Existence and multiplicity results for the fractional Schrodinger-Poisson systems

This paper is devoted to study the existence and multiplicity solutions for the nonlinear Schr"odinger-Poisson systems involving fractional Laplacian operator: \begin{equation}\label{eq*} \left{ \aligned &(-\Delta)^{s} u+V(x)u+ \phi u=f(x,u), \quad &$\text{in }$$\mathbb{R}$^3, &(-\Delta)^{t} \phi=u^2, \quad &$\text{in }$$\mathbb{R}$^3, \endaligned \right. \end{equation} where $(-\mathrm{\Delta }{)}^{\alpha }$ stands for the fractional Laplacian of order $\alpha \in (0,1)$. Under certain assumptions on $V$ and $f$, we obtain infinitely many high energy solutions for \eqref{eq*} without assuming the Ambrosetti-Rabinowitz condition by using the fountain theorem.