Morse index of layered solutions to the heterogeneous Allen–Cahn equation

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. We estimate the small eigenvalues of the linearized eigenvalue problem at uϵ when ϵ is small. As a consequence, we prove that the Morse index of uϵ is asymptotically given by [μ∗+o(1)]ϵ−(N−1)/2 with μ∗ a certain positive constant expressed in terms of parameters determined by the Allen–Cahn equation. Our estimates on the small eigenvalues have many other applications. For example, they may be used in the search of other non-radially symmetric solutions, which will be considered in forthcoming papers.