Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction

We consider the following elliptic system: \[ \{ {array}{l} ε ^2 Δ u-λ _1 u+μ _1 u^3 + β u v^2 =0 \ in \ Ω , ε ^2 Δ v-λ _2 v+μ _2 v^3 + β u^2 v =0 \ in \ Ω , u,v >0 \ in \ Ω , \ u=v=0 \ on \ ∂ Ω , {array} . \] where Ω ⊂ RN (N≤ 3) is a smooth and bounded domain, ε >0 is a small parameter, λ ₁, λ ₂, μ ₁, μ ₂ >0 are positive constants and β ≠ 0 is a coupling constant. We show that there exists an interval I=[a₀, b₀] and a sequence of numbers 0<β ₁ <β ₂ <...<β ₙ <... such that for any β ∈ (0, +∞ ) (I ∪ \ β ₁,..., β ₙ, ...\), the above problem has a solution such that both u and v develop a spike layer at the innermost part of the domain. Central to our analysis is the nondegeneracy of radial solutions in RN.