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

Communications on Pure and Applied MathematicsVolume 54, Issue 2 p. 229-258 On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions † Florin Catrina, Florin Catrina [email protected] Utah State University, Department of Mathematics and Statistics, 3900 Old Main Hill, Logan, Utah 84322-3900Search for more papers by this authorZhi-Qiang Wang, Zhi-Qiang Wang [email protected] Utah State University, Department of Mathematics and Statistics, 3900 Old Main Hill, Logan, Utah 84322-3900Search for more papers by this author Florin Catrina, Florin Catrina [email protected] Utah State University, Department of Mathematics and Statistics, 3900 Old Main Hill, Logan, Utah 84322-3900Search for more papers by this authorZhi-Qiang Wang, Zhi-Qiang Wang [email protected] Utah State University, Department of Mathematics and Statistics, 3900 Old Main Hill, Logan, Utah 84322-3900Search for more papers by this author First published: 21 December 2000 https://doi.org/10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-ICitations: 310 † Dedicated to Professor L. Nirenberg on the occasion of his 75th birthday AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6] where, for N ≥ 3, −∞ < a < (N − 2)/2, a ≤ b ≤ a + 1, and p = 2N/(N − 2 + 2(b − a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given. © 2001 John Wiley & Sons, Inc. Bibliography 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 Catrina, F.; Wang, Z.-Q. Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN. 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 Gidas, B.; Ni, W. M.; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in ℝN. 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 Talenti, G. Best constant in Sobolev inequality. Ann Mat Pura Appl (4) 110 (1976), 353–372. 24 Wang, Z.-Q. Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations. J Differential Equations 159 (1999), no. 1, 102–137. 25 Wang, Z.-Q.; Willem, M. Singular minimization problems. J Differential Equations 161 (2000), no. 2, 307–320. 26 Willem, M. Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, 24. Birkhäuser, Boston, 1996. 27 Willem, M. A decomposition lemma and critical minimization problems. Preprint. Citing Literature Volume54, Issue2February 2001Pages 229-258 ReferencesRelatedInformation