Abstract

--- - Let (M, g) be an n-dimensional compact Riemannian manifold. Let h be a smooth function on M and assume that it has a critical point xi is an element of M such that h(xi) = 0 and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics g(lambda) = u(lambda)(4/n-2) g, with u > 0, whose scalar curvature is the prescribed function h(lambda):= lambda(2) + h, where lambda is a small parameter. - In the positive case, i.e. when the scalar curvature R-g is strictly positive, we find a family of "bubbling" metrics g lambda, where u(lambda) blows up at the point xi and approaches zero far from xi as lambda goes to zero. - In the general case, if in addition we assume that there exists a non-degenerate conformal metric g(0) = u(0)(4/n-2) with u(0) > 0, whose scalar curvature is equal to h, then there exists a bounded family of conformal 4 metrics g(0,lambda) = u(0,lambda)(4/n-2) g, with u(0,lambda) > 0, which satisfies u(0,lambda) -> u(0) uniformly as lambda -> 0. Here, we build a second family of "bubbling" metrics g(lambda), where u(lambda) blows up at the point xi and approaches u(0) far from xi as lambda goes to zero. In particular, this shows that this problem admits more than one solution. (C) 2017 Elsevier Inc. All rights reserved.