Prescribed scalar curvature on compact Riemannian manifolds in the negative case (Q5961819)

Let \((V_n,g)\) be a smooth \((=C^\infty)\) compact \(n\)-dimensional Riemannian manifold \((n\geq 3)\), \(R\) its scalar curvature, \([g]\) the conformal class of \(g\) and \(\Delta=-g^{ij}\nabla_i\nabla_j\). The paper deals with the problem of determining functions which are the scalar curvature of some metric \(g'\in[g]\). If \(g'\) is taken in the form \(u^{4/(n-2)}g\) with \(u\in C^\infty(V_n)\), \(u>0\), and \(u\) is a solution of the equation NEWLINE\[NEWLINE{4(n-1)\over n-2}\Delta u+Ru=fu^{(n+2)/(n-2)},\;u>0,NEWLINE\]NEWLINE then the scalar curvature \(R'\) of \((V_n,g')\) is equal to \(f\).NEWLINENEWLINENEWLINEIn this paper, the case \(R<0\) is considered (also, without loss of generality, one may assume that \(R\) is constant, see \textit{H. Yamabe} [Osaka Math. J. 12, 21-37 (1960; Zbl 0096.37201)]). Sufficient conditions for the above equation to have a solution are pointed out.