In this paper, we study the existence and concentration of ground state solution for the Choquard equation$$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}-\Delta u+V(x)u=\left( \int _{{\mathbb {R}}^N}\frac{A(\epsilon y)|u(y)|^p}{|x-y|^\mu }\mathrm{d}y\right) A(\epsilon x)|u|^{p-2}u~~\text {in}~{\mathbb {R}}^{N},\\ &{}u\in H^1({\mathbb {R}}^N), \end{aligned} \end{array} \right. \end{aligned}$$where \(N\ge 2\), \(0<\mu <2\), \(\epsilon \) is a positive parameter. V is a \({\mathbb {Z}}^N\)-periodic function, and 0 lies in a gap of the spectrum of \(-\Delta + V\). \(A\in C({\mathbb {R}}^N)\) satisfies$$\begin{aligned} 0<\inf \limits _{x\in {\mathbb {R}}^N}A(x)\le \lim \limits _{|x|\rightarrow +\infty }A(x) <\sup \limits _{x\in {\mathbb {R}}^N}A(x). \end{aligned}$$

中文翻译：

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一类强不定方程组的基态解

在本文中，我们研究了Choquard方程$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ begin {aligned}＆{}-\ Delta u +的基态解的存在和集中V（x）u = \ left（\ int _ {{\\ mathbb {R}} ^ N} \ frac {A（\ epsilon y）| u（y）| ^ p} {| xy | ^ \ mu} \ mathrm {d} y \ right）A（\ epsilon x）| u | ^ {p-2} u ~~ \ text {in}〜{\ mathbb {R}} ^ {N}，\\＆{} u \ in H ^ 1（{\ mathbb {R}} ^ N），\ end {aligned} \ end {array} \ right。\ end {aligned} $$其中\（N \ ge 2 \），\（0 <\ mu <2 \），\（\ epsilon \）是正参数。V是\（{\ mathbb {Z}} ^ N \）的周期函数，而0在\（-\ Delta + V \）的频谱间隙中。\（A \ in C（{\ mathbb {R}} ^ N）\）满足$$ \ begin {aligned} 0 <\ inf \ limits _ {x \ in {\ mathbb {R}} ^ N} A（x）\ le \ lim \ limits _ {| x | \ rightarrow + \ infty} A （x）<\ sup \ limits _ {x \ in {\ mathbb {R}} ^ N} A（x）中。\ end {aligned} $$