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Topological tools for the prescribed scalar curvature problem on S n

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EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
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Bibliografia
  • [1] A. Ambrosetti, M. Badiale, Homoclinics: Poincaré-Melinkov type results via varitional approch, Ann.Inst.H.Poincaré Anal. Nonlinéaire 15 (1998) 233–252. http://dx.doi.org/10.1016/S0294-1449(97)89300-6
  • [2] Ambrosetti A., Garcia Azorero J., Peral A., Perturbation of \( - \Delta u + u\tfrac{{(N + 2)}} {{(N - 2)}} = 0 \) , the Scalar Curvature Problem in ℝ Nand related topics, Journal of Functional Analysis, 165 (1999), 117–149. http://dx.doi.org/10.1006/jfan.1999.3390
  • [3] A. Bahri, Critical point at infinity in some variational problems, Pitman Res. Notes Math, Ser 182, Longman Sci. Tech. Harlow 1989.
  • [4] A. Bahri, An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J. 81 (1996), 323–466. http://dx.doi.org/10.1215/S0012-7094-96-08116-8
  • [5] A. Bahri and P. Rabinowitz, Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 561–649.
  • [6] M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J. 84 (1996), 633–677. http://dx.doi.org/10.1215/S0012-7094-96-08420-3
  • [7] R. Ben Mahmoud, H. Chtioui, Existence results for the prescribed Scalar curvature on S3, Annales de l’Institut Fourier, 61 (2011), 971–986. http://dx.doi.org/10.5802/aif.2634
  • [8] R. Ben Mahmoud, H. Chtioui, Prescribing the Scalar Curvature Problem on Higher-Dimensional Manifolds, Discrete and Continuous Dynamical Systems A, 32 (2012), 1857–1879. http://dx.doi.org/10.3934/dcds.2012.32.1857
  • [9] R. Ben Mahmoud, H. Chtioui, A. Rigane On the prescribed scalar curvature problem on Sn: the degree zero case, C. R. Acad. Sci. Paris, Ser. I, 350, (2012), 583–586. http://dx.doi.org/10.1016/j.crma.2012.06.012
  • [10] A. Chang and P. Yang, A perturbation result in prescribing scalar curvature on S n, Duke Math. J. 64 (1991), 27–69. http://dx.doi.org/10.1215/S0012-7094-91-06402-1
  • [11] H. Chtioui, Prescribing the Scalar Curvature Problem on Three and Four Manifolds, Advanced Nonlinear Studies, 3 (2003), 457–470.
  • [12] H. Chtioui, R. Ben Mahmoud, D. A. Abuzaid, Conformal transformation of metrics on the n-sphere, Nonlinear Analysis: TMA, Volume 82 (2013) pp 66–81.
  • [13] H. Chtioui and A. Rigane, On the prescribed Q-curvature problem on Sn, C. R. Acad. Sci. Paris, Ser. I 348 (2010), 635–638. http://dx.doi.org/10.1016/j.crma.2010.03.018
  • [14] H. Chtioui and A. Rigane, On the prescribed Q-curvature problem on S n, Journal of Functional Analysis, 261 (2011), 2999–3043. http://dx.doi.org/10.1016/j.jfa.2011.07.017
  • [15] J. Kazdan and J. Warner, „Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatur”, Annals of Math.101(1975), p.317–331. http://dx.doi.org/10.2307/1970993
  • [16] Y.Y. Li, Prescribing scalar curvature on Sn and related topics, Part I, Journal of Differential Equations, 120 (1995), 319–410. http://dx.doi.org/10.1006/jdeq.1995.1115
  • [17] Y.Y. Li, Prescribing scalar curvature on Sn and related topics, Part II: existence and compactness, Comm. Pure Appl. Math. 49 (1996), 541–579. http://dx.doi.org/10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
  • [18] R. Schoen and Z. Zhang, Prescribed scalar curvature on the n-Sphere, Calc. Var. P.D.E. 4 (1996), 1–25. http://dx.doi.org/10.1007/BF01322307
  • [19] M. Stuwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z. 187, (1984), 511–517. http://dx.doi.org/10.1007/BF01174186
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Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-014-0443-9
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