摘要： In this paper we are interested in the following nonlinear Choquard equation

where , , , is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function such that is a nonempty bounded set with smooth boundary. If , is a constant such that the operator is non-degenerate, we prove the existence of ground state solutions which localize near the potential well for large enough and also characterize the asymptotic behavior of the solutions as the parameter goes to infinity. Furthermore, for any , we are able to prove the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where is the first eigenvalue of on with Dirichlet boundary condition.