On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions

Communications on Pure and Applied Mathematics

Volumn 54, Issue 2, 2001, Pages 229-258

  Catrina, Florin ^a   Wang, Zhi Qiang ^a  

^a UTAH STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035578399     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I     Document Type: Article

References (28)

1. Aubin, T. Problèmes isopérimétriques de Sobolev. J. Differential Geometry 11 (1976), no. 4, 573-598.
2. Berestycki, H.; Esteban, M. Existence and bifurcation of solutions for an elliptic degenerate problem. J. Differential Equations 134 (1997), no. 1, 1-25.
3. Brezis, H.; Lieb, E. H. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490.
4. Brezis, H.; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical exponents. Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477.
5. Caffarelli, L. A.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271-297.
6. Caffarelli, L. A.; Kohn, R.; Nirenberg, L. First order interpolation inequalities with weights. Compositio Math. 53 (1984), no. 3, 259-275.
7. Caldiroli, P.; Musina, R. On the existence of extremal functions for a weighted Sobolev embedding with critical exponent. Cal. Var. Partial Differential Equations 8 (1999), no. 4, 365-387.
8. Catrina, F.; Wang, Z.-Q. On the Caffarelli-Kohn-Nirenberg inequalities. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 437-442.
9. N. Ann. Inst. H. Poincaré Anal. Non Linéaire, in press.
10. Chen, W.; Li, C. Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), no. 3, 615-622.
11. Chou, K. S.; Chu, C. W. On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc. (2) 48 (1993), no. 1, 137-151.
12. Dautray, R.; Lions, J.-L. Mathematical analysis and numerical methods for science and technology. Vol. 1. Physical origins and classical methods. Spinger, Berlin, 1985.
13. Davies, E. B. A review of Hardy inequalities. Preprint.
14. N. Mathematical analysis and applications, Part A, 369-402. Advances in Mathematics Supplemetary Studies, 7a. Academic Press, New York-London, 1981.
15. Hardy, G. H.; Littlewood, J. E.; Pólya, G. Inequalities. Second edition. Cambridge, University Press, 1952.
16. Horiuchi, T. Best constant in weighted Sobolev inequality with weights being powers of distance from the origin. J. Inequal. Appl. 1 (1997), no. 3, 275-292.
17. Korevaar, N.; Mazzeo, R.; Pacard, F; Schoen, R. Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135 (1999), no. 2, 233-272.
18. Lieb, E. H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349-374.
19. Lin, C. S. Interpolation inequalities with weights. Comm. Partial Differential Equations 11 (1986), no. 14, 1515-1538.
20. Lin, S.-S. Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains. Trans. Amer. Math. Soc. 332 (1992), no. 2, 775-791.
21. Lions, P.-L. The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and Part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145 223-283.
22. Lions, P.-L. Concentration compactness principle in the calculus of variations. The limit case. I, II. Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201; 1 (1985), no. 2, 45-121.
23. Lions, P.-L. Concentration compactness principle in the calculus of variations. The limit case. I, II. Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201; 1 (1985), no. 2, 45-121.
24. Talenti, G. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110 (1976), 353-372.
25. Wang, Z.-Q. Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations. J. Differential Equations 159 (1999), no. 1, 102-137.
26. Wang, Z.-Q.; Willem, M. Singular minimization problems. J. Differential Equations 161 (2000), no. 2, 307-320.
27. Willem, M. Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, 24. Birkhäuser, Boston, 1996.
28. Willem, M. A decomposition lemma and critical minimization problems. Preprint.