Abstract

We consider the problem ε2 Δu + (u -a(x))(1 - u 2) = 0 in Ω, ∂u/∂v = 0 on ∂Ω, where Ω is a smooth and bounded domain in ℝ2, -1 < a(x) < 1. Assume that Γ = {x ∈ Ω, a(x) = 0} is a closed, smooth curve contained in Ω in such a way that Ω = Ω+ ∪ Γ ∪ Ω- and ∂a/∂ > 0 on Γ, where n is the outer normal to Ω+. Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103-131] proved the existence of an interior transition layer solution ue which approaches -1 in Ω- and +1 in Ω+, for all s sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in Ω- and -1 in Ω+ exists. In this paper, we answer this question affirmatively when n = 2, provided that e is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator. © 2007 Society for Industrial and Applied Mathematics.