LAUSR

Abstract In this paper, we consider the following nonlinear coupled elliptic systems ( A e ) { − e 2 Δ u + u = μ 1 u 3 + β u v 2 in  Ω , − e 2 Δ v + v = μ 2 v 3 + β u 2 v in  Ω , u > 0 , v > 0 in  Ω , ∂ u ∂ ν = ∂ v ∂ ν = 0 on  ∂ Ω , where e > 0 , μ 1 > 0 , μ 2 > 0 , β ∈ R , and Ω is a bounded domain with smooth boundary in R 3 . Due to Lyapunov–Schmidt reduction method, we proved that ( A e ) has at least O ( 1 e 3 | ln ⁡ e | ) synchronized and segregated vector solutions for e small enough and some β ∈ R . Moreover, for each m ∈ ( 0 , 3 ) there exist synchronized and segregated vector solutions for ( A e ) with energies in the order of e 3 − m . Our result extends the result of Lin, Ni and Wei [20] , from the Lin–Ni–Takagi problem to the nonlinear elliptic systems.