In this paper, we study the fractional Schrödinger-Poisson system (−Δ)su+V(x)u+K(x)ϕ|u|q−2u=h(x)f(u)+|u|2s∗−2u,in R3,(−Δ)tϕ=K(x)|u|q,in R3, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$ where s , t ∈ (0, 1), 3 < 4 s < 3 + 2 t , q ∈ (1, 2s∗ $\begin{array}{} \displaystyle 2^*_s \end{array}$/2) are real numbers, (− Δ ) s stands for the fractional Laplacian operator, 2s∗ :=63− 2s $\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$ is the fractional critical Sobolev exponent, K , V and h are non-negative potentials and V , h may be vanish at infinity. f is a C 1 -function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.

中文翻译：

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一类具有临界增长和消失势的分数阶薛定谔-泊松系统的基态解

在本文中，我们研究分数薛定谔-泊松系统 (−Δ)su+V(x)u+K(x)ϕ|u|q−2u=h(x)f(u)+|u|2s∗ −2u,in R3,(−Δ)tϕ=K(x)|u|q,in R3, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{ \it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^ {\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K( x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right。\end{array}$$ 其中 s , t ∈ (0, 1), 3 < 4 s < 3 + 2 t , q ∈ (1, 2s∗ $\begin{array}{} \displaystyle 2^*_s \ end{array}$/2) 是实数， (− Δ ) s 代表分数拉普拉斯算子，2s∗ :=63− 2s $\begin{array}{} \displaystyle 2^{*}_{s} :=\frac{6}{3-2s} \end{array}$ 是分数临界 Sobolev 指数，K , V 和 h 是非负电位，V , h 可能在无穷远处消失。f是满足适当增长假设的C 1 -函数。我们通过变分方法证明了上述分数式薛定谔-泊松系统具有正的和符号变化的最小能量解。