Multiple solutions for quasi-linear pdes involving the critical sobolev and hardy exponents

Transactions of the American Mathematical Society

Volumn 352, Issue 12, 2000, Pages 5703-5743

  Ghoussoub, N ^a   Yuan, C ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044523081     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-00-02560-5     Document Type: Article

References (31)

1. T. Aubin: Nonlinear Analysis on Manifolds. Mange-Ampere Equations, Springer-Verlag, 1982. MR 85j:58002
2. G. Bliss: An Integral Inequality, J. London Math. Soc., 5 (1930) 40-46.
3. L. Boccardo and F. Murât: Almost Everywhere Convergence of the Gradients of Solutions to Elliptic and Parabolic Equations, Nonlinear Analysis, TMA 19, (1992) 581-597. MR 93h:35061
4. H. Brezis, E. Lieb: A Relation Between Point Convergence of Functions and Convergence of Functional, Proc. Amer. Math. Soc. 88(1983), 486-490. MR 84c:28003
5. H. Brezis and L. Nirenberg: Positive Solutions of Nonlinear Elliptic Equations involving Critical Exponents, Communications on Pure and Applied Mathematics, 34 (1983), 437-477. MR 84h:35059
6. H. Brezis and L. Nirenberg: A Minimization Problem with Critical Exponent and Non-zero Data, in "Symmetry in Nature", Scuola Norm. Sup. Pisa (1989), 129-140.
7. L. Caffarelli, B. Gidas and J. Spruck: Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth, Comm. Pure Appl. Math. 42 (1989) 271-477. MR 90c:35075
8. L. Caffarelli, R. Kohn and L. Nirenberg: First Order Interpolation Inequality with Weights, Compositio Math. 53 (1984) 259-275. MR 86c:46028
9. G. Cerami, S. Solimini, and M. Struwe: Some Existence Results for Super-linear Elliptic Boundary Value Problems Involving Critical Exponents, J. Functional Analysis 69, 9 (1986) 289-306. MR 88b:35074
10. K. S. Chou and C. W. Chu: On the Best Constant for a Weighted Sobolev-Hardy Inequality, J. London Math. Soc. (2) 48 (1993) 137-151. MR 94h:46052
11. P. Drabek: Strongly Nonlinear Degenerated and Singular Elliptic Problems, Pitman Research Notes in Mathematics Series 343 (1996) 112-146. MR 97k:35091
12. P. Drabek and S. I. Pohozaev: Positive Solutions for the p-Laplacian: Application of the Fibering Method,. Proceeding of the Royal Society of Edinburgh, 127A (1997) 703-726. MR 98h:35061
13. J. P. Garcia Azorero and I. Peral Alonso: Hardy Inequalities and Some Critical Elliptic and Parabolic Problems, J. of Differential Equations 144 (1998) 441-476. MR 99f:35099
14. J. Garcia Azorero and I. Peral Alonso: Multiplicity of Solutions for Elliptic Problems with Critical Exponent or with a Non-symmetric Term, Trans. Amer. Math. Soc. 323 (1991), 877-895. MR 99g:35108
15. J. Garcia Azorero and I. Peral Alonso: Some Results about the Existence of a Second Positive Solution in a Quasi-linear Critical Problem, Indiana University Math. Journal 43 (1994), 941-957. MR 95i:35093
16. J. Garcia Azorero and I. Peral Alonso: Existence and Non-uniqueness for the p-Laplacian: Nonlinear Eigenvalues. Commun, in PDE, 12 (1987), 1389-1430. MR 89e:35058
17. N. Ghoussoub: Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press (1993). MR 95a:58021
18. D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag (1983). MR 86c:35035
19. M. Guedda, L. Veron: Local and Global Properties of Solutions of Quasi-linear Elliptic Equations, Journal of Differential Equations 76 (1988) 159-189. MR 89k:35078
20. M. Guedda, L. Veron: Quasi-linear Elliptic Equations Involving Critical Sobolev Exponents, Nonlinear Anal TMA, 13 (8) (1989) 879-902. MR 90h:35100
21. E. H. Lieb: Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Ann. of Math. 118 (1983) 349-374. MR 86i:42010
22. E. H. Lieb and M. Loss: Analysis, Graduate Studies in Mathematics Vol. 14 Amer. Math. Soc., Providence, RI, (1997). MR 98b:00004
23. M. Otani: Existence and Nonexistence of Nontrivial Solutions of Some Nonlinear Degenerated Elliptic Equations, Journal of Functional Analysis 76 (1988) 140-159. MR 89c:35067
24. E. S. Noussair, C. A. Swanson, J. Yang: Quasi-linear Elliptic Problems with Critical Exponents, Nonlin. Anal. 20 (1993) 285-301. MR 94b:35106
25. I. Peral: Multiplicity of Solutions for the p-Laplacian, Lecture Notes at the Second School on Nonlinear Functional Analysis and Applications to Differential Equations at ICTP of Trieste (April 21 - May 9, 1997).
26. M. Struwe: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag (1996). MR 98f:49002
27. G. Talenti: Best Constant in Sobolev Inequalities, Ann. Math. Pura Appl. 110 (1976), 353-372. MR 57:3846
28. G. Tarantello: Nodal Solutions of Semi-linear Elliptic Equations with Critical Exponent, Differential and Integral Equations, 5 (1992) 25-42. MR 92k:35109
29. P. Tolksdorf: Regularity for a More General Class of Quasi-linear Elliptic Equations, Journal of Differential Equations 51 (1984) 126-150. MR 85g:35047
30. J. Yang: Positive Solutions of Quasi-linear Elliptic Obstacle Problems with Critical Exponents, Nonlinear Analysis, TMA 25 (1995) 1283-1306. MR 96m:35120
31. C. Yuan: On Non-homogeneous Quasi-linear PDEs Involving the p-Laplacian and the Critical Sobolev Exponent, Ph.D. Dissertation, UBC 1998.