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

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials: { − ε 2 Δ u + ( 1 + δ P ( x ) ) u = μ 1 u 3 + β u v 2 i n Ω , − ε 2 Δ v + ( 1 + δ Q ( x ) ) u = μ 2 u 3 + β u 2 v i n Ω , u > 0 , v > 0 i n Ω , ∂ u ∂ v = ∂ u ∂ v = 0 o n ∂ Ω , where Ω is a smooth bounded domain in ℝ N for N = 2, 3, δ , ε , μ 1 and μ 2 are positive parameters, β ∈ ℝ, P ( x ) and Q ( x ) are two smooth potentials defined on Ω ¯ , the closure of Ω. Due to Liapunov-Schmidt reduction method, we prove that ( A ε ) has at least O (1/( ε |ln ε |) N ) synchronized and O (1/( ε |ln ε |) 2 N ) segregated vector solutions for ε and δ small enough and some β ∈ ℝ. Moreover, for each m ∈ (0, N ) there exist synchronized and segregated vector solutions for ( A ε ) with energies in the order of ε N - m . Our results extend the result of Lin et al. (2007) from the Lin-Ni-Takagi problem to the nonlinear Schrödinger elliptic systems with continuous potentials.