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November  2013, 12(6): 2923-2933. doi: 10.3934/cpaa.2013.12.2923

A double saddle-node bifurcation theorem

Ping Liu 1, , Junping Shi 2, and  Yuwen Wang 1,

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

Received  September 2010 Revised  July 2012 Published  May 2013

In this paper, we consider an abstract equation $F(\lambda,u)=0$ with one parameter $\lambda$, where $F\in C^p(\mathbb{R} \times X, Y)$, $p\geq 2$, is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a "double" saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation.
Keywords: two-mimensional kernel, Saddle-node bifurcation, Morse lemma..
Mathematics Subject Classification: Primary: 34C23, 35R15; Secondary: 47J1.
Citation: Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
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 "Bifurcation and Chaos: Analysis, Algorithms, Applications" (Würzburg, 1990), 1-18, Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991. doi: 10.1007/978-3-0348-7004-7_1.  Google Scholar

[2]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.  Google Scholar

[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), 3499-3505. doi: 10.1090/S0002-9939-03-06906-5.  Google Scholar

[6]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.  Google Scholar

[7]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.  Google Scholar

[8]

S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel, J. Differential Equations, 220 (2006), 234-258. doi: 10.1016/j.jde.2005.02.008.  Google Scholar

[9]

P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.  Google Scholar

[10]

P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[11]

M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000.  Google Scholar

[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), 145-170. doi: 10.1016/0022-1236(84)90085-5.  Google Scholar

[13]

J. P. Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math., 55 (2002), 815-830. doi: 10.1002/cpa.3027.  Google Scholar

[14]

S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions, J. Funct. Anal., 55 (1984), 247-275. doi: 10.1016/0022-1236(84)90012-0.  Google Scholar

[15]

C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem, Math. Programming (Ser. A), 47 (1990), 117-141. doi: 10.1007/BF01580856.  Google Scholar

[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), 1-4.  Google Scholar

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 "Bifurcation and Chaos: Analysis, Algorithms, Applications" (Würzburg, 1990), 1-18, Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991. doi: 10.1007/978-3-0348-7004-7_1.  Google Scholar

[2]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.  Google Scholar

[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), 3499-3505. doi: 10.1090/S0002-9939-03-06906-5.  Google Scholar

[6]

M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98. doi: 10.1002/cpa.3160320103.  Google Scholar

[7]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.  Google Scholar

[8]

S. Krömer, T. J. Healey and H. Kielhöfer, Bifurcation with a two-dimensional kernel, J. Differential Equations, 220 (2006), 234-258. doi: 10.1016/j.jde.2005.02.008.  Google Scholar

[9]

P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.  Google Scholar

[10]

P. Liu, J. P. Shi and Y. W. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[11]

M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000.  Google Scholar

[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), 145-170. doi: 10.1016/0022-1236(84)90085-5.  Google Scholar

[13]

J. P. Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math., 55 (2002), 815-830. doi: 10.1002/cpa.3027.  Google Scholar

[14]

S. D. Taliaferro, Bifurcation at multiple eigenvalues and stability of bifurcating solutions, J. Funct. Anal., 55 (1984), 247-275. doi: 10.1016/0022-1236(84)90012-0.  Google Scholar

[15]

C. A. Tiahrt and A. B. Poore, A bifurcation analysis of the nonlinear parametric programming problem, Math. Programming (Ser. A), 47 (1990), 117-141. doi: 10.1007/BF01580856.  Google Scholar

[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), 1-4.  Google Scholar

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