Geometry of phase space and solutions of semilinear elliptic equations in a ball

Transactions of the American Mathematical Society

Volumn 359, Issue 9, 2007, Pages 4073-4087

  Dolbeault, Jean ^a   Flores, Isabel ^b,c  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSIDAD DEL BÍO BÍO   (Chile)

^c UNIVERSITY OF CHILE   (Chile)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70349207467     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-07-04397-8     Document Type: Article

References (34)

1. F.V. Atkinson, H. Brezis, H., L.A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations 85 (1990), no. 1, 151-170. MR1052332 (91e:35021)
2. R. Bamon, I. Flores, M. del Pino, Ground states of semilinear elliptic equations: a geometric approach, Ann. Inst. H. Poincare Anal. Non Lineaire 17 (2000), 551-581. MR1791878 (2001k:35082)
3. G.R. Belickit, Equivalence and normal forms of germs of smooth mappings, Russ. Math. Surv. 33 (1978), 101-177. MR0490708 (80k:58017)
4. G.R. Belickit, Normal forms, invariants and local mappings, Naukova Dumka, Kiev, 176 (1979). MR0537763 (81h:58014)
5. R. Benguria, J. Dolbeault, M.J. Esteban, Classification of the solutions of semilinear elliptic problems in a ball, J. Differential Equations 167 (2000), no. 2, 438-466. MR1793200 (2001k:35083)
6. H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure and Appl. Math. 36 (1983) no. 4, 437-477. MR0709644 (84h:35059)
7. C. Budd, J. Norbury, Semilinear elliptic equations and supercritical growth, J. Differential Equations 68 (1987) no. 2, 169-197. MR0892022 (88i:35056)
8. C.B. Clemons, C.K.R.T. Jones, A geometric proof of the Kwong-McLeod uniqueness result, SIAM J. Math. Anal. 24 (1993), 436-443. MR1205535 (94f:35047)
9. G. Cerami, S. Solimini, M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), no. 3, 289-306. MR0867663 (88b:35074)
10. M. Del Pino, J. Dolbeault, M. Musso, "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differential Equations 193 (2003), no. 2, 280-306. MR1998635 (2004h:35077) (Pubitemid 37048986)
11. M. Del Pino, J. Dolbeault, M. Musso, A phase plane analysis of the "Multi-bubbling" phenomenon in some slightly supercritical equations, Monatsh. Math. 142 (2004), no. 1-2, 57-79. MR2065022 (2005h:35103) (Pubitemid 38808994)
12. M. Del Pino, J. Dolbeault, M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl. (9) 83 (2004), no. 12, 1405-1456. MR2103187 (2005h:35123) (Pubitemid 39498851)
13. J. Dolbeault, M. Esteban, M. Ramaswamy, Radial singular solutions of a critical problem in a ball, Differential Integral Equations 15 (2002), no. 12, 1459-1474. MR1920255 (2003f:35102)
14. L. Erbe, M. Tang, Uniqueness of positive radial solutions of AM+K(\X\)Y(M) = 0, Differential Integral Equations 11 (1998), no. 4, 663-678. MR1666214 (99j:35046)
15. L. Erbe, M. Tang, Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball, J. Differential Equations 138 (1997), no. 2, 351-379. MR1462272 (98k:35069) (Pubitemid 127341167)
16. R. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math. 2 (1931), 259-288.
17. B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the Maximum Principle, Commun. Math. Phys. 68 (1979), 209-243. MR0544879 (80h:35043)
18. M. Hirsch, C. Pugh, M. Schub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, New York (1977). MR0501173 (58:18595)
19. M.K. Kwong, Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. A.M.S. 333 (1992), 339-363. MR1088021 (92k:35102)
20. J. Jacobsen, K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations 184 (2002), 283-298. MR1929156 (2003g:34036)
21. R. Johnson, X.-B. Pan, Y. Yi, The Melnikov method and elliptic equations with critical exponent, Indiana Univ. Math. J. 43 (1994), 1045-1077. MR1305959 (95k:35068)
22. D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241-269. MR0340701 (49:5452)
23. F. Merle, L.A. Peletier, Positive solutions of elliptic equations involving supercritical growth, Proc. Roy. Soc. EdinburghSect. A 118 (1991), no. 1-2, 49-62. MR1113842 (92g:35074)
24. F. Merle, L.A. Peletier, J. Serrin, A bifurcation problem at a singular limit, Indiana Univ. Math. J. 43 (1994), no. 2, 585-609. MR1291530 (95j:35024)
25. W.-M. Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differential Equations 50 (1983), no. 2, 289-304. MR0719451 (85i:35060)
26. T. Ouyang, J. Shi, A bifurcation approach to the exact multiplicity of positive solutions of AM + f (M) = 0 on the unit ball, Dynamical systems and differential equations, Vol. II (Springfield, MO, 1996). MR1721186 (2000j:35105)
27. T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem. II, J. Differential Equations 158 (1999), no. 1, 94-151. MR1721723 (2001b:35117)
28. J. Palis, W. de Melo, Geometric theory of dynamical systems: an introduction, Springer-Verlag, New York, Heidelberg, Berlin (1982). MR0669541 (84a:58004)
29. S.I. Pohozaev, On the eigenfunctions of the equation Au + Xf(u) = 0, Dokl. Akad. Nauk SSSR 165 (1965), 36-39. MR0192184 (33:411)
30. J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), no. 3, 897-923. MR1803216 (2002d:35072)
31. J. Shi, Exact multiplicity of positive solutions to superlinear problems, Elec. Jour. Diff. Equa. Conf. 10 (2002), 257-265. MR1976648 (2004d:35079)
32. P.N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations 6 (1993), no. 3, 663-670. MR1202564 (94b:35122)
33. L.-Q. Zhang, Uniqueness of positive solutions to semilinear elliptic equations. (Chinese), Acta Math. Sci. (Chinese) 11 (1991), no. 2, 130-142. MR1129746 (92j:35055)
34. p = 0 in a ball, Comm. Partial Differential Equations 17 (1992), nos. 7-8, 1141-1164. MR1179281 (94b:35125)