Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems

In this paper, we consider the following nonlinear coupled elliptic systems(Aε){−ε2Δu+u=μ1u3+βuv2in Ω,−ε2Δv+v=μ2v3+βu2vin Ω,u>0,v>0in Ω,∂u∂ν=∂v∂ν=0on ∂Ω, where ε>0, μ1>0, μ2>0, β∈R, and Ω is a bounded domain with smooth boundary in R3. Due to Lyapunov-Schmidt reduction method, we proved that (Aε) has at least O(1ε3|ln⁡ε|) synchronized and segregated vector solutions for ε small enough and some β∈R. Moreover, for each m∈(0,3) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of ε3−m. Our result extends the result of Lin, Ni and Wei [20], from the Lin-Ni-Takagi problem to the nonlinear elliptic systems.