Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
HTML articles powered by AMS MathViewer
- by Didier Smets
- Trans. Amer. Math. Soc. 357 (2005), 2909-2938
- DOI: https://doi.org/10.1090/S0002-9947-04-03769-9
- Published electronically: December 29, 2004
- PDF | Request permission
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.References
- Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
- Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 5, Aii, A279–A281. MR 407905
- A. Bahri, Critical points at infinity in some variational problems, Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1019828
- A. Bahri and H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 1–100. MR 1390310
- A. Bahri and J.-M. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal. 95 (1991), no. 1, 106–172. MR 1087949, DOI 10.1016/0022-1236(91)90026-2
- Abbas Bahri and Yan Yan Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\textbf {R}^N$, Rev. Mat. Iberoamericana 6 (1990), no. 1-2, 1–15. MR 1086148, DOI 10.4171/RMI/92
- Gabriele Bianchi and Henrik Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18–24. MR 1124290, DOI 10.1016/0022-1236(91)90099-Q
- Gabriele Bianchi and Henrik Egnell, A variational approach to the equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ in $\mathbf R^n$, Arch. Rational Mech. Anal. 122 (1993), no. 2, 159–182. MR 1217589, DOI 10.1007/BF00378166
- Haïm Brézis and Tosio Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151. MR 539217
- Haïm Brézis and Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. MR 699419, DOI 10.1090/S0002-9939-1983-0699419-3
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
- Giovanna Cerami, Donato Fortunato, and Michael Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, 341–350 (English, with French summary). MR 779872, DOI 10.1016/S0294-1449(16)30416-4
- Olivier Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327–346. MR 1697448, DOI 10.1007/s002080050297
- Olivier Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré C Anal. Non Linéaire 19 (2002), no. 2, 125–142 (English, with English and French summaries). MR 1902741, DOI 10.1016/S0294-1449(02)00095-1
- Olivier Druet, Emmanuel Hebey, and Frédéric Robert, A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 19–25. MR 1988868, DOI 10.1090/S1079-6762-03-00108-2
- Zheng-Chao Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 2, 159–174 (English, with French summary). MR 1096602, DOI 10.1016/S0294-1449(16)30270-0
- Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279. MR 1340298, DOI 10.1215/S0012-7094-95-07906-X
- Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. MR 365409
- Yan Yan Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations 120 (1995), no. 2, 319–410. MR 1347349, DOI 10.1006/jdeq.1995.1115
- Yanyan Li, Prescribing scalar curvature on $S^n$ and related problems. II. Existence and compactness, Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597. MR 1383201, DOI 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
- Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50. MR 1681811, DOI 10.1142/S021919979900002X
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
- Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
- Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517. MR 760051, DOI 10.1007/BF01174186
- Susanna Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), no. 2, 241–264. MR 1364003
- Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
- Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
Bibliographic Information
- Didier Smets
- Affiliation: Laboratoire J.L. Lions, Université Pierre & Marie Curie, 175 rue du chevaleret, 75013 Paris, France
- Email: smets@ann.jussieu.fr
- Received by editor(s): February 11, 2002
- Received by editor(s) in revised form: January 13, 2004
- Published electronically: December 29, 2004
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2139932