A double saddle-node bifurcation theorem

Ping Liu; Junping Shi; Yuwen Wang

doi:10.3934/cpaa.2013.12.2923

Article Contents

2013, Volume 12, Issue 6: 2923-2933. Doi: 10.3934/cpaa.2013.12.2923

This issue Previous Article Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems Next Article Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition

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
2.
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Received: September 2010

Revised: July 2012

Published: November 2013

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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.

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Mathematics Subject Classification: Primary: 34C23, 35R15; Secondary: 47J15.

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