Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where \(2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, \(\lambda>0\), \(N\geq 3\), \(0