Remarks on elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities
HTML articles powered by AMS MathViewer
- by Gongbao Li and Shuangjie Peng
- Proc. Amer. Math. Soc. 136 (2008), 1221-1228
- DOI: https://doi.org/10.1090/S0002-9939-07-09229-5
- Published electronically: December 18, 2007
- PDF | Request permission
Abstract:
We give some regularity results of the solutions and a Liouville type theorem to singular elliptic equations involving the Caffarelli-Kohn- Nirenberg inequalities.References
- F. V. Atkinson, H. Brezis, and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations 85 (1990), no. 1, 151–170. MR 1052332, DOI 10.1016/0022-0396(90)90093-5
- F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349–365. MR 915493, DOI 10.1016/0022-0396(87)90156-2
- 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
- Daomin Cao and Shuangjie Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations 193 (2003), no. 2, 424–434. MR 1998962, DOI 10.1016/S0022-0396(03)00118-9
- Daomin Cao and Shuangjie Peng, Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth, Ann. Mat. Pura Appl. (4) 185 (2006), no. 2, 189–205. MR 2214132, DOI 10.1007/s10231-005-0150-z
- 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: 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
- Louis Dupaigne, A nonlinear elliptic PDE with the inverse square potential, J. Anal. Math. 86 (2002), 359–398. MR 1894489, DOI 10.1007/BF02786656
- Alberto Ferrero and Filippo Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001), no. 2, 494–522. MR 1876652, DOI 10.1006/jdeq.2000.3999
- Veronica Felli and Matthias Schneider, A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type, Adv. Nonlinear Stud. 3 (2003), no. 4, 431–443. MR 2017240, DOI 10.1515/ans-2003-0402
- 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 and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
- Enrico Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999), no. 2, 407–426. MR 1705383, DOI 10.1006/jdeq.1998.3589
- D. Kang, G. Li and S. Peng, Positive solutions and critical dimensions for the elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities, preprint.
- 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
Bibliographic Information
- Gongbao Li
- Affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China
- Shuangjie Peng
- Affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China
- MR Author ID: 635770
- Received by editor(s): August 2, 2006
- Published electronically: December 18, 2007
- Additional Notes: This work was partially supported by NSFC (10571069,10631030), the Key Project of the Chinese Ministry of Education (107081), and NCET-07-0350.
- Communicated by: David S. Tartakoff
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1221-1228
- MSC (2000): Primary 35J60, 35B33
- DOI: https://doi.org/10.1090/S0002-9939-07-09229-5
- MathSciNet review: 2367096