Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities
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- by Chang-Shou Lin and Zhi-Qiang Wang PDF
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Erratum: Proc. Amer. Math. Soc. 132 (2004), 2183-2183.
Abstract:
We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.References
- Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
- Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
- L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
- Florin Catrina and Zhi-Qiang Wang, On the Caffarelli-Kohn-Nirenberg inequalities, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 437–442 (English, with English and French summaries). MR 1756955, DOI 10.1016/S0764-4442(00)00201-9
- Florin Catrina and Zhi-Qiang Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), no. 2, 229–258. MR 1794994, DOI 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I
- F. Catrina and Z.-Q. Wang, Asymptotic Uniqueness and Exact Symmetry of $k$-bump Solutions for a Class of Degenerate Elliptic Problems, Discrete Contin. Dynam. Systems, Added Vol. (2001), 80-87.
- Jann-Long Chern and Chang-Shou Lin, The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations 187 (2003), no. 2, 240–268. MR 1949440, DOI 10.1016/S0022-0396(02)00080-3
- Kai Seng Chou and Chiu Wing Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993), no. 1, 137–151. MR 1223899, DOI 10.1112/jlms/s2-48.1.137
- V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differential Equations, 191 (2003), 121-142.
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
- Toshio Horiuchi, Best constant in weighted Sobolev inequality with weights being powers of distance from the origin, J. Inequal. Appl. 1 (1997), no. 3, 275–292. MR 1731336, DOI 10.1155/S1025583497000180
- Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
- Chang Shou Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations 11 (1986), no. 14, 1515–1538. MR 864416, DOI 10.1080/03605308608820473
- Chang-Shou Lin, Locating the peaks of solutions via the maximum principle. I. The Neumann problem, Comm. Pure Appl. Math. 54 (2001), no. 9, 1065–1095. MR 1835382, DOI 10.1002/cpa.1017
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations 18 (2003), 57-75.
- Z.-Q. Wang and M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203 (2003), 550-568.
- M. Willem, A decomposition lemma and critical minimization problems, preprint.
Additional Information
- Chang-Shou Lin
- Affiliation: Department of Mathematics, National Chung Cheng University, Chiayi, Taiwan
- MR Author ID: 201592
- Zhi-Qiang Wang
- Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322
- MR Author ID: 239651
- Received by editor(s): October 30, 2002
- Published electronically: January 16, 2004
- Communicated by: David S. Tartakoff
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1685-1691
- MSC (2000): Primary 35B33; Secondary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-04-07245-4
- MathSciNet review: 2051129