文摘
In this paper, we consider the following nonlinear coupled elliptic systemsequation(AεAε){−ε2Δu+u=μ1u3+βuv2in Ω,−ε2Δv+v=μ2v3+βu2vin Ω,u>0,v>0in Ω,∂u∂ν=∂v∂ν=0on ∂Ω, where ε>0ε>0, μ1>0μ1>0, μ2>0μ2>0, β∈Rβ∈R, and Ω is a bounded domain with smooth boundary in R3R3. Due to Lyapunov–Schmidt reduction method, we proved that (AεAε) has at least O(1ε3|lnε|) synchronized and segregated vector solutions for ε small enough and some β∈Rβ∈R. Moreover, for each m∈(0,3)m∈(0,3) there exist synchronized and segregated vector solutions for (AεAε) with energies in the order of ε3−mε3−m. Our result extends the result of Lin, Ni and Wei [20], from the Lin–Ni–Takagi problem to the nonlinear elliptic systems.