Segregated vector solutions with multi-scale spikes for nonlinear coupled elliptic systems

In this paper, we consider the following nonlinear coupled elliptic system(Aε){−ε2Δu+P(x)u=μ1u3+βuv2in RN,−ε2Δv+Q(x)v=μ2v3+βu2vin RN,u>0,v>0in RN,u→0,v→0as |x|→+∞, where ε>0 is a parameter, μ1,μ2>0 and β>0 are constants, and P(x) and Q(x) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov-Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε0>0 such that for any 0<ε<ε0, (Aε) has a segregated vector solution where (uε,vε) and uε is trapped in a neighborhood of the nondegenerate critical points of P(x) as well as the zero sets of P(x), and vε is trapped in a neighborhood of the nondegenerate critical points of Q(x) as well as the zero sets of Q(x). Moreover, the amplitudes of uε (res vε) around the nondegenerate critical points and the zero sets of P(x) (res Q(x)) are of a different order compared with ε. To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously.