On conformal deformations of metrics on \(S^n\) (Q1270312)

The authors' main result is: Let us suppose that \(Q>0\) is a smooth function on \(S^n\), satisfying the nondegeneracy condition \(\Delta Q(x)\neq 0\) whenever \(\nabla Q(x)= 0\) and \(\deg(G, B^{n+1},0)\neq 0\), then the equation \[ P_nw+ (n-1)!= Qe^{nw}\quad\text{on }S^n \] has a solution where \(P_n\) is an \(n\)th order conformal invariant operator on \(S^n\), \(g_w= e^{2w}\cdot g_0\) is a conformal change of the metric \(g_0\) of \(S^n\), \(B^{n+1}\) is the unit ball in \(\mathbb{R}^{n+1}\); \(G:B^{n+1}\to \mathbb{R}^{n+1}\) is a map associated to the function \(Q\) by using the action of the conformal group of \(S^n\).