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

Abstract

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. We estimate the small eigenvalues of the linearized eigenvalue problem at u... when ∈ is small. As a consequence, we prove that the Morse index of u... is asymptotically given [μ*+o(1)]∈... 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.