Stable and finite morse index solutions on Rn or on bounded domains with small diffusion

Transactions of the American Mathematical Society

Volumn 357, Issue 3, 2005, Pages 1225-1243

  Dancer, E N ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 14844289582     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-04-03543-3     Document Type: Article

References (44)

1. G. Alberti, L. Ambrosio and X. Cabré, "On a long-standing conjecture of E. De Giorgi : symmetry in 3D for general nonlinearities and a local minimality property", Acta Applicandae Math. 65 (2001), 9-33. MR2002f:35080
2. 3 and a conjecture of De Giorgi", J. Amer. Math Soc 13 (2000), 725-739. MR2001g:35064
3. A. Bahri and P. Lions, "Solutions of superlinear elliptic problems and their indices", Comm Pure Appl Math 45 (1992), 1205-1215. MR93m:35077
4. H. Berestycki, L. Caffarelli and L. Nirenberg, "Further qualitative properties for elliptic equations in unbounded domains", Ann Scuola Norm Sup Pisa 25 (1997), 69-94. MR2000e:35053
5. H. Berestycki, L. Caffarelli and L. Nirenberg, "Inequalities for second order elliptic equations with applications to unbounded domains", Duke Math J. 81 (1996), 467-494. MR97h:35054
6. R. Casten and C. Holland, "Instability results for reaction diffusion equations with Neumann boundary conditions", J. Diff Eqns 27 (1978), 266-273. MR80a:35064
7. P. Clement and G. Sweers, "Existence and multiplicity results for a semilinear elliptic eigenvalue problem", Ann Scuola Norm Sup Pisa 14 (1987), 97-121. MR89j:35053
8. n", Ann Scuola Norm Sup Pisa 30 (2001), 535-563. MR2003g:35057
9. E.N. Dancer, "On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large", Proc London Math Soc 53 (1986), 429-452. MR88c:35061
10. E.N. Dancer, "The effect of domain shape on the number of positive solutions of certain nonlinear equations", J. Diff Eqns 74 (1988), 120-156. MR89h:35256
11. E.N. Dancer, "On positive solutions of some singularly perturbed problems where the non-linearity changes sign", Top Methods Nonlinear Anal 5 (1995), 141-175. MR96i:35037
12. E.N. Dancer, "A note on the method of moving planes", Bull Austral Math Soc 46 (1992), 425-434. MR93m:35080
13. E.N. Dancer, "Multiple fixed points of positive maps", J. Reine Ang Math 371 (1986), 46-66. MR88b:58020
14. E.N. Dancer, "Infinitely many turning points for some supercritical problems", Ann. Mat Pura Appl (4) 178 (2000), 225-233. MR2002g:35077
15. E.N. Dancer, "Superlinear problems on domains with holes of asymptotic shape and exterior problems", Math Zeit 229 (1998), 475-491. MR2000c:35058
16. E.N. Dancer and Z.M. Guo, "Some remarks on the stability of sign changing solutions", Tohoku Math J. 47 (1995), 199-225. MR97c:35091
17. E.N. Dancer and P. Hess, "Stability of fixed points for order preserving discrete time dynamical systems", J. Reine Ang. Math 419 (1991), 125-139. MR92i:47088
18. E.N. Dancer and J. Wei, "On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem", Proc Royal Soc Edinburgh A 127 (1997), 691-701. MR98i:35012
19. E.N. Dancer and J. Wei, "On the location of spikes of solutions with two sharp layers for a similarly perturbed semilinear Dirichlet problem", J. Diff Eqns 157 (1999), 82-101. MR2000j:35017
20. E.N. Dancer and S. Yan, "On the profile of the sign changing mountain pass solutions for an elliptic problem", Trans Amer Math Soc. 354 (2002), 3573-3600. MR2003d:35082
