Solutions with transition layer and spike in an inhomogeneous phase transition model

Abstract

We consider the following singularly perturbed elliptic problem ε 2 Δ u ˜ + ( u ˜ − a ( y ˜ ) ) ( 1 − u ˜ 2 ) = 0 in Ω , ∂ u ˜ ∂ n = 0 on ∂ Ω , where Ω is a bounded domain in R 2 with smooth boundary, − 1 < a ( y ˜ ) < 1 , ε is a small parameter, n denotes the outward normal of ∂ Ω. Assume that Γ = { y ˜ ∈ Ω : a ( y ˜ ) = 0 } is a simple closed and smooth curve contained in Ω in such a way that Γ separates Ω into two disjoint components Ω + = { y ˜ ∈ Ω : a ( y ˜ ) > 0 } and Ω − = { y ˜ ∈ Ω : a ( y ˜ ) < 0 } and ∂ a ∂ ν > 0 on Γ, where ν is the outer normal to Ω − . We will show the existence of a solution u ε with a transition layer near Γ and a downward spike near the maximum points of a ( y ˜ ) whose profile looks like u ε → C < 1 at a point P ε , u ε → 1 in Ω + ∖ P ε , u ε → − 1 in Ω − , as ε → 0 .