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, $\lambda>0$, $N\geq 3$, $0<\alpha< N$, $A: \mathbb{ R}^{ N}\rightarrow \mathbb{ R}^{ N}$ is an $C^1$, $\mathbb{ Z}^N$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Under suitable assumptions on different types of nonlinearities $f$, namely, $f(x,u)=\left(\frac{ 1}{ |x|^{ \alpha}}*|u|^{ p}\right)|u|^{ p-2} u$ for $(2N-\alpha)/N<p<2^{ *}_{ \alpha}$, then $f(u)=|u|^{ p-1} u$ for $1<p<2^*-1$ and $f(u)=|u|^{ 2^* - 2}u$ (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 $\lambda$.
Categories: math.AP