Existence and multiplicity results for the fractional Schrodinger-Poisson systems

Abstract

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 $(-\Delta)^{\alpha}$ 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.