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

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 \(\lambda \).

在本文中，我们考虑以下磁性非线性Choquard方程

在Hardy–Littlewood–Sobolev不等式的意义上，\（2 _ {\ alpha} ^ {*} = \ frac {2N- \ alpha} {N-2} \）是关键指数，\（\ lambda> 0 \），\（N \ ge 3 \），\（0 <\ alpha <N \），\（A：\ mathbb {R} ^ {N} \ rightarrow \ mathbb {R} ^ {N} \）是一个\（C ^ 1 \） ，\（\ mathbb {Z} ^ N \）为周期向量电势和V是给定为一个周期性电势的扰动的连续标量势。考虑不同类型的非线性f，即\（f（x，u）= \ left（\ frac {1} {| x | ^ {\ alpha}} * | u | ^ {p} \ right）| u | ^ {p-2} U \）为\（（2 N - \阿尔法）/ N <p <2 ^ {*} _ {\阿尔法} \） ，然后\（f（u）= | u | ^ {p-1} u \）表示\（1 <p <2 ^ *-1 \）和\（f（u）= | u | ^ {2 ^ *- 2} u \）（其中\（2 ^ * = 2N /（N-2）\）），如果p属于某些区间，则通过变分方法证明该方程至少存在一个基态解N和\（\ lambda \）。