Morse index and symmetry-breaking bifurcation of positive solutions to the one-dimensional Liouville type equation with a step function weight

\begin{equation*} \left{ \begin{array}{l} u’’ + λh(x,α) e^u = 0, \quad x \in (-1,1), \[1ex] u(-1) = u(1) = 0, \end{array} \right. \end{equation*} where $\lambda >0$, $0<\alpha <1$, $h(x,\alpha )=0$ for $\mathrm{\mid }x\mathrm{\mid }<\alpha$, and $h(x,\alpha )=1$ for $\alpha \le \mathrm{\mid }x\mathrm{\mid }\le 1$. We compute the Morse index of positive even solutions, and then we prove the existence of an unbounded connected set of positive non-even solutions emanating from a symmetry-breaking bifurcation point.