Toda system and cluster phase transition layers in an inhomogeneous phase transition model

Abstract

We consider the following singularly perturbed elliptic problem

ε²Δũ+(ũ−a(y^˜))(1−ũ²)=0 in Ω,  ∂ũ/∂ν=0 on ∂Ω,

where Ω is a bounded domain in R² with smooth boundary, −1<a(y^˜)<1, ε is a small parameter, ν 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/∂ν₀>0 on Γ, where ν₀ is the outer normal of Ω₊, pointing to the interior of Ω₋. For any fixed integer N=2m+1≥3, we will show the existence of a clustered solution u_ε with N-transition layers near Γ with mutual distance O(ε|log ε|), provided that ε stays away from a discrete set of values at which resonance occurs. Moreover, u_ε approaches 1 in Ω₋ and −1 in Ω₊. Central to our analysis is the solvability of a Toda system.