A note on scalar curvature type problems in dimension 4

Let (M, g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem gu + a(x)u = f (x)u3 always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem. © 2008 Elsevier Inc. All rights reserved