We deal with the following fractional Schrödinger-Poisson equation with magnetic field$$\varepsilon^{2s}(-{\Delta})_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{{2}_{s}^{*}-2}u \quad \text{ in } \mathbb{R}^{3}, $$where ε > 0 is a small parameter, \(s\in (\frac {3}{4}, 1)\), t ∈ (0, 1), \({2}_{s}^{*}=\frac {6}{3-2s}\) is the fractional critical exponent, \((-{\Delta })^{s}_{A}\) is the fractional magnetic Laplacian, \(V:\mathbb {R}^{3}\rightarrow \mathbb {R}\) is a positive continuous potential, \(A:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}\) is a smooth magnetic potential and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a subcritical nonlinearity. Under a local condition on the potential V, we study the multiplicity and concentration of nontrivial solutions as \(\varepsilon \rightarrow 0\). In particular, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.

中文翻译：

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具有磁场和临界增长的分数阶Schrödinger-Poisson方程的多重性和集中结果

我们用磁场$$ \ varepsilon ^ {2s}（-{\ Delta}）_ {A / \ varepsilon} ^ {s} u + V（x）u + \ varepsilon ^ {{}}处理以下分数阶Schrödinger-Poisson方程-2t}（| x | ^ {2t-3} * | u | ^ {2}）u = f（| u | ^ {2}）u + | u | ^ {{2} _ {s} ^ {* } -2} u \ quad \ text {in} \ mathbb {R} ^ {3}，$$，其中ε > 0是一个小参数，\（s \ in（\ frac {3} {4}，1） \） ，吨∈（0，1），\（{2} _ {S} ^ {*} = \压裂{6} {3-2s} \）是分数临界指数，\（（ - {\德尔塔}）^ {s} _ {A} \）是分数磁拉普拉斯算子，\（V：\ mathbb {R} ^ {3} \ rightarrow \ mathbb {R} \）是正连续电位，\（A： \ mathbb {R} ^ {3} \ rightarrow \ mathbb {R} ^ {3} \）是平滑的磁势，\（f：\ mathbb {R} \ rightarrow \ mathbb {R} \）是次临界非线性。在电势V的局部条件下，我们研究非平凡解的多重性和浓度为\（\ varepsilon \ rightarrow 0 \）。特别是，我们将非平凡解的数量与电位V达到最小值的集合的拓扑联系起来。