Chapter 3 Bifurcation and related topics in elliptic problems

Handbook of Differential Equations: Stationary Partial Differential Equations

Volumn 2, Issue , 2005, Pages 127-209

  Du, Yihong ^a  

^a UNIVERSITY OF NEW ENGLAND   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33746682335     PISSN: 18745733     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5733(05)80011-7     Document Type: Chapter

References (101)

1. Ali S.W., and Cosner C. Models for the effects of individual size and spatial scale on competition between species in heterogeneous environments. Math. Biosci. 127 (1995) 45-76
2. Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976) 620-709
3. Aris R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts (1975), Oxford University Press
4. Bebernes J., and Eberly D. Mathematical Problems from Combustion Theory (1989), Springer-Verlag, London
5. Blat J., and Brown K.J. Bifurcation of steady-state solutions in predator-prey and competition systems. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 21-34
6. Blat J., and Brown K.J. Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal. 17 (1986) 1339-1353
7. Bony J.M. Principe du maximum dans les espaces de Sobolev. C. R. Math. Acad. Sci. Paris Sér. A 265 (1967) 333-336
8. Brown K.J., Ibrahim M.M.A., and Shivaji R. S-shaped bifurcation curves. Nonlinear Anal. 5 (1981) 475-486
9. Cantrell R.S., and Cosner C. Should a park be an island?. SIAM J. Appl. Math. 53 (1993) 219-252
10. Cantrell R.S., Cosner C., and Hutson V. Ecological models, permanence and spatial heterogeneity. Rocky Mountain J. Math. 26 (1996) 1-35
11. Castro A., and Shivaji R. Uniqueness of positive solutions for a class of elliptic boundary value problems. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984) 267-269
12. Conway E., Hoff D., and Smoller J. Large time behaviour of solutions of systems of nonlinear reaction- diffusion equations. SIAM J. Appl. Math. 35 (1978) 1-16
13. Crandall M.G., and Rabinowitz P.H. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971) 321-340
14. Crandall M.G., and Rabinowitz PH. Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Ration. Mech. Anal. 52 (1973) 161-180
15. Dancer E.N. On the structure of solutions of an equation in catalysis theory when a parameter is large. J. Differential Equations 37 (1980) 404-437
16. Dancer E.N. On the indices of fixed points of mappings in cones and applications. J. Math. Anal. Appl. 91 (1983) 131-151
17. Dancer E.N. On positive solutions of some pairs of differential equations. Trans. Amer. Math. Soc. 284 (1984) 729-743
18. Dancer E.N. On positive solutions of some pairs of differential equations, II. J. Differential Equations 60 (1985) 236-258
19. Dancer E.N. The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Differential Equations 74 (1988) 120-156
20. Dancer E.N. On the existence and uniqueness of positive solutions for competing species models with diffusion. Trans. Amer. Math. Soc. 326 (1991) 829-859
21. Dancer E.N. Upper and lower stability and index theory for positive mappings and applications. Non- linear Anal. 17 (1991) 205-217
22. Dancer E.N. Some remarks on classical problems and fine properties of Sobolev spaces. Differential Integral Equations 9 (1996) 437-446
23. Dancer E.N., and Du Y. Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differential Equations 114 (1994) 434-475
24. Dancer E.N., and Du Y. Effects of certain degeneracies in the predator-prey model. SIAM J. Math. Anal. 34 (2002) 292-314
25. Dancer E.N., Lopez-Gomez J., and Ortega R. On the spectrum of some linear noncooperative elliptic systems with radial symmetry. Differential Integral Equations 8 (1994) 515-523
26. del Pino M. Positive solutions of a semilinear elliptic equation on a compact manifold. Nonlinear Anal. 22 (1994) 1423-1430
27. Deimling K. Nonlinear Functional Analysis (1985), Springer-Verlag, New York
28. de Mottoni P., and Rothe F. Convergence to homogeneous equilibrium state for generalized Volterra- Lotka systems. SIAM J. Appl. Math. 37 (1979) 648-663
29. Du Y. The structure of the solution set of a class of nonlinear eigenvalue problems. J. Math. Anal. Appl. 170 (1992) 567-580
30. Du Y. Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory. SIAM J. Math. Anal. 32 (2000) 707-733
31. Du Y. Effects of a degeneracy in the competition model, Part I and Part II. J. Differential Equations 181 (2002) 92-164
32. Du Y. Realization of prescribed patterns in the competition model. J. Differential Equations 193 (2003) 147-179
