Radial prescribing scalar curvature on $R{P}^n$

Abstract

The study of radial prescribing scalar curvature of X.Xu and P.C.Yang [2] in 1993 showed a nonexistence result on $S^2$. Later in 1995, W.Chen and C.Li [2] generalized the nonexistence result to higher dimensions. G.Bianchi and E.Egnell [1] suggested that there may exist some non-negative smooth radial function which cannot be scalar curvature in the standard conformal class of $R{P}^n.$ However, in our paper, we prove that all smooth radial non-negative smooth functions which are positive on the pole could be prescribing scalar curvatures. We consider the quotient $\frac{v_{\lambda }}{V_{\lambda }}$ rather than the difference $v_{\lambda }-V_{\lambda }$ as in [1]. This trick yields a concise argument of the existence. Consequently, their counter examples stated in [1] cannot be true on $R{P}^n.$