We consider the prescribing scalar curvature equation \begin{equation} -\lap u + \frac{n(n-2)}{4} u = \frac{n-2}{4(n-1)} R(x)u^{ \frac{n+2}{n-2}} \label{eq0} \end{equation} on $S^n$ for $n \geq 3$. In the case R is rotationally symmetric, the well-known Kazdan-Warner condition implies that a necessary condition 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: \begin{equation} \mbox{ $R'(r)$ changes signs in the region {\em where $R$ is positive.} } \label{01} \end{equation} 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 sufficient condition for (1) to have a solution. This essentially answers question (b). We also generalized this result to non-symmetric functions R. Here the additional `flatness condition' is a standard assumption which has been used by many authors to guarantee the existence of a solution. In particular, for $n=3$, `non-degenerate' functions satisfy this condition.

Based on Theorem 3 in Chen \& Li, 1995, we also show that for some rotationally symmetric R, (1) is solvable while none of the solutions is rotationally symmetric. This is interesting in the studying of symmetry breaking.