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

## TL;DR

This paper investigates the existence of ground state solutions for a nonlinear magnetic Choquard equation with critical Hardy-Littlewood-Sobolev exponent, considering periodic potentials and various nonlinearities, using variational methods.

## Contribution

It establishes new existence results for ground states of a magnetic Choquard equation with critical exponent and periodic potentials, extending previous work to more general nonlinearities.

## Key findings

- Existence of at least one ground state solution under certain conditions.
- Results depend on the nonlinearity exponent p and parameters N, λ.
- Applicable to a range of nonlinearities including power-type and critical cases.

## Abstract

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 $\lambda$.