On a critical Schrödinger system in [formula omitted] with steep potential wells

Abstract

In this paper, we study the following two-component elliptic system: Δu−(λa(x)+a0)u+u3+βv2u=0inR4,Δv−(λb(x)+b0)v+v3+βu2v=0inR4,(u,v)∈H1(R4)×H1(R4),where a0,b0∈R are constants, λ>0 and β∈R are parameters, a(x),b(x)≥0 are potentials. By using variational methods, we prove the existence and nonexistence of general ground state solutions (maybe semi-trivial) and ground state solutions of the above system under some further conditions on the potentials a(x),b(x) and the parameters λ,β. It is worth pointing out that the cubic nonlinearities and the coupled terms of the above system are all of critical growth owing to the Sobolev embedding theorem. Furthermore, by introducing some ideas which are different from that in the literature, the phenomenon of phase separation of ground state solutions of the above system is also obtained without any symmetry conditions.