Affiliations: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China. E-mail: [email protected] | College of Mathematics and Computational Sciences, Shenzhen University, Nanhai Ave 3688, Shenzhen, 518060 China. E-mail: [email protected]
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 uε 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, uε approaches 1 in Ω− and −1 in Ω+. Central to our analysis is the solvability of a Toda system.
Keywords: singularity perturbed elliptic equation, inhomogeneous phase transition model, clustered interfaces, Toda system
