On a class of coupled critical Hartree system with deepening potential

In this paper, we are interested in the following coupled critical Hartree system where λ > 0, β ≥ 0, , 0 < μ < N, N ≥ 4, are the upper critical exponents due to the Hardy‐Littlewood‐Sobolev inequality and the nonnegative potential function has potential as well. By using variational methods, we prove the existence of ground state solutions and also characterize the asymptotic behavior of the solutions as the parameter λ goes to infinity. Furthermore, we are able to find the existence of multiple semitrivial weak solutions by the Lusternik‐Schnirelmann category theory.