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Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems
A double saddle-node bifurcation theorem
1. | Y.Y. Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025, China, China |
2. | Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187 |
References:
[1] |
E. L. Allgower, K. Böhmer and M. Zhen, A complete bifurcation scenario for the $2$-d nonlinear Laplacian with Neumann boundary conditions on the unit square,, in, (1990), 1.
doi: 10.1007/978-3-0348-7004-7_1. |
[2] |
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", Springer-Verlag, (1982).
doi: 10.1007/978-1-4613-8159-4. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.
doi: 10.1007/BF00282325. |
[5] |
M. del Pino, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square,, Proc. Amer. Math. Soc., 131 (2003), 3499.
doi: 10.1090/S0002-9939-03-06906-5. |
[6] |
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, Comm. Pure Appl. Math., 32 (1979), 21.
doi: 10.1002/cpa.3160320103. |
[7] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs,", Applied Mathematical Sciences, 156 (2004).
|
[8] |
S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel,, J. Differential Equations, 220 (2006), 234.
doi: 10.1016/j.jde.2005.02.008. |
[9] |
P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution,, Comment. Math. Prace Mat., 45 (2005), 145.
|
[10] |
P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573.
doi: 10.1016/j.jfa.2007.06.015. |
[11] |
M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations,", Springer Series in Computational Mathematics, 28 (2000).
|
[12] |
P. Rabier, A generalization of the implicit function theorem for mappings from $R^{n+1}$ into $R^n$ and its applications,, J. Funct. Anal., 56 (1984), 145.
doi: 10.1016/0022-1236(84)90085-5. |
[13] |
J. P. Shi, Saddle solutions of the balanced bistable diffusion equation,, Comm. Pure Appl. Math., 55 (2002), 815.
doi: 10.1002/cpa.3027. |
[14] |
S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions,, J. Funct. Anal., 55 (1984), 247.
doi: 10.1016/0022-1236(84)90012-0. |
[15] |
C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem,, Math. Programming (Ser. A), 47 (1990), 117.
doi: 10.1007/BF01580856. |
[16] |
J. F. Wang, J. P. Shi and Y. W. Wang, Bifurcation from the second eigenvalue of a class of semilinear elliptic equations,, (Chinese) Natur. Sci. J. Harbin Normal Univ., 21 (2005), 1.
|
show all references
References:
[1] |
E. L. Allgower, K. Böhmer and M. Zhen, A complete bifurcation scenario for the $2$-d nonlinear Laplacian with Neumann boundary conditions on the unit square,, in, (1990), 1.
doi: 10.1007/978-3-0348-7004-7_1. |
[2] |
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", Springer-Verlag, (1982).
doi: 10.1007/978-1-4613-8159-4. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.
doi: 10.1007/BF00282325. |
[5] |
M. del Pino, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square,, Proc. Amer. Math. Soc., 131 (2003), 3499.
doi: 10.1090/S0002-9939-03-06906-5. |
[6] |
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory,, Comm. Pure Appl. Math., 32 (1979), 21.
doi: 10.1002/cpa.3160320103. |
[7] |
H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs,", Applied Mathematical Sciences, 156 (2004).
|
[8] |
S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel,, J. Differential Equations, 220 (2006), 234.
doi: 10.1016/j.jde.2005.02.008. |
[9] |
P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution,, Comment. Math. Prace Mat., 45 (2005), 145.
|
[10] |
P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573.
doi: 10.1016/j.jfa.2007.06.015. |
[11] |
M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations,", Springer Series in Computational Mathematics, 28 (2000).
|
[12] |
P. Rabier, A generalization of the implicit function theorem for mappings from $R^{n+1}$ into $R^n$ and its applications,, J. Funct. Anal., 56 (1984), 145.
doi: 10.1016/0022-1236(84)90085-5. |
[13] |
J. P. Shi, Saddle solutions of the balanced bistable diffusion equation,, Comm. Pure Appl. Math., 55 (2002), 815.
doi: 10.1002/cpa.3027. |
[14] |
S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions,, J. Funct. Anal., 55 (1984), 247.
doi: 10.1016/0022-1236(84)90012-0. |
[15] |
C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem,, Math. Programming (Ser. A), 47 (1990), 117.
doi: 10.1007/BF01580856. |
[16] |
J. F. Wang, J. P. Shi and Y. W. Wang, Bifurcation from the second eigenvalue of a class of semilinear elliptic equations,, (Chinese) Natur. Sci. J. Harbin Normal Univ., 21 (2005), 1.
|
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