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

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**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 $\mathbb {R}$.## References

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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
- 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
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2745649