Some uniform estimates for scalar curvature type equations

Overview & Synopsis

We consider the prescribed scalar curvature equation on an open set of ‫ޒ‬ n , − u = V u (n+2)/(n−2) + u n/(n−2) with V ∈ C 1,α (0 < α ≤ 1), and we prove the inequality sup K u × inf u ≤ c where K is a compact set of. In dimension 4, we have an idea on the supremum of the solution of the prescribed scalar curvature if we control the infimum. For this case we suppose the scalar curvature C 1,α (0 < α ≤ 1).