received by the editors October 30, 2002. 2000 Mathematics Subject Classification. Primary 35B33; Secondary 46E35. Key words and phrases. Weighted Sobolev inequalities, Extremal functions, Exact symmetry. c 2004 American Mathematical Society 1685