On a critical Schrodinger system in R4 with steep potential wells

In this paper, we study the following two-component elliptic system: {Delta u - (lambda a(x) + a(0))u + u(3) + beta v(2)u = 0 in R-4, Delta v - (lambda b(x) + b(0))v + v(3) + beta u(2)v = 0 in R-4, (u, v) is an element of H-1(R-4) x H-1(R-4), where a(0), b(0) is an element of R are constants, lambda > 0 and beta is an element of 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 lambda, beta. 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. (C) 2019 Elsevier Ltd. All rights reserved.