Nonlinear perturbations of a periodic magnetic Choquard equation with Hardy–Littlewood–Sobolev critical exponent

Abstract

In this paper, we consider the following magnetic nonlinear Choquard equation

$$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ \mathbb {R}^N, \end{aligned}$$

where $2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $\lambda >0$, $N\ge 3$, $0<\alpha < N$, $A: \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$ 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. Considering 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 $\lambda$.