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

Abstract

We consider the following singularly perturbed elliptic problem e4ũ + ( ũ− a(ỹ) ) (1− ũ) = 0 in Ω, ∂ũ ∂ν = 0 on ∂Ω, where Ω is a bounded domain in R with smooth boundary, −1 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 ue with N -transition layers near Γ with mutual distance O(e| log e|), provided that e stays away from a discrete set of values at which resonance occurs. Moreover, ue approaches 1 in Ω− and −1 in Ω+. Central to our analysis is the solvability of a Toda system.