# American Institute of Mathematical Sciences

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

## 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

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.
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

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##### 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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