Abstract In this paper, we study the singularly perturbed fractional Choquard equation ε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y))|x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3, $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, 2μ,s⋆=6−μ3−2s $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

中文翻译：

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具有临界增长的分数阶 Choquard 方程解的多重性和浓度行为

摘要 本文研究奇异摄动分数阶Choquard方程ε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y)) |x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3, $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta })^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{ \mu,s}}+F(u(y))}{|xy|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1} {2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ 其中 ε > 0 是一个小参数, (−△)s 表示 s 阶分数拉普拉斯算子 ∈ (0, 1), 0 < μ < 3, 2μ,s⋆=6−μ3−2s $2_{\mu ,s}^{\star }= \frac{6-\mu }{3-2s}$ 是 Hardy-Littlewood-Sobolev 不等式和分数拉普拉斯算子意义上的临界指数。F 是 f 的原语，f 是一个连续的亚临界项。在施加于电位 V 的局部条件下，我们研究了正解的数量与势能达到其最小值的集合的拓扑之间的关系。在证明中，我们应用了变分方法、惩罚技术和 Ljusternik-Schnirelmann 理论。