On axisymmetric solutions of the conformal scalar curvature equation on \(S^n\) (Q5954408)

Given a smooth function \(K\) on the \(n\)-dimensional sphere, the problem is to find a Riemannian metric conformal with the canonical metric and having \(K\) as its scalar curvature. Under stereographic projection it is equivalent to some partial differential equation depending on \(K\). The first question is how the properties of \(K\) near its critical points affect the blow-up behavior of a sequence of solutions of the equation. Some technical results on this are given. It is shown that the degree among the radial solutions is equal to zero when some exponents of flatness of \(K\) are bounded in a suitable way. The radial solutions used produce some simple proofs of technical results on the behavior and bounds for the solutions of the partial differential equation of the problem.