33. Du Y. Spatial patterns for population models in a heterogeneous environment. Taiwanese J. Math. 8 (2004) 155-182
34. Du Y. Asymptotic behavior and uniqueness results for boundary blow-up solutions. Differential Integral Equations 17 (2004) 819-834
35. Du Y., and Brown K.J. Bifurcation and monotonicity in competition reaction-diffusion systems. Non- linear Anal. 23 (1994) 1-13
36. Y. Du and Z.M. Guo The degenerate logistic model and a singularly mixed boundary blow-up problem, Discrete Contin. Dyn. Syst., to appear.
37. Preprint. Du Y., and Hsu S.B. A diffusive predator-prey model in heterogeneous environment (2003)
38. Du Y., and Huang Q. Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J. Math. Anal. 31 (1999) 1-18
39. Du Y., and Li S.J. Positive solutions with prescribed patterns in some simple semilinear equations. Differential Integral Equations 15 (2002) 805-822
40. Du Y., and Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 (1998) 390-440
41. Du Y., and Lou Y. Proof of a conjecture for the perturbed Gelfand equation from combustion theory. J. Differential Equations 173 (2001) 213-230
42. N by a squeezing method involving boundary blow-up solutions. J. London Math. Soc. 64 (2001) 107-124
43. Du Y., and Ouyang T. Bifurcation from infinity induced by a degeneracy in semilinear equations. Adv. Nonlinear Stud. 2 (2002) 117-132
44. Fraile J.M., Koch-Medina P., Lopez-Gomez J., and Merino S. Elliptic eigenvalue problems and un- bounded continua of positive solutions of a semilinear equation. J. Differential Equations 127 (1996) 295-319
45. Garcia-Melian J., Gomez-Renasco R., Lopez-Gomez J., and Sabina de Lis J.C. Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs. Arch. Ration. Mech. Anal. 145 (1998) 261-289
46. Gidas B., Ni W.-M., and Nirenberg L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979) 209-243
47. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equation of Second Order (2001), Springer-Verlag, Berlin
48. Hastings S.P., and McLeod J.B. The number of solutions to an equation from catalysis. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985) 15-30
49. Healey T.J., and Kielhofer H. Separation of global solution branches of elliptic systems with symmetry via nodal properties. Nonlinear Anal. 21 (1993) 665-684
50. Hess P. Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Math. (1991), Longman, Berlin
51. Hirsch M. Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383 (1988) 1-53
52. Hsu S.B. Steady states of a system of partial differential equations modeling microbial ecology. SIAM J. Math. Anal. 14 (1983) 1130-1138
53. Hsu S.B., Smith H.L., and Waltman P. Competitive exclusion and coexistence for competitive systems on ordered Banach spaces. Trans. Amer. Math. Soc. 348 (1996) 4083-4094
54. Hutson V., Lou Y., and Mischaikow K. Spatial heterogeneity of resources versus Lotka-Volterra dynamics. J. Differential Equations 185 (2002) 97-136
55. Hutson V., Lou Y., Mischaikow K., and Polacik P. Competing species near a degenerate limit. SIAM J. Math. Anal. 35 (2003) 453-491
56. Hutson V., Mischaikow K., and Polacik P. The evolution of dispersal rates in a heterogeneous time- periodic environment. J. Math. Biol. 43 (2001) 501-533
57. Joseph D.D., and Lundgren T.S. Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49 (1973) 241-269
58. Kan-on Y. Bifurcation structure of stationary solutions of a Lotka-Volterra competition model with diffusion. SIAM J. Math. Anal. 29 (1998) 424-436
59. Kan-on Y., and Yanagida E. Existence of nonconstant stable equilibria in competition diffusion equations. Hiroshima Math. J. 23 (1993) 193-221
60. Kato T. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972) 133-148
61. Kielhofer H. Bifurcation Theory. An Introduction with Applications to PDEs (2004), Springer-Verlag, Harlow
62. Kishimoto K., and Weinberger H.F. The spatial homogeneity of stable equilibria of some reaction- diffusion systems on convex domains. J. Differential Equations 58 (1985) 15-21
63. Koman P., and Leung A.W. A general monotone scheme for elliptic systems with applications to ecological models. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986) 315-325
64. Korman P., and Li Y. On the exactness of an S-shaped bifurcation curve. Proc. Amer. Math. Soc. 127 (1999) 1011-1020
65. Krasnoselski M.A. Topological Methods in the Theory of Nonlinear Integral Equations (1965), Macmillan, New York
