In this paper, we study the following nonlinear Choquard equation

$$\begin{aligned} (-\Delta )^{s}u+V(x)u=(I_{\alpha }*|u|^{2^{*}_{\alpha ,s}})|u|^{2^{*}_{\alpha ,s}-2}u, \quad u\in \mathcal {D}^{s,2}(\mathbb {R}^N), \end{aligned}$$(0.1)

where \(s\in (0,1)\), \(N>2s\), \(0<\alpha <N\) and \(2^{*}_{\alpha ,s}=\frac{N+\alpha }{N-2s}\) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We showed that Eq. (0.1) has at least one bound state solution if \(\Vert V(x)\Vert _{L^{\frac{N}{2s}}}\) is suitably small, by proving a version to the Fractional operator in \(\mathbb {R}^N\) of the Global Compactness result due to Struwe (see Struwe 1984)

中文翻译：

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具有 Hardy-Littlewood-Sobolev 临界指数的分数阶 Choquard 方程的束缚态解

在本文中，我们研究以下非线性 Choquard 方程

$$\begin{aligned} (-\Delta )^{s}u+V(x)u=(I_{\alpha }*|u|^{2^{*}_{\alpha ,s}}) |u|^{2^{*}_{\alpha ,s}-2}u, \quad u\in \mathcal {D}^{s,2}(\mathbb {R}^N), \end {对齐}$$ (0.1)

其中\(s\in (0,1)\) , \(N>2s\) , \(0<\alpha <N\)和\(2^{*}_{\alpha ,s}=\frac {N+\alpha {N-2s}\)是 Hardy-Littlewood-Sobolev 不等式意义上的临界指数。我们证明了方程。如果\(\Vert V(x)\Vert _{L^{\frac{N}{2s}}}\)适当小，则(0.1) 至少有一个边界状态解，通过向分数运算符证明一个版本在\(\mathbb {R}^N\)由于 Struwe 导致的全局紧凑性结果（参见 Struwe 1984）