Prescribing scalar curvature on S-n

We consider the prescribing scalar curvature equation (1) -Deltau + n(n- 2)/4u = n-2/4(n-1)R(x)(un+2/n-2) on S-n n for n greater than or equal to 3. In the case R is rotationally sy mmetric, the well-known Kazdan-Warner condition implies that a necessary co ndition for ( 1) to have a solution is: R > 0 somewhere and R' (r) changes signs. Then, (a) is this a sufficient condition? (b) If not, what are the necessary and sufficient conditions? These have been open problems for decades. In Chen & Li, 1995, we gave question ( a) a negative answer. We showed that a necessary condition for ( 1) to have a solution is: (2) R'(r) changes signs in the region where R is positive. Now is this also a sufficient condition? In this paper, we prove that if R( r) satisfies the 'flatness condition', then ( 2) is the necessary and suffi cient condition for ( 1) to have a solution. This essentially answers quest ion ( b). We also generalized this result to non-symmetric functions R. Her e the additional 'flatness condition' is a standard assumption which has be en used by many authors to guarantee the existence of a solution. In partic ular, for n = 3, non-degenerate functions satisfy this condition. Based on Theorem 3 in Chen & Li, 1995, we also show that for some rotationa lly symmetric R, ( 1) is solvable while none of the solutions is rotational ly symmetric. This is interesting in the studying of symmetry breaking.