Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

We study the coupled Choquard type system with lower critical exponents {-Delta u+lambda(1)(x)u=mu(1)((I)alpha*|u|N+alpha/N)|u|alpha/N-1u+beta(I-alpha*|v|N+alpha/N)|u|alpha/N-1u, x is an element of R-N, -Delta v+lambda(2)(x)v=mu(2)(I-alpha*|v|N+alpha/N)|v|alpha/N-1v+beta(I-alpha*|u|N+alpha/N)|v|alpha/N-1v,x is an element of R-N, u,v is an element of H-1(R-N), where N >= 3, mu(1),mu(2),beta>0, and lambda(1)(x), lambda 2(()x) are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on lambda(1), lambda(2).