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Transactions of the American Mathematical Society

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Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities


Author: Didier Smets
Journal: Trans. Amer. Math. Soc. 357 (2005), 2909-2938
MSC (2000): Primary 35J10, 35J70, 58J37
DOI: https://doi.org/10.1090/S0002-9947-04-03769-9
Published electronically: December 29, 2004
MathSciNet review: 2139932
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Abstract: We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.


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  • 1. T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296. MR 0431287 (55:4288)
  • 2. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), 279-281. MR 0407905 (53:11675)
  • 3. A. Bahri, Critical points at infinity in some variational problems, Pitman Res. Notes in Math. Ser. 182, Longman Scientific & Techn., 1989. MR 1019828 (91h:58022)
  • 4. A. Bahri and H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, Topics in geometry, 1-100, Progr. Nonlinear Diff. Eqns Appl. 20, Birkhäuser, Boston, 1996. MR 1390310 (97c:53056)
  • 5. A. Bahri and J.-M. Coron, The scalar-curvature problem on standard three dimensional sphere, J. of Funct. Anal. 95 (1991), 106-172.MR 1087949 (92k:58055)
  • 6. A. Bahri and Y.Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in ${R}\sp N$, Rev. Mat. Iberoamericana 6 (1990), 1-15. MR 1086148 (92b:35054)
  • 7. G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. of Funct. Anal. 100 (1991), 18-24. MR 1124290 (92i:46033)
  • 8. G. Bianchi and H. Egnell, A variational approach to the equation $\Delta u +Ku^{(N+2)/(N-2)}=0$ in $\mathbb{R} ^N$, Arch. Rat. Mech. Anal. 122 (1993), 159-182. MR 1217589 (94g:35079)
  • 9. H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures et Appl. 58 (1979), 137-151. MR 0539217 (80i:35135)
  • 10. H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. MR 0699419 (84e:28003)
  • 11. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure and Appl. Math. 36 (1983), 437-477. MR 0709644 (84h:35059)
  • 12. G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. IHP Anal. Non Lin. 1 (1984), 341-350. MR 0779872 (86e:35016)
  • 13. O. Druet, The best constants problem in Sobolev inequalities, Mathematische Annalen, 314 (1999), 327-346. MR 1697448 (2000d:58033)
  • 14. O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Annales de l'Institut Henri Poincaré, Analyse non linéaire 19 (2002), 125-142. MR 1902741 (2003f:35104)
  • 15. O. Druet, E. Hebey and F. Robert, A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth, Electronic Research Announcements of the AMS, 9 (2003), 19-25. MR 1988868 (2004c:58046)
  • 16. Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincaré, Analyse non linéaire 8 (1991), 159-174. MR 1096602 (92c:35047)
  • 17. E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Mathematical Journal 79 (1995), 235-279. MR 1340298 (96c:53057)
  • 18. J.L. Kazdan and F.W. Warner, Scalar curvature and conformal deformation of Riemannian strructure, Journal of Differential Geometry 10 (1975), 113-134. MR 0365409 (51:1661)
  • 19. Y.Y. Li, Prescribing scalar curvature on $\mathbb{S} ^n$ and related problems, Part I, J. Diff. Eqns 120 (1995), 319-410.MR 1347349 (98b:53031)
  • 20. Y.Y. Li, Prescribing scalar curvature on $\mathbb{S} ^n$ and related problems, Part II, Comm. Pure Appl. Math. 49 (1996), 541-597.MR 1383201 (98f:53036)
  • 21. Y.Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Comm. Contemp. Math. 1 (1999), 1-50. MR 1681811 (2000m:53051)
  • 22. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Ibero Americana 1 (1985), 45-121. MR 0850686 (87j:49012)
  • 23. P. Sacks and K. Uhlenbeck, The existence of minimal immersions of the 2-spheres, Ann. Math. 113 (1981), 1-24. MR 0604040 (82f:58035)
  • 24. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495.MR 0788292 (86i:58137)
  • 25. R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, 120-154, Lecture Notes in Mathematics, Springer, 1989. MR 0994021 (90g:58023)
  • 26. M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), 511-517. MR 0760051 (86k:35046)
  • 27. S. Terracini, On positive solutions to a class of equations with a singular coefficient and critical exponent, Adv. Dif. Eqns 2 (1996), 241-264. MR 1364003 (97b:35057)
  • 28. M. Willem, Minimax Theorems, Progr. Nonlinear Diff. Eqns Appl. 24, Birkhäuser, Boston, 1996. MR 1400007 (97h:58037)
  • 29. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37. MR 0125546 (23:A2847)

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Additional Information

Didier Smets
Affiliation: Laboratoire J.L. Lions, Université Pierre & Marie Curie, 175 rue du chevaleret, 75013 Paris, France
Email: smets@ann.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-04-03769-9
Keywords: Hardy potential, critical Sobolev exponent, prescribed scalar curvature
Received by editor(s): February 11, 2002
Received by editor(s) in revised form: January 13, 2004
Published electronically: December 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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