On the critical Choquard equation with potential well

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.