We consider the boundary value problem \(\begin{aligned} \left\{ \begin{array}{l} u'' + \lambda h(x,\alpha ) e^u = 0, \quad x \in (-1,1), \\ u(-1) = u(1) = 0, \end{array} \right. \end{aligned}\) 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.
