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

Abstract

\begin{equation*} \left\{ \begin{array}{l} u'' + \lambda h(x,\alpha) 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 $|x|<\alpha$, and $h(x,\alpha)=1$ for $\alpha \le |x| \le 1$. 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.