Multiplicity results for the scalar curvature equation

Isabel Flores, Matteo Franca

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1 Citation (Scopus)

Abstract

© 2015 Elsevier Inc. This paper is devoted to the study of positive radial solutions of the scalar curvature equation, i.e. Δu(x) + K(|x|)uσ-1(x) = 0 where σ = 2n/n-2and we assume that K(|x|) = k(|x|ε) and k(r) ∈ C1is 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 stateswith 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.
Original languageEnglish
Pages (from-to)4327-4355
Number of pages29
JournalJournal of Differential Equations
DOIs
Publication statusPublished - 1 Jan 2015

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