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Nondegeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domain


Author: Yasuhito Miyamoto
Journal: Proc. Amer. Math. Soc. 139 (2011), 975-984
MSC (2010): Primary 35B32, 35P15; Secondary 35J61, 35J15
DOI: https://doi.org/10.1090/S0002-9939-2010-10616-0
Published electronically: July 30, 2010
MathSciNet review: 2745649
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Abstract: Let $ \Omega$ be a planar domain such that $ \Omega$ is symmetric with respect to both the $ x$- and $ y$-axes and $ \Omega$ satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on $ \Omega$, $ \nu_2(\Omega)$, is simple, and the corresponding eigenfunction is odd with respect to the $ y$-axis. Let $ f\in C^3$ be a function such that

$\displaystyle f'(0)>0, f'''(0)<0, f(-u)=-f(u) \textrm{and}\ \frac{d}{du}\left(\frac{f(u)}{u}\right)<0 \textrm{for} u>0. $

Let $ \mathcal{C}$ denote the maximal continua consisting of nontrivial solutions, $ \{(\lambda,u)\}$, to

$\displaystyle \Delta u+\lambda f(u)=0 \ \textrm{in} \ \Omega,\qquad u=0 \ \textrm{on} \ \partial\Omega $

and emanating from the second eigenvalue $ (\nu_2(\Omega)/f'(0),0)$. We show that, for each $ (\lambda,u)\in\mathcal{C}$, the Morse index of $ u$ is one and zero is not an eigenvalue of the linearized problem. We show that $ \mathcal{C}$ consists of two unbounded curves, each curve is parametrized by $ \lambda$ and the closure $ \overline{\mathcal{C}}$ is homeomorphic to $ \mathbb{R}$.


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Additional Information

Yasuhito Miyamoto
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
Email: miyamoto@math.titech.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2010-10616-0
Keywords: Global branch, bifurcation, sign-changing solution, planar symmetric domain, second eigenvalue
Received by editor(s): March 23, 2010
Published electronically: July 30, 2010
Additional Notes: This work was partially supported by Grant-in-Aid for Young Scientists (B) (Subject No. 21740116).
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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