Ground states of a nonlinear Schrdinger system with nonconstant potentials

In the case where either the potentials Vj,μj andβare periodic or Vj are well-shaped andμj andβare anti-well-shaped,existence of a positive ground state of the Schrdinger system -Δu1+V1(x)u1=μ1(x)u31+β(x)u1u22in RN,-Δu2+V2(x)u2=β(x)u21u2+μ2(x)u32in RN,uj∈H1(RN),j=1,2,where N=1,2,3,is proved provided thatβis either small or large in terms of Vj andμj.The system with constant coefficients has been studied extensively in the last ten years,and the nonconstant coefficients case has seldom been studied.It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.