In this paper we are interested in the following nonlinear Choquard equation $$ -\Delta u+(\lambda V(x)-\beta)u =\big(|x|^{ -\mu}\ast |u|^{ 2_{ \mu}^{ \ast}}\big)|u|^{ 2_{ \mu}^{ \ast}-2}u\hspace{ 4.14mm}\mbox{ in}\hspace{ 1.14mm} \mathbb{ R}^N, $$ where $\lambda,\beta\in\mathbb{ R}^+$, $0<\mu<N$, $N\geq4$, $2_{ \mu}^{ \ast}=(2N-\mu)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V\in \mathcal{ C}(\mathbb{ R}^N,\mathbb{ R})$ such that $\Omega :=\mbox{ int} V^{ -1}(0)$ is a nonempty bounded set with smooth boundary. If $\beta>0$ is a constant such that the operator $-\Delta +\lambda V(x)-\beta$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{ -1}(0)$ for $\lambda$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $\lambda$ goes to infinity. Furthermore, for any $0<\beta<\beta_{ 1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $\beta_{ 1}$ is the first eigenvalue of $-\Delta$ on $\Omega$ with Dirichlet boundary condition.
Categories: math.AP