GROUND STATES OF COUPLED CRITICAL CHOQUARD EQUATIONS WITH WEIGHTED POTENTIALS

In this paper, we are concerned with the following coupled Choquard type system with weighted potentials {-Delta u + V-1(x)u = mu(1)(I-alpha*[Q(x)vertical bar u vertical bar(N+alpha/N)])Q(x)vertical bar u vertical bar(alpha/N-1)u + beta(I-alpha*[Q(x)vertical bar v vertical bar(N+alpha/N)])Q(x)vertical bar u vertical bar(alpha/N-1)u, {-Delta v + V-2(x)v = mu(2)(I-alpha*[Q(x)vertical bar v vertical bar(N+alpha/N)])Q(x)vertical bar v vertical bar(alpha/N-1)v + beta(I-alpha*[Q(x)vertical bar u vertical bar(N+alpha/N)])Q(x)vertical bar v vertical bar(alpha/N-1)v, u, v is an element of H-1 (R-N), where N >= 3, mu(1), mu(2), beta > 0 and V-1(x), V-2(x) are nonnegative functions. Via the variational approach, one positive ground state solution of this system is obtained under some certain assumptions on V-1(x), V-2(x) and Q(x). Moreover, by using Hardy's inequality and one Pohozaev identity, a non-existence result of non-trivial solutions is also considered.