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 -(del+iA(x))(2)u+V(x)u = (1/vertical bar x vertical bar(alpha)* vertical bar u vertical bar 2*alpha)vertical bar u vertical bar 2*alpha - (2)u + lambda f(u) in R-N, where 2*alpha = 2N-alpha/N-2 is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, lambda > 0, N >= 3, 0 < alpha < N, A : R-N -> R-N is an C-1, Z(N)-periodic vector potential and V is a continuous scalar potential given as a perturbation of a periodic potential. Considering different types of nonlinearities f, namely f(x,u) = (1/vertical bar x vertical bar(alpha)*vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u for (2N - alpha)/N < p < 2*alpha, then f(u) = vertical bar u vertical bar(p-1) u for 1 < p < 2* - 1 and f(u)=vertical bar u vertical bar(2)*(-2)u (where 2* = 2N/(N - 2)), we prove the existence of at least one ground-state solution for this equation by variational methods if p belongs to some intervals depending on N and lambda.