Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation (Q2187874)

The authors are interested in the Neumann problem of a \(1\)D stationary Allen-Cahn equation with a nonlocal term \[-du_{xx}=(1-u^2)\Big(u-\frac {\mu}2\int^1_{-1} u\,dx\Big), \; x\in (-1,1),\] \[v_x(\pm1)=0, \;v_x\geq 0 \text{ for } x\in (-1,1).\] In their previous papers, they obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, they give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.