Multiplicity results for the scalar curvature equation

Abstract

This paper is devoted to the study of positive radial solutions of the scalar curvature equation, i.e. !u(x) +K(|x|)uσ−1(x) = 0 where σ = 2n n−2 and we assume that K(|x|) = k(|x|ε) and k(r) ∈ C1 is bounded and ε >0 is small. It is known that we have at least a ground state with fast decay for each positive critical point of k for ε small enough. In fact if the critical point k(r0) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k(r0) is a minimum we have an arbitrarily large number of ground states with fast decay. The results are obtained using Fowler transformation and developing a dynamical approach inspired by Melnikov theory. We emphasize that the presence of subharmonic solutions arising from zeroes of Melnikov functions has not appeared previously, as far as we are aware. © 2015 Elsevier Inc. All rights reserved. MSC: 35J60; 34C37; 34E15 Keywords: Critical exponent; Ground state; Fowler transformation; Singular perturbations; Melnikov theory