Number of synchronized and segregated solutions for linear coupled elliptic systems
Ke Jin, Zifei Shen · Complex Variables and Elliptic Equations · 2020
In this paper, we study the linear coupled elliptic equations: ({Pε})−ε2Δu+u=u3+λvin Ω,−ε2Δv+v=v3+λuin Ω,u>0,v>0in Ω,∂u∂n=∂v∂n=0on ∂Ω, where Ω is a smooth bounded domain in R3, ϵ is a small parameter and λ is a coupling constant. Due to Lyapunov–Schmidt reduction method, we prove that (Pε) has at least O1ε3|lnε|3 synchronized and O1ε6|lnε|6 segregated solutions for ϵ sufficiently small and some λ. Moreover, for each m∈(0,3) there exist synchronized and segregated solutions for (Pε) with energies of order ε3−m.