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

Let uϵ be a single layered radially symmetric unstable solution of the Allen–Cahn equation −ϵ2Δu=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 uϵ when ϵ is small, we construct solutions of the form uϵ+vϵ, with vϵ 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.