Contrast structures in equations of elliptic type (Q1802575)

The authors consider a boundary value problem for the singularly perturbed elliptic equation with a cubic nonlinearity of the form \(\varepsilon u_{xx}+u_{yy}=u(u^ 2-a^ 2)\), \(a\)=const. This equation is shown to possess not only the standard solution with two boundary layers, but also a solution with periodic contrasting structures characterized by slow and rapid variations of the solution. The results obtained can be applied to the self-organization theory.