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Multiplicity of ground states for the scalar curvature equation

  • Francesca Dalbono
  • Matteo FrancaEmail author
  • Andrea Sfecci
Article
  • 16 Downloads

Abstract

We study existence and multiplicity of radial ground states for the scalar curvature equation
$$\begin{aligned} \Delta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n, \quad n>2, \end{aligned}$$
when the function \(K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+\) is bounded above and below by two positive constants, i.e. \(0<{\underline{K}} \le K(r) \le {\overline{K}}\) for every \(r > 0\), it is decreasing in (0, 1) and increasing in \((1,+\infty )\). Chen and Lin (Commun Partial Differ Equ 24:785–799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio \({\overline{K}}/{\underline{K}}\) is smaller than some computable values.

Keywords

Scalar curvature equation Ground states Fowler transformation Invariant manifold Shooting method Bubble tower solutions Phase plane analysis Multiplicity results 

Mathematics Subject Classification

35J61 37D10 34C37 

Notes

Acknowledgements

F. Dalbono would like to express her gratitude to the “Centro de Matemática, Aplicaç\(\tilde{\text{ o }}\)es Fundamentais e Investigaç\(\tilde{\text{ a }}\)o Operacional” of the University of Lisbon for its hospitality. M. Franca wishes to honour prof. R. Johnson who recently passed away, for his generosity and careful guide. He was the one who introduced the author to the study of this subject and to research in mathematics.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PalermoPalermoItaly
  2. 2.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly
  3. 3.Dipartimento di Matematica e GeoscienzeUniversità degli Studi di TriesteTriesteItaly

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