Variational problems with singular perturbation

In this paper, we construct the local minimum of a certain variational problem which we take in the form inf∫Ω{ϵ2kg2|∇w|2+14ϵf2g4(1−w2)2}dx, where ϵϵ is a small positive parameter and Ω⊂RnΩ⊂Rn is a convex bounded domain with smooth boundary. Here f,g,k∈C3(Ω)f,g,k∈C3(Ω) are strictly positive functions in the closure of the domain Ω̄. If we take the inf over all functions H1(Ω)H1(Ω), we obtain the (unique) positive solution of the partial differential equation with Neumann boundary conditions (respectively Dirichlet boundary conditions).We wish to restrict the inf to the local (not global) minimum so that we consider solutions of this Neumann problem which take both signs in ΩΩ and which vanish on (n−1)(n−1) dimensional hypersurfaces Γϵ⊂ΩΓϵ⊂Ω. By using a ΓΓ-convergence method, we find the structure of the limit solutions as ϵ→0ϵ→0 in terms of the weighted geodesics of the domain ΩΩ.