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Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains

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Abstract

In this paper we prove that, for every positive integer k, there exists a contractible bounded domain Ω in ℝN with N≥3, where the problem (*) (see Introduction) has at least k solutions.

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References

  1. T. Aubin: “Equations differentielles nonlinéaires et problème de Yamabe concernant la courbure scalaire”. J. Math. Pures Appl.55 (1976), 269–293

    Google Scholar 

  2. A. Bahri, J.M. Coron: “Sur une équation elliptique nonlinéaire avec l'exposant critique de Sobolev”. C.R. Acad. Sci. Paris301 (1985), 345–348

    Google Scholar 

  3. A. Bahri, J.M. Coron: “On a nonlinear elliptic equation involving the critical Sobolev exponent. The effect of the topology of the domain”. Comm. Pure Appl. Math., Vol.XLI, (1988), 253–294

    Google Scholar 

  4. H. Brezis: “Elliptic equations with limiting Sobolev exponents. The impact of topology”. Comm. Pure Appl. Math.39 (1986), S.17-S.39

    Google Scholar 

  5. H. Brezis, E. Lieb: “A relation between pointwise convergence of functions and convergence of functionals”. Proc. Amer. Math. Soc.,88 (1983), 486–490

    Google Scholar 

  6. H. Brezis, L. Nirenberg: “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents”. Comm. Pure Appl. Math.36 (1983), 437–477

    Google Scholar 

  7. A. Capozzi, D. Fortunato, G. Palmieri: “An existence result for nonlinear elliptic problems involving critical Sobolev exponents”. Ann. Inst. H. Poincaré. Analyse Nonlinéaire2 (1985), 463–470

    Google Scholar 

  8. G. Cerami, D. Fortunato, M. Struwe: “Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents”. Ann. Inst. H. Poincaré. Analyse Nonlinéaire1 (1984), 341–350

    Google Scholar 

  9. J.M. Coron: “Topologie et cas limite des injections de Sobolev”. C.R. Acad. Sc. Paris,299, series I (1984), 209–212

    Google Scholar 

  10. W. Ding: “On a conformally invariant elliptic equation on ℝN”. Comm. Math. Phys.107 (1986), 331–335

    Google Scholar 

  11. W. Ding: “Positive solutions of\(\Delta u + u\frac{{n + 2}}{{n - 2}} = 0\) on contractible domains”. To appear

  12. D. Fortunato, E. Jannelli: “Infinitely many solution for some nonlinear elliptic problems in symmetrical domains”. Proc. of the Royal Soc. Edinburgh,105 A (1987), 205–213

    Google Scholar 

  13. B. Gidas: “Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations”, in “Nonlinear Differential equations in Engineering and Applied Sciences” (R.L. Sternberg Ed.), Dekker, New York 1979, 255–273

    Google Scholar 

  14. B. Gidas, W.M. Ni, L. Nirenberg: “Symmetry of positive solutions of nonlinear elliptic equations in ℝN”, in “Mathematical Analysis and Applications”, Part A (L. Nachbin ed.), 370–401, Academic Press, 1981

  15. J. Kazdan, F. Warner: “Remarks on some quasilinear elliptic equations”. Comm. Pure Appl. Math.28 (1975), 567–597

    Google Scholar 

  16. E. Lieb: “Sharp constants in the Hardy-Littlewood Sobolev inequality and related inequalities”. Annals of Math.118 (1983), 349–374

    Google Scholar 

  17. P.L. Lions: “Applications de la méthode de concentration-compacité à l'existence de fonctions exstrémales”. C.R. Acad. Sc. Paris296 (1983), 645–64

    Google Scholar 

  18. P.L. Lions: “The concentration-compactness principle in the Calculus of Variations. The limit case”. Rev. Mat. Iberoamericana1 (1985) 45–121 and 145–201

    Google Scholar 

  19. A. Marino: “La biforcazione nel caso variazionale”. Conferenze Sem. Mat. Univ. Bari (1973)

  20. M. Obata: “The conjectures conformal transformation of Riemannian manifolds”. J. Diff. Geom.6 (1971), 247–258

    Google Scholar 

  21. S. Pohozaev: “Eigenfunctions of the equation Δu+λf(u)=0”. Soviet Math. Dokl.6 (1965), 1408–1411

    Google Scholar 

  22. P. Rabinowitz: “Some aspects of nonlinear eigenvalue problems”. Rocky Mount. J. Math.3 (1973), 161–202

    Google Scholar 

  23. M. Struwe: “A global compactness result for elliptic boundary value problems involving limiting nonlinearities”. Math. Z.187 (1984), 511–517

    Google Scholar 

  24. G. Talenti: “Best constants in Sobolev inequality”. Annali di Mat. Pura e Appl.110 (1976), 353–372

    Google Scholar 

  25. H. Yamabe: “On a deformation of Riemannian structures on compact manifold”. Osaka Math. J.12 (1960), 21–37

    Google Scholar 

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  1. Dipartimento di Matematica dell'Università, Via F. Buonarroti,2, 56100, Pisa, Italy

    Donato Passaseo

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  1. Donato Passaseo
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Passaseo, D. Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math 65, 147–165 (1989). https://doi.org/10.1007/BF01168296

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  • DOI: https://doi.org/10.1007/BF01168296

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