A double saddle-node bifurcation theorem

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.
[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.
[9]	P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.
[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.
[11]	M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 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), 145-170. 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), 815-830. doi: 10.1002/cpa.3027.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.
[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.
[9]	P. Liu and Y. W. Wang, The generalized saddle-node bifurcation of degenerate solution, Comment. Math. Prace Mat., 45 (2005), 145-150.
[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.
[11]	M. Zhen, "Numerical Bifurcation Analysis for Reaction-diffusion Equations," Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 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), 145-170. 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), 815-830. doi: 10.1002/cpa.3027.
[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.
[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.
[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.

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