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

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 $\mathbb {R}$.