Three positive solutions for the indefinite fractional Schrödinger-Poisson systems

Abstract

In this paper, we are concerned with the following fractionalSchrödinger-Poisson systems with concave-convex nonlinearities: \begin{equation*} \begin{cases} (-\Delta )^{s}u+u+\mu l(x)\phi u=f(x)|u|^{p-2}u+g(x)|u|^{q-2}u & \text{in }\mathbb{R}^{3}, \\ (-\Delta )^{t}\phi =l(x)u^{2} & \text{in }\mathbb{R}^{3},% \end{cases} \end{equation*} where ${1}/{2}< t\leq s< 1$, $1< q< 2< p< \min \{4,2_{s}^{\ast }\}$, $2_{s}^{\ast }={6}/({3-2s})$, and $\mu > 0$ is a parameter, $f\in C\big(\mathbb{R}^{3}\big)$ is sign-changing in $\mathbb{R}^{3}$ and $g\in L^{p/(p-q)}\big(\mathbb{R}^{3}\big)$. Under some suitable assumptions on $l(x)$, $f(x)$ and $g(x)$, we explore that the energy functional corresponding to the system is coercive and bounded below on $H^{\alpha }\big(\mathbb{R}^{3}\big)$ which gets a positive solution. Furthermore, we constructed some new estimation techniques, and obtained other two positive solutions. Recent results from the literature are generally improved and extended.