The Scalar-Curvature Problem on the Standard Three- Dimensional Sphere

Let (S3, c) be the standard 3-sphere, i.e., the 3-sphere equipped with the standard metric. Let K be a C2 positive function on S3. The Kazdan-Warner problem [l] is the problem of finding suitable conditions on K such that K is the scalar curvature for a metric g on S3 conformally equivalent to c. The metric g then reads g=u4c and u is a positive function on S3 satisfying the partial differential equation-8 Au + 6u = K(x) us u > 0. Let L =-8 Au + 624 be the conformal Laplacian. The same problem can be formulated for any compact Riemannian manifold (M " , g). Since this problem has been formulated, there have been some partial answers (see [3-7, 181). Obstructions have also been pointed out [l, 21. The main difficulty, arising when one tries to solve equations of type (4), consists of the failure of the Palais-Smale condition. We show, in this paper, how this difficulty may be overcome in the case Eq. (1). Our method consists of studying the critical points at infinity of the variational problem, in 106