Singularity and decay estimates in superlinear problems via liouville-type theorems, I: Elliptic equations and systems

Duke Mathematical Journal

Volumn 139, Issue 3, 2007, Pages 555-579

  Poláčik, Peter ^a   Quittner, Pavol ^b   Souplet, Philippe ^c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b Physics and Informatics   (Slovakia)

^c UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34748907866     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-07-13935-8     Document Type: Article

References (38)

1. M.-F. BIDAUT-VÉRON, Local and global behavior of solutions of quasilinear elliptic equations of Emden-Fowler type, Arch. Rational Mech. Anal. 107 (1989), 293-324. MR 1004713
2. _, "Initial blow-up for the solutions of a semilinear parabolic equation with source term" in Équations aux dérivé es partielles et applications, Gauthier-Villars, Paris, 1998, 189-198. MR 1648222
3. _, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations 5 (2000), 147-192. MR 1734540
4. M.-F. BIDAUT-VÉRON and T. RAOUX, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations 21 (1996), 1035-1086. MR 1399190
5. M.-F. BIDAUT-VÉRON and L. VÉRON, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489-539;
6. Erratum, Invent. Math. 112 (1993), 445. MR 1134481; MR 1213110
7. I. BIRINDELLI and E. MITIDIERI, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A128 (1998), 1217-1247. MR 1664101
8. J. BUSCA and R. MANÁSEVICH, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J. 51 (2002), 37-51. MR 1896155
9. L. A. 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), 271-297. MR 0982351
10. W. X. CHEN and C. LI, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622. MR 1121147
11. E. N. DANCER, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), 425-434. MR 1190345
12. _, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z. 229 (1998), 475-491. MR 1658565
13. D. G. DE FIGUEIREDO, "Semilinear elliptic systems" in Nonlinear Functional Analysis and Applications to Differential Equations (Trieste, Italy, 1997), World Sci., River Edge, N.J., 1998, 122-152. MR 1703530
14. D. G. DE FIGUEIREDO and P. L. FELMER, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 387-397. MR 1310633
15. D. G. DE FIGUEIREDO and B. SIRAKOV, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann. 333 (2005), 231-260. MR 2195114
16. D. G. DE FIGUEIREDO and J. YANG, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305-2321. MR 1876419
17. B. GIDAS and J. SPRUCK, A priori bounds for positive solutions of a nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883-901. MR 0619749
18. _, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598. MR 0615628
19. B. HU, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differential Integral Equations 9 (1996), 891-901. MR 1392086
20. J. B. KELLER, On solutions of Δu = f(u), Comm. Pure Appl. Math. 10 (1957), 503-510. MR 0091407
21. Y. LI and L. ZHANG, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27-87. MR 2001065
22. P.-L. LIONS, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), 441-550. MR 0605060
23. E. MITIDIERI, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125-151. MR 1211727
24. N, Differential Integral Equations 9 (1996), 465-479. MR 1371702
25. E. MITIDIERI and S. I. POKHOZHAEV, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities (in Russian), Tr. Mat. Inst. Steklova 234 (2001), 1-384;
26. English translation in Proc. Steklov Inst. Math. 234, no. 3 (2001), 1-362. MR 1879326
27. R. OSSERMAN, On the inequality Δu ≥ f(u), Pacific J. Math. 7 (1957), 1641-1647. MR 0098239
28. P. POLÁČIK and P. QUITTNER, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Anal. 64 (2006), 1679-1689. MR 2197355
29. P. POLÁČIK, P. QUITTNER, and PH. SOUPLET, Singularity and decay estimates in superlinear problems via Liouville-type theorems, II. Parabolic equations, Indiana Univ. Math. J. 56 (2007), 879-908. MR 2317549
30. W. REICHEL and H. ZOU, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000), 219-243. MR 1740363
31. J. SERRIN, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302. MR 0170096
32. J. SERRIN and H. ZOU, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations 9 (1996), 635-653. MR 1401429
33. _, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), suppl., 369-380. MR 1645728
34. _, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79-142. MR 1946918
35. λ Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 889-901. MR 1939089
36. P. TOLKSDORF, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. MR 0727034
37. H. ZOU, A priori estimates for a semilinear elliptic systems without variational structure and their applications, Math. Ann. 323 (2002), 713-735. MR 1924277
38. _, A priori estimates and existence for strongly coupled semilinear cooperative elliptic systems, Comm. Partial Differential Equations 31 (2006), 735-773. MR 2233039