In this paper, we study the following class of fractional Choquard-type equations

$$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$

where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(I_{\mu }\) is the Riesz potential with \(0<\mu <1\), and F is the primitive function of f. We use variational methods and minimax estimates to study the existence of solutions when f has critical exponential growth in the sense of Trudinger–Moser inequality.

中文翻译：

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具有临界指数增长的$$ \ mathbb {R} $$ R中分数分数Choquard型方程的解的存在性

在本文中，我们研究以下几类分数式Choquard型方程

$$ \ begin {aligned}（-\ Delta）^ {1/2} u + u = \ Big（I_ \ mu * F（u）\ Big）f（u），\ quad x \ in \ mathbb {R }，\ end {aligned} $$

其中\（（-\ Delta）^ {1/2} \）表示1 / 2-拉普拉斯算子，\（I _ {\ mu} \）是具有\（0 <\ mu <1 \）的Riesz势。和˚F是的原始功能˚F。我们使用变分方法和极小极大估计来研究当t具有Trudinger-Moser不等式意义上的临界指数增长时，解的存在性。