# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Nondegeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domainHTML articles powered by AMS MathViewer

by Yasuhito Miyamoto
Proc. Amer. Math. Soc. 139 (2011), 975-984 Request permission

## 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 $f’(0)>0,\ f''’(0)<0,\ f(-u)=-f(u)\ \textrm {and}\ \frac {d}{du}\left (\frac {f(u)}{u}\right )<0\ \textrm {for}\ u>0.$ Let $\mathcal {C}$ denote the maximal continua consisting of nontrivial solutions, $\{(\lambda ,u)\}$, to $\Delta u+\lambda f(u)=0\ \ \textrm {in}\ \ \Omega ,\qquad u=0\ \ \textrm {on}\ \ \partial \Omega$ 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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