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Communications on Pure and Applied Analysis (CPAA)
 

A double saddle-node bifurcation theorem

Pages: 2923 - 2933, Volume 12, Issue 6, November 2013      doi:10.3934/cpaa.2013.12.2923

 
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Ping Liu - Y.Y. Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025, China (email)
Junping Shi - Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, United States (email)
Yuwen Wang - Y.Y.Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025, China (email)

Abstract: 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:  Saddle-node bifurcation, two-mimensional kernel, Morse lemma.
Mathematics Subject Classification:  Primary: 34C23, 35R15; Secondary: 47J15.

Received: September 2010;      Revised: July 2012;      Available Online: May 2013.

 References