Asymptotic behavior of ground state solutions for nonlinear Schrödinger systems

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Consider the nonlinear Schrödinger system −Δu1+V1(x)u1=μ1(x)u13+β(x)u1u22inRN,−Δu2+V2(x)u2=β(x)u12u2+μ2(x)u23inRN,uj∈H1(RN),j=1,2,where N=1,2,3, and the potentials Vj,μj,β are periodic or Vj are well-shaped and μj,β are anti-well-shaped. When the coupling coefficient β is either small or large in terms of Vj and μj, existence of a positive ground state solution was proved in Liu and Liu (2015). In this paper, we describe the asymptotic behavior of ground state solutions when |β|L∞(RN) tends to zero or minRNβ tends to +∞.