Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents

We consider the following Choquard equation \begin{document}$ \label{modelv11} \begin{cases} -(a\!+\!\varepsilon \int_{\Omega}|\nabla u|^2)\Delta u\! = \!\left( \int_{\Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^\mu}dy\right)|u|^{2^{*}_{\mu}-2}u \!+\! \lambda f(x)|u|^{q-2}u \quad in \quad \Omega,\\ u\! = \!0 \qquad \qquad \qquad \qquad \qquad on \quad \partial\Omega, \end{cases} $\end{document} where \begin{document}$ \lambda $\end{document} is a real parameter, \begin{document}$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2}(0<\mu is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under some suitable assumptions on \begin{document}$ \lambda, \; \mu $\end{document} , via the constrained minimizer method and concentration compactness principle, we prove that this system has multiple of solutions, and one of which is a positive ground state solution. Moreover, by using an abstract result due to K.-C Chang, we admit infinitely many pairs of distinct solutions. In addition, we prove the nonexistence result by Pohožaev identity when \begin{document}$ \lambda<0 $\end{document} . The main results extend and complement the earlier works in the literature.