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)=0Δu(x)+K(|x|)uσ−1(x)=0 where σ=2n/(n−2 ) and we assume that K(|x|)=k(|x|ε)K(|x|)=k(|x|ε) and k(r)∈C1k(r)∈C1 is bounded and ε>0ε>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)k(r0) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k(r0)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.

Multiplicity results for the scalar curvature equation

Franca Matteo
Membro del Collaboration Group
2015

Abstract

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)=0Δu(x)+K(|x|)uσ−1(x)=0 where σ=2n/(n−2 ) and we assume that K(|x|)=k(|x|ε)K(|x|)=k(|x|ε) and k(r)∈C1k(r)∈C1 is bounded and ε>0ε>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)k(r0) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k(r0)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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/723511
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