In this paper, we consider the following fractional Schrödinger–Poisson system with singularity \begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*} where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.

中文翻译：

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具有奇异性的分数阶薛定谔-泊松系统：存在性、唯一性和渐近行为

在本文中，我们考虑以下具有奇异性的分数薛定谔-泊松系统\begin{方程*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma }, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right。\end{方程*}其中 0 <γ< 1, λ > 0 和 0 <s≤吨< 1 与 4s+ 2吨> 3. 在某些假设下五和F，我们证明了正解的存在性、唯一性和单调性你λ使用变分法。我们还给出了收敛性你λ当 λ 被视为正参数时，λ → 0。