Existence and multiplicity of non-trivial solutions for fractional Schrödinger–Poisson systems with a combined nonlinearity

In this paper, we are concerned with the following fractional Schrödinger–Poisson system: \(\begin{aligned} \left\{ \begin{array}{ll} \displaystyle (-\Delta )^s u + V(x)u +\lambda \phi u=K(x){|u|}^ {q-2}u+ f(x,u), &{}\qquad \ x \in \mathbb {R}^3 \\ \displaystyle (-\Delta )^t\phi = u^2,&{} \qquad \ x \in \mathbb {R}^3\\ \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {(-\mathrm{\Delta })}^{s}u+V(x)u+\lambda \varphi u=K(x){|u|}^{q-2}u+f(x,u),}\hfill & x\in {\mathbb{R}}^3\hfill \\ {\displaystyle {(-\mathrm{\Delta })}^t\varphi =u^2,}\hfill & x\in {\mathbb{R}}^3\hfill \end{array}\right)\end{array}$$ where \(\lambda >0\) $\lambda >0$ is a constant, \(s,t \in (0,1]\) $s,t\in (0,1]$ , \(2t+4s>3\) $2t+4s>3$ , \(1<q<2\) $1<q<2$ and f(x, u) is linearly bounded in u at infinity. With some assumptions on K, V and f we get the existence and multiplicity of non-trivial solutions with the help of the variational methods.