In this paper, we consider the following fractional Choquard equation: ${\epsilon }^{2s}{(-\Delta )}^{s}u+V\left(x\right)u={\epsilon }^{-\alpha }(I_{\alpha }\ast F\left(u\right))F^{{}^{\prime }}\left(u\right),x\in {\mathbb{R}}^N,$where $N⩾3$, $s\in (0,1]$ and $\alpha \in (0,N)$, $\epsilon >0$ is a parameter, $I_{\alpha }$ is the Riesz potential, and $F\left(u\right)≔\frac{1}{2_{\alpha }^{♯}}{\left|u\right|}^{2_{\alpha }^{♯}}+\frac{1}{2_{\alpha }^{\ast }}{\left|u\right|}^{2_{\alpha }^{\ast }}$, where $2_{\alpha }^{♯}=\frac{N+\alpha }{N}$ and $2_{\alpha }^{\ast }=\frac{N+\alpha }{N-2s}$ are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. Firstly, by the refined Sobolev inequality with Morrey norm, we show a generalization of Lions type theorem. Secondly, combining this theorem with variational methods, we show the multiplicity and concentration of positive solutions for above equation. Moreover, the multiplicity and concentration results are obtained in the case where $F\left(u\right)$ has the lower critical exponent $2_{\alpha }^{♯}$, which remains unsolved in the existing literature.

在本文中，我们考虑以下分数Choquard方程： ${\epsilon }^{2s}{（-\Delta ）}^sü+V（X）ü={\epsilon }^{-\alpha }（{\mathrm{一世}}_{\alpha }\ast F（ü））F^{{}^{\prime }}（ü），X\in {\mathbb{[R}}^{ñ}，$哪里 $ñ⩾3$， $s\in （0，\mathrm{1个}]$ 和 $\alpha \in （0，ñ）$， $\epsilon >0$ 是一个参数， ${\mathrm{一世}}_{\alpha }$ 是里斯势， $F（ü）≔\frac{\mathrm{1个}}{2_{\alpha }^{♯}}{\left|ü\right|}^{2_{\alpha }^{♯}}+\frac{\mathrm{1个}}{2_{\alpha }^{\ast }}{\left|ü\right|}^{2_{\alpha }^{\ast }}$，在哪里 $2_{\alpha }^{♯}=\frac{ñ+\alpha }{ñ}$ 和 $2_{\alpha }^{\ast }=\frac{ñ+\alpha }{ñ-2s}$在Hardy–Littlewood–Sobolev不等式的意义上，是较低和较高的临界指数。首先，通过带有Morrey范数的精确Sobolev不等式，我们展示了Lions型定理的一个推广。其次，将该定理与变分方法相结合，我们证明了上述方程正解的多重性和集中性。此外，在以下情况下获得多重性和集中性结果$F（ü）$ 临界指数较低 $2_{\alpha }^{♯}$，这在现有文献中仍未解决。