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

TL;DR

This paper analyzes the Morse index of positive solutions to a one-dimensional Liouville type equation with a step function weight, revealing symmetry-breaking bifurcations and the existence of non-even solutions.

Contribution

It computes the Morse index of solutions and proves the existence of a connected set of non-even solutions emerging from a bifurcation point.

Findings

Morse index of positive even solutions is explicitly computed.

Existence of an unbounded connected set of non-even solutions is established.

Symmetry-breaking bifurcation point is identified.

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.