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

Let u(epsilon) be a single layered radially symmetric unstable solution of the Allen-Cahn equation -epsilon(2 Delta)u = u(u-a(vertical bar x vertical bar))(1-u) over the unit ball with Neumann boundary conditions. We estimate the small eigenvalues of the linearized eigenvalue problem at u(epsilon) when epsilon is small. As a consequence, we prove that the Morse index of u(epsilon) is asymptotically given by [mu* +o(1)]epsilon(-(N-1)/2) with mu* 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. (C) 2007 Elsevier Inc. All rights reserved.