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The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

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Abstract

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem

$${-\varepsilon^2\Delta_g u + u = u^{p-1}\,{\rm in}\,M,}$$

are generated by stable critical points of the scalar curvature of g, provided \({\varepsilon}\) is small enough. Here p > 2 if N = 2 and \({2 < p < 2^{*} = {2N \over N-2}}\) if N ≥ 3.

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata “U.Dini”, Università di Pisa, via F. Buonarroti 1/c, 56100, Pisa, Italy

    Anna Maria Micheletti

  2. Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161, Roma, Italy

    Angela Pistoia

Authors
  1. Anna Maria Micheletti
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  2. Angela Pistoia
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Correspondence to Anna Maria Micheletti.

Additional information

The authors are supported by Mi.U.R. project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

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Micheletti, A.M., Pistoia, A. The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. Calc. Var. 34, 233–265 (2009). https://doi.org/10.1007/s00526-008-0183-4

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  • DOI: https://doi.org/10.1007/s00526-008-0183-4

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