In this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities:

$\{\begin{array}{cc}\left(a+b{\left[u\right]}_s^2\right){\left(-\Delta \right)}^{s}u+u+\varphi \left(x\right)u=\left({\left|x\right|}^{-\mu }\ast F\left(u\right)\right)f\left(u\right)& \mathrm{in}{ℝ}^3\text{,}\\ {\left(-\Delta \right)}^t\varphi \left(x\right)=u^2& \mathrm{in}{ℝ}^3\text{,}\end{array}$

where

${\left[u\right]}_s^2={\int }_{{ℝ}^3}{\left|{\left(-\Delta \right)}^{\frac{s}{2}}u\right|}^{2}dx={\iint }_{{ℝ}^3\times {ℝ}^3}\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^2}{{\left|x-y\right|}^{3+2s}}dxdy\text{,}$

$s,t\in \left(0,1\right)$ with $2t+4s>3,0<\mu <3-2t,$$f:{ℝ}^3\times ℝ\to ℝ$ satisfies a Carathéodory condition and (−Δ)^s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

中文翻译：

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非线性分数基尔霍夫-薛定ding-泊松系统的基态解

在本文中，我们研究了以下具有一般非线性的分数阶Kirchhoff-Schrödinger-Poisson系统的基态解的存在：

$\{\begin{array}{cc}（\mathrm{一种}+b{\left[ü\right]}_s^2）{（-\Delta ）}^sü+ü+\varphi （X）ü=（{\left|X\right|}^{-\mu }\ast F（ü））F（ü）& 在{ℝ}^3\text{，}\\ {（-\Delta ）}^{Ť}\varphi （X）={ü}^2& 在{ℝ}^3\text{，}\end{array}$

哪里

${\left[ü\right]}_s^2={\int }_{{ℝ}^3}{\left|{（-\Delta ）}^{\frac{s}{2}}ü\right|}^{2}dX={\iint }_{{ℝ}^3\times {ℝ}^3}\frac{{\left|ü（X）-ü（ÿ）\right|}^2}{{\left|X-ÿ\right|}^{3+2s}}dXdÿ\text{，}$

$s，Ť\in （0，\mathrm{1个}）$ 与 $2Ť+4s>3，0<\mu <3-2Ť，$$F：{ℝ}^3\times ℝ\to ℝ$满足Carathéodory条件，并且（-Δ）^s是分数拉普拉斯算子。本文有两个新颖之处。首先，方程中的非局部项设置了有界Cerami序列无法收敛的障碍。其次，非线性项f不满足Ambrosetti-Rabinowitz增长条件和单调假设。因此，Nehari流形方法在我们的设置中不再起作用。为了克服这些困难，我们使用Pohozǎev型流形获得上述系统的Pohozǎev型基态解的存在。