66. Lin C.S., and Ni W.-M. A counterexample to the nodal domain conjecture and a related semilinear equation. Proc. Amer. Math. Soc. 102 (1988) 271-277
67. Lions P.L. A remark on Bony's maximum principle. Proc. Amer. Math. Soc. 88 (1983) 503-508
68. Lopez-Gomez J. On the structure of the permanence region for competing species models with general dijfusivities and transport effects. Discrete Contin. Dyn. Syst. 2 (1996) 525-542
69. Lopez-Gomez J. Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, Electron. J. Differ. Equ. Conf. 5 (2000) 135-171
70. Lopez-Gomez J., and Pardo R.M. Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case. Differential Integral Equations 6 (1993) 1025-1031
71. Lopez-Gomez J., and Pardo R.M. Invertibility of linear noncooperative elliptic systems. Nonlinear Anal. 31 (1998) 687-699
72. Lopez-Gomez J., and Sabina de Lis J. Coexistence states and global attractivity for some convective diffusive competing species models. Trans. Amer. Math. Soc. 347 (1995) 3797-3833
73. Lou Y., and Ni W.-M. Diffusion, self-diffusion and cross diffusion. J. Differential Equations 131 (1996) 79-131
74. Lou Y., and Ni W.-M. Diffusion vs cross-diffusion: An elliptic approach. J. Differential Equations 154 (1999) 157-190
75. Marcus M., and Veron L. Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Lineaire 14 (1997) 237-274
76. Matano H. Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984) 645-673
77. Matano H., and Mimura M. Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. 19 (1983) 1049-1079
78. Mimura M. Stationary patterns of some density-dependent diffusion systems with competitive dynam- ics. Hiroshima Math. J. 11 (1981) 621-635
79. Mimura M., and Kawasaki K. Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9 (1980) 49-64
80. Mimura M., and Sakamoto K. Multi-dimensional transition layers for an exothermic reaction-diffusion system in long cylindrical domains. J. Math. Sci. Univ. Tokyo 3 (1996) 109-179
81. u on circular domains. Math. Ann. 299 (1994) 1-15
82. Nakashima K. Multiple existence of spatially inhomogeneous steady-states for competition diffusion systems. Adv. Math. Sci. Appl. 2 (1999) 973-991
83. Ni W.-M., and Nussbaum R.D. Uniqueness and nonuniqueness for positive radial solutions of GDu+ f (u, r) = 0. Comm. Pure Appl. Math. 38 (1985) 67-108
84. Ouyang T. On the positive solutions of the semilinear equation u+λu+hup =0 on compact manifolds. Trans. Amer. Math. Soc. 331 (1992) 503-527
85. Ouyang T., and Shi J. Exact multiplicity of positive solutions for a class of semilinear problems. J. Dif- ferential Equations 146 (1998) 121-156
86. Ouyang T., and Shi J. Exact multiplicity of positive solutions for a class of semilinear problems, II. J. Differential Equations 158 (1999) 94-151
87. Pao C.V. On nonlinear reaction-diffusion systems. J. Math. Anal. Appl. 87 (1982) 165-198
88. Parter S.V. Solutions of a differential equation arising in chemical reaction processes. SIAM J. Appl. Math. 2 (1974) 687-716
89. Protter M.H., and Weinberger H.F. Maximum Principles in Differential Equations (1967), Prentice-Hall, New York
90. Rabinowitz P.H. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971) 487-513
91. Sattinger D.H. Topics in Stability and Bifurcation Theory. Lecture Notes in Math. (1972), Springer- Verlag, Englewood Cliffs
92. Shi J. Persistence and bifurcation of degenerate solutions. J. Funct. Anal. 169 (1999) 494-531
93. Shivaji R. Remarks on an S-shaped bifurcation curve. J. Math. Anal. Appl. 111 (1985) 374-387
94. RI. Smith H.L. Monotone Dynamical Systems. Math. Surveys and Monographs. Amer. Math. Soc. Providence (1995)
95. Taira K. Semilinear elliptic boundary-value problems in combustion theory. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 1453-1476
96. Tang M. Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 705-717
97. Wang S.-H. On S-shaped bifurcation curves. Nonlinear Anal. 22 (1994) 1475-1485
98. Wang S.-H. Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws. Proc. Roy. Soc. London Ser. A 454 (1998) 1031-1048
99. Wiebers H. S-shaped bifurcation curves of nonlinear elliptic boundary value problems. Math. Ann. 270 (1985) 555-570
100. Wiebers H. Critical behavior of nonlinear elliptic boundary value problems suggested by exothermic reactions. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986) 19-36
101. Yamada Y. Stability of steady-states for prey-predator diffusion equations with homogeneous Dirichlet conditions. SIAM J. Math. Anal. 21 (1990) 327-345