21. E.N. Dancer and S. Yan, "Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem", Top. Methods in Nonlinear Analysis 14 (1999), 1-38. MR2001b:35106
22. E.N. Dancer and S. Yan, "Interior and boundary peak solutions for a mixed boundary value problem", Indiana Math J. 48 (1999) 1177-1212. MR2001f:35146
23. E.N. Dancer and K. Schmitt, "On positive solutions of semilinear elliptic problems", Proc Amer Math Soc 101 (1987), 445-452. MR89a:35017
24. M. Del Pino and P. Felmer, "Spike layered solutions of singularly perturbed elliptic problems", Indiana Math J. 48 (1999), 883-898. MR2001b:35027
25. N. Ghoussoub and G. Gui, "On a conjecture of De Giorgi and some related problems", Math Ann 311 (1998), 481-491. MR99j:35049
26. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, Berlin, 1977. MR57:13109
27. C. Gui and J. Wei, "Multiple interior peaks for some singularly perturbed Neumann problems", J. Diff Eqns 158 (1999), 1-27. MR2000g:35035
28. P. Hess, "On multiple solutions of nonlinear elliptic eigenvalue problems", Comm Partial Diff Eqns 6 (1981), 951-961. MR82j:35062
29. H. Hofer, "A note on the topological degree at a critical point of mountain pass type", Proc Amer Math Soc 90 (1984), 309-315. MR85a:58015
30. J. Jang, "On spike solutions of singularly perturbed semilinear Dirichlet problems", J. Diff Eqns 114 (1994), 370-395. MR95i:35099
31. R. Kohn and P. Sternberg, "Local minimizers and singular perturbations", Proc Royal Soc Edinburgh A 111 (1989), 69-84. MR90c:49021
32. n", Comm Partial Diff Eqns 18 (1993), 1043-1054. MR95c:35026
33. W.M. Ni, I. Takagi and J. Wei, "On the location and profile of spike layer solutions to a singularly perturbed semilinear Dirichlet problem : intermediate solutions", Duke Math J. 94 (1998), 597-618. MR99h:35011
34. W.M. Ni and J. Wei, "On the location and profile of spike layer solutions to singularly perturbed semilinear Dirichlet problems", Comm Pure App Math 48 (1995), 731-768. MR96g:35077
35. G. Sweers, "On the maximum of solutions for a semilinear elliptic problem", Proc Royal Soc Edinburgh A 108 (1988), 357-370. MR89h:35122
36. J. Wei, "On the construction of single peaked solutions to a singularly perturbed semilinear Dirichlet problem", J. Diff Eqns 129 (1996), 315-333. MR97f:35015
37. J. Wei, "On the interior spike solutions for some singular perturbation problems", Proc Royal Soc Edinburgh A 128 (1998), 849-874. MR99h:35012
38. J. Wei, "On the effect of domain geometry in singular perturbation problems", Diff Integral Eqns 13 (2000), 15-45. MR2002c:35032
39. S. Yan, "On the number of interior multipeak solutions for singularly perturbed Neumann problems", Top Methods Nonlinear Anal 12 (1999), 61-78. MR2001c:35024
40. E.N. Dancer, Yihong Du, "Some remarks on Liouville type results for quasilinear elliptic equations", Proc Amer Math Soc 131 (2003), 1891-1899.
41. Ha Dang, P. Fife, L. Peletier, "Saddle solutions of the bistable diffusion equation", Z. Angew Math Phys 43 (1992), 984-998. MR94b:35041
42. H. Matano, "Asymptotic behaviour and stability of solutions of semilinear diffusion equations", Publ Res Int Math Sci 15 (1977), 401-454. MR80m:35046
43. M. Schatzman, "On the stability of the saddle solution of Allen Cahn's equation", Proc Royal Soc Edinburgh A125 (1995), 1241-1275. MR96j:35085
44. Junping Shi, Saddle solutions of the balanced bistable diffusion equation", Comm Pure Appl Math 55 (2002), 815-830. MR2003b:35068