Secondary bifurcations in semilinear ordinary differential equations

We consider the Neumann problem for the equation $u_{xx}+\lambda f(u)=0$ in the punctured interval $(-1,1) \setminus \{0\}$, where $\lambda>0$ is a bifurcation parameter and $f(u)=u-u^3$. At $x=0$, we impose the conditions $u(-0)+au_x(-0)=u(+0)-au_x(+0)$ and $u_x(-0)=u_x(+0)$ for a constant $a>0$ (the symbols $+0$ and $-0$ stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval $(-1,1)$. First we prove that odd solutions and even solutions form families of branches $\{ \mathcal{C}^o_k\}_{k \in \mathbb{N}}$ and $\{ \mathcal{C}^e_k\}_{k \in \mathbb{N}}$, respectively. Both $\mathcal{C}^o_k$ and $\mathcal{C}^e_k$ bifurcate from the trivial solution $u=0$. We then show that $\mathcal{C}^e_k$ contains no other bifurcation point, while $\mathcal{C}^o_k$ contains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions on $f(u)$ for the same assertions to hold are also given.