The heterogeneous Allen–Cahn equation in a ball: Solutions with layers and spikes

Let u be a single layered radially symmetric unstable solution of the Allen–Cahn equation − 2Δu=u(u−a(x))(1−u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u when is small, we construct solutions of the form u +v , with v non-radially symmetric and close to zero in the unit ball except near one point x0 such that x0 is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.