Symmetry-breaking bifurcation for the Moore-Nehari differential equation (Q1729788)

In this paper, the authors deal with Moore-Nehari type bifurcation problems second order nonlinear boundary value problem of the type \[ \begin{aligned} &u^{''}+h(x,\lambda)u^{p}=0,\quad u>0,\, x\in(-1,1),\, u(-1)=u(1)=0,\\ &p>1,\quad\lambda\in(0,1),\\ &h(x,\lambda)=0\text{ for } |x|<\lambda, h(x,\lambda)=1\text{ for }\lambda\leq|x|\leq 1. \end{aligned} \] Based on the Schauder fixed point theorem a unique even positive solution is characterized. Next, relying on the Morse index of the obtained unique positive solution (denoted by $U(x,\lambda)$ in the paper) and a bifurcation criterion due to the Rabinowitz and Schmitt, is proven that a non-even positive solution bifurcates from the even positive solution $U(x,\lambda)$ at $\lambda_* (p)$.