The scalar curvature equation on 2- and 3-spheres (Q1315044)

The problem of finding a metric on \(S^ n\) of prescribed scalar curvature \(R\) which is conformally equivalent to the standard metric has received a great deal of attention, but is still not completely resolved. It is well known that the problem reduces to solving a certain nonlinear elliptic equation on \(S^ n\). Here the authors derive some a priori estimates for solutions of this equation in the cases \(n=2,3\) under appropriate conditions on \(R\). This enables them to use the continuity method to obtain some existence results which generalize those of \textit{S.-Y. A. Chang} and \textit{P. C. Yang} [Acta Math. 159, 215-259 (1987; Zbl 0636.53053)] for \(S^ 2\), and \textit{A. Bahri} and \textit{J. M. Coron} [J. Funct. Anal. 95, No. 1, 106-172 (1991; Zbl 0722.53032)] for \(S^ 3\).